liens code de la propriété ... -...
TRANSCRIPT
AVERTISSEMENT
Ce document est le fruit d'un long travail approuvé par le jury de soutenance et mis à disposition de l'ensemble de la communauté universitaire élargie. Il est soumis à la propriété intellectuelle de l'auteur. Ceci implique une obligation de citation et de référencement lors de l’utilisation de ce document. D'autre part, toute contrefaçon, plagiat, reproduction illicite encourt une poursuite pénale. Contact : [email protected]
LIENS Code de la Propriété Intellectuelle. articles L 122. 4 Code de la Propriété Intellectuelle. articles L 335.2- L 335.10 http://www.cfcopies.com/V2/leg/leg_droi.php http://www.culture.gouv.fr/culture/infos-pratiques/droits/protection.htm
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Æ 0;E/$1#.-J$&D|:1#.-.-!#*o0Å/$5K!) $&D$0Å+-y56/$/$.-!# ,*+-$.Å!)YiÇ()!#.7*/0;/$+]%|Ès*, I:$.70LLu4*$5#$ m0¨!)Yi()!#.7*/0-/$+T%Hu4E*,&H.;!VÉO^ÊzT^ _ ^!#+-+-14"$*/$$+ZqB%$+T56.716:$+T*,0;!#*,.-$+NM16]0-* ,*+-$.-!#*o0Z !V1#.-&'!# ,*,0-/m0 ?>Ci*,+70;$"$9%9"$16&D!#"0-*,a"A!<0;*,16+T0;1#.-1)cd%!# ,$+Z%9"$$+Z()!).-*,/0;/$+$^u41#*,0 Ag, 0 "$1#&'!)"0;*oa"A!)0-*1#'0;1#.-1)cd%!# ,% Ag, 0 ^YQ3&'1#0-.-$.-!%!)+r ,Z0-E/$16.7JA&DÁË?Ì8ÍË24 +-"nE/$&3!Ã!#I:/$ *,$Î*o(K$.7+-$ V+-. Ag, 0 +A>C/0;$%Ï"A!#1#*2$&'$0|$ÏÏ+;"hE/$&3!Ã+-$&D*o
!#I:/$ ,*$ G0+-. Ag, 0 ^ _ >@!#%E/$.7$"$m%Å%.-!#:A!#]*,(6$.-+-$ H•
%!#+ G0$+70Z24!)+;*jda*m0Z 8!<0;
+-. Ag, 0 gt&'!#*,+H!)+a*,#^OÐs!):$ 1#+H24>C+-"nE/$&'!¨$5#.-1#:$+H+-$&'*j~+70h!)I ,3$+-0HÑ+-"nE/$&'!$Ò56.716:$+Q *,+-+;H%1#0Á ,$+saI.-$+Q+-16K0ÁiY0-$+;*,16Ò%>CÒ56.716:a*O/0;!# ,B!).QÒ+-"nE/$&'!°+-$&D*o!#I:/$ ,*$^ekl T$+-0'!)0-.-$ T%¨+-Å%$&'!#%$.'+$>C* Ti*+70-¨Ó+-"nE/$&'! G +-$&'*j~+-0;!#I ,Å+-. Ag, 0 2*/0;$% ,H+-"nE/$&'!°!#I:/$ *,$Ô*,(K$.7+-$ ?g:0s0-$ t24 ?>@!#%E/$.7$"$H% H•
%!)+ G +;1#*,0sa*,$0s 8!<0;+-. Ag, 0 ^
"!$#&%(')' +*-,.!$/.*-0214365&/758'60:9.;&50<0<5>=?5& $021@'A)*(' "B5C;8*( ")DE'5GFIHKJMLNMHKLOJ2P7QP>RSUT(P<VWL
QPYX[Z\SURO]PIR^_` D = 1 a b bc0<5&'' bB5C!<#& <B40d*-0" <143W*('0&^_=?1e <5&!<*.%!<#&;&1@ <#>)*W' "B@5C0<f#8-g!<h&/75Cijkjml an *(%%_5&B(' o 21@/G%B@5&/G5&'0o=?5:Bp*q;&/7%( 2*('0<5:'5&02!$5:)DE'r <;If#&/.*q <5&/G1\gs <02*(cB@5:5& <0B5r <;If#&/.*t <5&/G1\gu*(c_#&B@15&'v#&w*(Bx9tBKDy'1('v)5& G;&(/G%_ 2*W'0<5& G;&''5Mz5& G)5rBKDy#&B#&/75&'0G'5&02!$5)5;If*W=?5:|c!<5 a~e(v a C H RF Ag, 0 ZpPNdVW"VFdF NMSKq D = 1 PIH H ZpP:F L6FMK]WJ LPYLORSUTWPMJdF8PMZ RSPMH__Z@VWHQPJ2V-RO]
pFILOJ Ag, 0 SUH Ag, 0 LORP[N "V6NMHKS N2VWHKS RH J - Q6V-ZpP:QP Ag, 0 C H RF G0ZpPFN2M.VrF8PMSU$VIMZ\SPIRFMLOJ Ag, 0 LOS"MHuPMRQ ZpPFN2M.V¡V6IMZ\SPIRLORSUT(PIJdF8PMZ¢FMLOJ Ag, 0 ¤£ Z¥P$¦SpFIH§PLORFNdI.V¨PIR]WJ LPdF7F8PMSUFIHuV6ZpP G FMLOJ Ag, 0 Q RH"ZVtN " F8VWRHuP.RPILOHKJ2PrPdFMH G0 © s]6VWZªVWLFNdM.VAV6dMZ\SPMRLORSUT(PMJ2FPMZVWL?$QPdFdFILF«Q6P Ag, 0 PMHxH§PMZ LPZ<¬ VQMJ2PMRN2P«Q6P H Q6VWROF G F SUH RSPPMH_ZVWH§P+FMLOJ Ag, 0
ª!$#&%(')' '#8w*W0<1@35&/G5&'60®9YB*C=5& $0<1('«%!<#&;&#&)5&'6025b)*W' ªB5;8*( ª(`g ≥ 2
50 D 6= 1 a¯'v°m*(1408^H•
'5 8Dy#025&')²±2*(/G*(1@ «5&'³'´)!2*W%58*W³)5 w!$%_5& .|'1 «50«%B*W0< « <! Ag, 0 ^=?5&B@B5=?5µ <140B*¡;&/7%*(;0<1@|;8*-021@'02(!<W¶)*(B@5 Ag, 0 )5 Ag, 0 ^¢50q/G·&/75r*W%!<h& q;If*W'w5&/G5&'60)5µc*( <59Spec(C) a¸ *²)14¹#!<5&';&5%*W!!<*(%%_(!$07*(;8*( )5B*¡ $02!<;0<!<5µ)5'1@358*()5rº:B1@'w5&'3O15&'60
)¡°m*(140:=?5q <1 D 6= 1 ^1@B¢5Mz1@ $025q)5& :;8*W!$0<5& :B@?;8*WB5& "°»!$/G5&B@B5& +)5 Ag, 0 <!+B5& <=5&BB5& Hs
5& $05MzO0<5&' 21@'''r02!$1@3O1*(B@5Y)5[ 2(' w(!2*W)#Y#0d*WB5[%*(!x 2*7%*(!¼021@5Y <5&/G1\gu*(c_#&B1@5&''5 a½ 5& 5Mz?025&' <1(' '5 <5[%!<(B('w5&'0x%*( *WOzr;&/7%*(;0<1@|;8*-021@' a¾ (15&'60
n ≥ 3'5&'6021@5&!50 Ag,n '5®;&(/G%*W;0214|;*-021@'q0<!$(¶)*WB5®) <;df#&/.*")5¥/G?)B@5& Ag,n)5& C3W*W!<1@#02#& >*(c_#&B1@5&''5& >%!$1';&1@%*(B@5&/G5&'0C%(Bp*W!<1@ <#&5& >/>'15& C)DE'5. $02!$;02!$5«)5.'143658*(%!$1'Og
;&1%*WB55&'n ¿EÀoÁÂÃ ^02f aÄuÅ>aÇÆaÉÈÊKaË *(' :Bp*%!$%_( 214021@'ijÌjml^' :/G('0<!<' :=?Dy1B5MzO1 $025'
<;If#&/.*.5&'Aw!$%_5& " <5&/714gs $0d*WcB5 G <! Ag,n =?1e#0<5&')¡B5C <;If#&/.*«*Wc_#&B1@5&'¡'1@35&!< <5&Be <! Ag,n ^50x0<5&B=5[B5[w(!<(%5[)5ng0<!$ <1(' G[n]
<(1@0o|'150x%Bp*-0 <! Ag,n a½ /7/G5A*(%%B@1;8*-021@'´)5AB*Í;&' $02!$;021@'´)5A;&(/G%*W;0214|;8*W0<1' G02(!<W¶)*(B@5& &^ª'( «)('''
'5Y'(365&B@B5Y%!$5&365Y)5[BKDy5Mz1 $0<5&';&5Y) < ¼gsw(!<(%5[;8*('('1=?5[%(!)5& o°m*(/71B@B5& )5:3(*W!<1@#02#& *(c_#&B@15&''5& &^)5¥9bÎbcch& 50¤ÏÐ?!<*('5 ¿EѵÒÓÃÔÊ 50¢9Î"')!<58*-0<02*b50¤Õ:*W <c*(!$!<1 ¿EѵÖÃ×?ÊKa 021\gB1@ <' e'51@)#&5)5ª9 Å 1@';&58'60¢Ø¤1B@B('1 a ¾ %%( <' e=5 D = g Ù 'Y%*(!<B@5ª*WB(!< e)5oFMHmJMLNMHKLOJ2PoQPRSUTWP2VWLrQP SPs]PMZ a ¾ 140 Ag, 0 Ú !$5& 2% a Ag Û '5b;&(/G%*W;0214|;8*W0<1('.0<!$(¶)*WB5)5 Ag, 0 Ú !<5& <% a )5 Ag Û ab( Y)#|'1@ 2 <(' YB@5GB@15&¨!$)1'*W1!$5 Aord
g
)5 Ag × Spec(Fp)50YB57B1@5&Í(!<)1@'*(1@!<5Mgs/>B@0<1%B@1;8*-0214°
Aord−mult
g, 0
)5 Ag, 0×Spec(Fp) a ¥/7'602!$' ¥=5"B*C <5&;021@'µ)5 Aordg
)*(' Aord−multg, 0
)''#&5%*(!oB5['Ü6*()/G(!<%f1@ </G5[)5YÝ!<(c5&'1@ x 8Dy#025&')5&''5[ <5&;021@'
Aord
g −→ Aord−mult
g, 0 .
b( +5&'¡)#&)1@ <' "B5C0<f#&!$h&/G5> <143(*W'0+w(!2ÞW;&5q9«B*tß" <!<;&('35&!<w(5&';&5q)A0<c_5"à_^_)#&/G('0<!<#&5)*(' ¿yáqâÂã ^02f aäaEåaÉæÊKa
! "!#%$'&)(+*-, .0/21436587:9<;>=@?3 Aord
g
?A/B=C1 Aord−mult
g, 0
9D=E?BFC9D7GFC=H9I14;>JK;>LNMOP9I18JK323Q=7RLS3T?A361U ;>9189D= /4VA361W187RLQ9<587D1X?361Y7RFZQ361X?3[5SO/58FP=)?3[56361\?38F%]^1_56O`8Ja/!1X?A/>=1YbI361[3S1cM/563S1YL89dVB9R?361X/41614;4589R`61e Ag 387 e Ag, 0 fgihijSklnm%oqpnrshtram%ujSovhtwRxyh8z|vjqhnunht~G|Sm%|NovmEht_%um%uPht|qo-uAmd|quBht|vjqoq8j~jSEh
~wdhnmov~uBovh * mBjSovhXrsljSkm~h\uh\EhnovrahjB|t~hXovlnn|qhnoiP%uAjSdjSBjqdhnrahnuAjnhAm%|qdu%%hnmuAjSoq%doqhnrshnuj0hjY+yP * uu0um%|ram%ujSovmu|hnuBEhnu~dnhaPhaEm%o
g = 2m%
30nhnoj6Buhn|¡ ¢m%oqrshn|ramP~w%doqhn|
n|q~Bwhn|[~hs_%oq%jqlnoqd|vjSdPhpB|S|qm_dlnhn| Ag,n
|qh£oqhnwdpnhnuj¡hnu¤_%oq%jqlnoqd|qjqhsuPwdwh *¦¥ %u|whXjqklnmovpnrah[|qdB%uAj_m%uum%jqh
Cusp(n, k,M)wh[%oqm% h[~hn| ¢movrahn|Wram~wIBovhn|§n|q~Bwhn|W~h
udh_%novhnradhnoY
p~h¡ m%~|
khjYsnmPh¨anhnuAjS|§~%u|§u
Z[1/n]NrsmP~wdh
M*
! "!#%$'&)©[*-, ª;>9<38=7g = 2
;>F3387k > g + 1
?38FB]3Q=7R9<38LS16«[387n ≥ 3
FC=¬38=7R9<38La=;!=?B9 U 9I189RZbI3M/BL p f ª9 g = 2
«i;>=18F!MAM;4143p > 2
3Q7189g = 3
«i;>=18F4MAM;41_3p > 5 f .03XJ®;!LcMO9I18J®3¡?356O/>=CV38JK38=7?3GZ6/41439D=E?BFC9D7FC=¯9I1_;>J®;>LNMO9I18J®3
Fp ⊗ Cusp(n, k,Z[1/n])∼−→ Cusp(n, k,Fp) .
° xyu¯~hn|XdujSloq±jS|\~hGnhj[%Ehnu~nhhn|qjX~h~muuhno\u¯h8zhnrswhm²¯w®lnm%raljqoqdhG~hn|Xnm%r£%jqd_>jSdmu|§~lnEhnu~+~h¡rKBupnovh¡hn|S|qhnujqhnwdwh¡~+6km³z~h
Σ*
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1NramBjSÄ ¢| * ° §%ojSdhYÆ * Æhn|qjnmu|vjqdjqlnh~hoSBEhnw|
|qo^whn|^|qQklnrKB||qhnra³c%Elnwdhnu| * ° )%ojSdhÇÆ *³È jqoSBdjqh®~hn|1Nrsm%jqd D|hj~hKwhnoq|^|vjSovjSovhn|~h
udh_%'%oq%km%oqdPhn| * ° ¾BovjSdh¾Æ *É hn|qjGnmu|S%novlnhÇwI¾nm%u|qjqoqjqm%uÊ~h+Ë+rs Dmoq~ * m|ÌuAjSovmP~|qm%u|GwÇ_>jSln%movh®~hn|
1crsm%jqd D|G~hË+r£ ¢mov~-Phn|vjGuhKhnoq|qm%uÍw||vÌCujqkljSdPh
~hGwI®_Bjqlnm%oqdhDDpol
~hsÎ%wÄjSdu|Xhj^gik% *¥ %u|XwI®%ojSdhÆ *Ï um%|[rsmuAjSovmu|XwRx%wdlnovnÄjSl~ÐQkBraÐ~hrsmP~whn|s~hn|
1Nrsm%jqd D|arud|a~xyuh|vjSovjSovh~hudh_%Ð%oq%km%oqdPh%hjahnu
~muum%u|§uh¡~hn|qnoqjqm%u+h8zwdndjqh *¥ Bu|twh|qhnnmu~6k%djqoqh%Pum%|tnm%u|qjqoqd|qmu|t~hn|i4Bovjqhn|twdmP_%wdhn|i ¢m%oqrshnwwdhn|t~hn|tnm%raBjSÄ_>
jSdmu|Ñjqmovm%Ò<~Bwhn| *A¥ Bu|ÅwI[%ojSdh ÈP* Æum|Ånm%raBjSÄmu|Å%ojSdhnwwdhnrahnuAjhjÅ~h§rKBupnovhWjSm%oqdPhwhn|\6k%rs|\~hrsmP~wdhn|\~h
1NramBjSÄ ¢|nEhnu¾jSdwd|6Buj\whno~hn|Snovjqm%u¾nmrsrah^jqmov|qhnoq|\|qm|\~hn|
jSm%oqhn|Ó %xhn|vjXwI®PxyuAjShnoChnuAj[wdhGQkm%ÄzÇnm%rGduBjqmdoqhΣ*¥ Bu|XwBovjqh ÈP*ÔÈ um%|Xnmrswljqmu|
nhn|Ynm%raBjSÄ_Bjqm%u|Y%ojSdhnwwdhn|Ywdhwdmua~h|vjqoS>jShn|nEmo§Emmdo§hÕ h_jqhno§uhnm%u|vjSovjSdmu~haË)r£ ¢m%oq~ *0Ö u¯oqhnrKBoqh£PhGwhGovmEhG~h
p<jqmov|qm%u¤~¯Pm%jqhnuAj¡wI+Ë+rs Dmov~¤hn|vj[uh
~muulnh%wdlnovPh%-Pih8z|vjqh|qowhQkBraÍ~hramP~wdhn|£_%BujGnmrswljqm%u¡Ó-nhnwIÇ|nx×_lnoqhnoShn|q|qhnujSdhnw *¥ Bu|wI%ojSdh ÈP*×É Cum%|W%ovm4zCrsmu|whXovln|qwdjSBj~hXwIGnm%u|vjSovjSdmu~hË+rs Dmov~ *g\xyhn|vj0wRxylj6B hÑwIw|0~lnwd_>jShÑ~hÅwWnmu|vjqoqjSdmu¡|S%u|¦udh_%_PovhnPdhnovj0wIwd|S|qÄjSlÑ~hn|0QkBra|~hram~whn| * u£hÕhjn%dwC ¢%jnmujqoqØ%whnowRx%ovm4zCraBjSdmu£Em%ouh%|~hEhnov~oqh~xdu ¢m%oqraBjqm%u *° xd|qmrsmovk|qrah~hiÙmP~Boq>ÚPEhnunhnoÛDoqm% *%Ü Â Ü Â:ÝÞ hnovrahj-nhnm%ujSovØwdhXÛ¢ovm *YÜ Âß ÂRÝCÞ 4rKB|uxh8zC|vjqh[PhX|qo~hn|WB|Shn|wd|q|Shn| * uPh[~hXwIGnm%raBjSÄ_Bjqm%u~h Ag, 0
um%|Wnm%u|qjqoqd|qmu|
! "$#&%')(*,+-')#./ "$021$2"435!+76)985"$#5 ;: p < +-#5"$1$')#.=> "?#. @5ABCBDCBFEHGJILK226MNO1$217 ')1$PQK2R85"$#5 JR21?+SR:#502T!68502U"$')PQ5@VR!1S6W 5"4+-')/XQ@ZY[#51S:0\')1-1$#51S621]K&5"4+-21S6)#QK&!62156)8.02U"?'P21&@_^26)621`"?26D,+-'a3_21cb Ag, 0 de!"-!'2+c5K2f')+-02"?g+c1$'H6]85"$#5 J]: p < +$#."?1$'#5fde&3!5'a+ 51h0+$0T "$021$2"43_0T6#5"$1h:T6ide! "$#&%')(*,+-')#.@Vj!1S6j+$"$#5'1$'k2(lRKnm! ')+$"$5[#.1SK2#51?+-"?'1$#.1S6)21SK2#.(l 5K+$')\K&,+-')#.1]+$#."?#5op:!6212@qrW !"?+$'
s @)tu21?+vK2#.1-5K2"?02u5w"$2K2#566)2(x2_+:21K25"4+-21v6#QK&!621!68502U"?'P212[8."$y5K2ubz6zK2#51?+-"?K+-')#.:l"$26!+$'#51:dZ02PQ')3!!62K2l0+|!6212@~M#51+$'6)'1$#51/b#53_&!6D6)'1$1$')+$0l:51T6DK2#51?+-"?K+-')#.1-51')3.&5Ff6FdZ'1$#5(x#5"$ m')1-(lj:jM#Q:!'"$ <4 J2K22"T=> "?#. @CBiEBiEHGn@^9#.(l(x Ag, 0 21?+] "$#5 "$1$" Ag [J'6r1$x+:/3.02"$'a\2"6* "$#5 "$+-0/:O6D*K2#5(x !K+-'a\K&!+$'#51-51'a3_&!=> "?#. @CBFEBDBAG|@^26"?2 #51$1$"6)K2#5+$"$56:6ide! "$#&%')(*,+-')#. J2"$(l'1 5"6id')1$#.(l#."$ m')1$(x:M#Q:!'"$ < J2K22"2@Vj51M6Dx 5"4+-') s @X[J#.1MK2#.1?+$"$'1$#51/K2#.(l 5K+$')\K&,+-')#.j!(x026')#."?02MPQ'rd 51r:1-')8.65"?')+$0;:d#5"$')8.')+-#5"$')PQ5@L#5"rK226[#.1r"?2(x 652#51r6;Knm#5'a%T: Σ !"~K2&6)'_:d:02K2#.(l J#.1$')+$'#5 #56).02:"-!6
S:dZvK22"4+|!'K2#.(l 6%HK2#5'PQ5@QVjM 612[Q6/K2#.(OU',+-#5'"?:M6
1?+-"$!+$')\K&,+-')#.: Ag, 0 1$f6'a+TU')2 61T!'1$02(l2+T1$" SPQf1$" Σ @Vj!1R6Dv 5"4+-') s @ s [#.15:! +-#51h6DfK2#.1?+-"?K+-')#. "?02K202:2+$/b Ag,n,0 @qPQ,+-"?'k2(lK|m5 'a+-"?21?+lK2#51-5K2"?0b6DuK2#51?+-"?K+-'#5:6uK2#5(x !K+-'a\K,+-')#.(l'')(*!6
: Ag,n,0 J#5" n ≥ 3 @VR!16* !"?+$'/Y@)t.[J#51:0\'1$1$#.1M621M#."?(x21(l#Q:6D!'"?21: '2852651$1$#QK2')0221b Ag,n,0 @H6)1$'2"$19:'a02"?2K2211$#5+]b#5+$2"9 5";"-! #5"?+5K&!1:MUJ#.M"?02:K+-')#.+-"$5'a+-0:!1v ¡S¢f£¤[7K|m@r¥¦F@q~§ 61O')(x J#."4+|!+-21?+O6D§#5+$'#5z'm!U')+-26)6:x¨©,FªL«v:d>#5"$(lx(l#Q:6D!'"?5[PQ'c21?+/fi!(x'6)6 (k1, . . . , ks)
: s 2_+-')2"$1"?26D,+-'a>12@~M#.1T:0\'1$1$#.1T6:03_26)#. J2(l2+T:l¬#5"$')2" <4 5K2#5U'`:21j>#."?(x21R(x#:6D!'"?21!1$1-#K2'0221Tb Ag,n,0 +R:02(x#._+-"?#.1;3.2"$1$'#5O:O "?'K2') ;:M®QKnm2"@,VR!176DN 5"4+-')9Y@¯XQ[L#517K2#.1?+-"?'1$#5176DNK2#.(l 5K+$')\K&,+-')#.(x')'(x56) A∗
g,n, 0: Ag,n,0 @q~x:'a02"?2K2/ "?'K2' !6 5"N"-! J#."?+!°K&51M:U#5/"?02:K+-')#.21?+6id2(l 6#5'N:±i5')1$K2&5 ⊗i ωi !²6)'2³: ω0
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0/L,-27
g = 2,-2
3 M N ,-2#O2580'$0'#),-# &+#P7)H+#)2$Q5>!R58#P &TS<U&+/U&+7) VCXWY!"7)79,-KZD[\^] _V`!"0#)0L:2&E &+#a5)UH+,-79I+3B&+#P bc!"<2$d!R580',-e &O$!F*+,1U,"C3B,1$,1f-0'&(*+,-UH+79&+5)&TJ1!"$d!"g$&+#;&+h*G!17)!1*5)H+7)0'#9580':<2&
p i ZkjlGmLnV`o&5FZqpOrtsuV`Av M
! "#%$&'( *)+,- .
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35476*8:9<;=9?>@47A*BDCFEF;HGI6*JHK*8:L*MON@6*BD8F8FJP6*8RQS;<8R8TCUETK*9OCUK*ET;V8RL*;=6*BDWP;V4HK'A@47EU47>*JPEFBDXYK*;<8R8FK*ERQS;<81 Z\[ JHCFBD]^8<_`aJHK*8 [ JH6ICFEFJP6*8bXYK*;cQD;<8d9?>@4 [ A*8dXYK*BeA@4HEF4 [gf CUET;<6ICd9<;<8dJPhIiF;OCU8d8FJH6ICj4HQDkPM<h*ETBSXYK*;<8<G7;OCdQS;<8dL*M<9<ETBDWPJP6*8;mlnA*QDBS9<BoCU; [ ;<6ICV_
p@qrp@qtsvuxwzyY|~g<eU|*yY^Y JHBDC
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FbF*+\+n_ =68F9?>*M [ 48F; [ B Z 4HhM<QDBS;<6 G→ S;<8TC K*6t8F9?>*M [ 4;<6tkPETJPK*A;<8 QDBS8F8F;HGn8FM ZA@4HETMHG¢¡5N@h*ET;<8kHM<J [ MOCUETBSXYK*; [ ;<6IC9<JH6*6*;mln;<8<GHCF;<Q*XYK*; A+JHK*EdCFJPKnC s ∈ S GPQS45N@h*EF; Gs
8FJHBDCj;OlYCU;<6*8FBDJP6L£¤K*6*;W747EFBDMOCUM¥47h+M<QDBS;<6*6*;
AsA@47EK*6CFJPET;
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U Ç ÅÛ ÇeÜ Ä?ÓmÆÞÝIÄmÅnßVÄàÝIÄ S ÙáÄ?Æ G ÄmÆG′ ÝIÄ Ç7â ßãÈUÉ@ÑmÌ'ËãßäßãÄ?ÌÃ^åTËPÒ?ÑmÏæøÄmÅeßäß Ç Ó S çè ÄÌé¨ÓÍ*ÉYÃßmÌ'ÄÝIÄêÓÊÄÊß?ÆrÓmÃrÈmÆrøÂxÅëß Ç Ã Ü Ë7Å@Æ:ÿÅ+Ý Ç Ã^Æ Ç ÅÃSßãÂxÌé¨ÓÍ*ÉYÃßmÌ'Ä
HomS(G,G′)
∼−→ HomU(GU , G′U).
JHBS;<6¢CSK*6.9?>@4 [ A4HQDkPM<h*ETBSXYK*;ì6*íêCF>*M<EFBD;<6;OC G → S
K*68F9?>*M [ 4î8F; [ B Z 4HhM<QSBD;<6_aïJHK*ECUJHKnCs ∈ S
G:QD;ëCUJHEF;Ts
L*MON@6*BDCéK*6°]×4HBD8F9<;ã47K°MOCÊ47QS;Þ;<6Z Z\[ JIL*K*QS;<8éQSBDh*EF;<8éL*;ÞEU476*kN@6*Ba8FK*E s;<6ðAJP8U476IC
Xs = Homs(Ts,Gm).ñv;CU>*M<JHE f<[ ;ê8FK*BDW74H6IC [ JH6ICFEF;êXYK*;ê9<;<89<JH6*8FCFEFK*9OCFBSJH6*88U;êET;<9<JPQDQS;<6¢CV_¦R§ ¨*©¨ªH«¬+\+¸òÕ¯m±¤²c³g´*µ*¶¸·ó wRqb¼xq½qp µ À _ ôÊÏõÄ â Ãß?ÆöÄ Ç Å Ç Å@ÃÔ Ç Ä÷<Ë7ÃßãÈÊÄFË Ç Ñ?ÆË7ÏÄäÈÊÂxÅnß?ÆrÓ Ç ÈmåÆrÃrÒmÏÄ
Xß Ç Ó S ÆöÄ?ÏÔ Ç ÄcÍ@Â Ç ÓêÆöÂ Ç Æ%Í@¨Ã^Å@Æ s Ý¢Ä S ٠¨ÅË7Ã^Æ X ×S s = Xs ç
ø 6,L*BDCcXYK*;G;<8TCcL*;ÓFË7ÅeÐÆöÂxÓ?ÃrÔ Ç ÄõÈÊÂxÅeßmÆË7Å*Æ:8FBQD;Î]47BS8F9<;V47KtMOCÊ47QS; X ;<8TCcQSJI9V4HQD; [ ;<6ICc9<JH6@8TCU4H6¢C8FK*E
S_
! "$##%!&'()%+*#,-./10 -2 34!10 .5* 6$
798;:=<>:?"@BAC@:DFEGBEG HJILKNMPORQSTVUXW2YZ;Y>[]\3[ ^[`__acbed f%g"hji-klS m konqpsrt-uovwkxylzrt-hVi-k|i-l G→ S
m kncp~rt-uvnqi-uRh+Cvt-hVikJ$i|v3k!,lg"hj m i'psg"knlBv3klBPsi2hn-li m klg;siT → S m knqpsrt-uov
vthVi-kA→ S
i-l m kiuov3khVsi9>tsp-h+~i G psg;uRuii2likn-hVg;kn m S0 −→ T −→ G −→ A −→ 0.
3 1 ¡¢¡£¡L ~¡¤ ~¥¦¡=§¡£¥¨©o=~¡©ª¬«B3¨1¡¢¡®~3¯ ~°=®1±¡£°= ~= ¡®°2$ °= ¨²q³¡£w´ljh+i-§£°=©o©ª¡wµ=®P¡-¶¡©oµ1¡
Gb·¸¡²£=¡5¢%¹º1°2©ª°2®µ¥©o¡5¢%¹N¡-¶! ¡q1°2
0 −→ T −→ G −→ A −→ 0°2 µ3®~3©o¨L ®¨¡µ3®
Ext1Z(A, T )
§±»¡ ¨¯3e¼HomZ(X,At)
µ3®½+°2®©1¡9¢¡¾/3®°3 ¬«¿,¡`b
_[^[ÀÂÁÃ1 ÄsÅ YÆcÇzÈBÃ
É o¡5¢°2¡5£s¥=©ªµ3¯=¨®1±¡S¡L Â`¹º°=o°= ¡ CbS
£q3 ¨¯2°=®¡5¢¡Ê£°2©ªµ¡-¶¡Ê°2®¨Ê¢¡½`3£¡q3¶ËÍÌÌcËΡ|¯2®°2µ¡²3¨¡P~®
S¡L DbS
ª£q" ~¨¯=°2®¡¢¨®¬³¨¡3~°$£¨¡=b É |¢¬ ²±$%¹º½`3£¡q3|¨L s3¡®
S¡ 5´~tsnciv m q¹N%¡ 51°£q=1¡©o¡ 5£q°= s3$ ²¼³3=1¡®5¢35 Z
«Í©o°$¢¡1®¡w¢¡w®=¯RÏjb
Ð'ÑÒD]@BAC@:DFEGzÓ>GBEÔI-KºÕÖÁ×ÍØ!ÙTVUXÚ»Û$È~[/_$S>[`_[j_acbed ÜÝ1«Í©o°3 ~¬½
M → S¡ '¢°=¨¡Þ¢¸¹N
£¥¨©Ý¡©o«B=¨1¡R¢¡ß®=¯ °2®±¡ß£°2 ~=$ G→ S
§"¢%¹º®¨¡q3Y → S
¡L X¢%¹NR£°=©oµ1¡-¶¡¢¡ CbS
£°=£q¡ ~®¨w¡|¢¡¯2®¨ −1¡L
0
M = [Y → G].
àá=âÞã8cä2å]áæEGzÓ>G`Óbed çy°2©ª©o¡² °2y¡y±=«B= °2©ª°2®µ¥©o¡y¢¡Ý£¡Ý£°2©ªµ¡-¶¡Ý°2 ߢ¡ß=!« ~°=©o°=®µ¥1©o¡ªèCKêéë$ìeQÙTj§eíqîbðïb1íqñ$ò§°2ε¡ ¢¬ó¨®¡©o©ª¡$ £°2¢¨®¡®
M£°=©o©ª¡Î°=ô¡L
¢¡ CbS°=|¢¡ DbS
b
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ª£q3 ¨¯2°=®1¡w¢¡1«Í©ª°= 1½z5®
Sb·¸¡9$i-k~i¢%¹N
1«Í©ª°= 1½
M¡ ²`¹º¡$ ¡®
rgZ(Y ) + dimS G .õ °2¬
0 → T → G → A → 0`¹º¡-¶! ¡°=௛¨LϬ
G~®
Sb·¸¡
1«Í©o°3 ~¬½
M¡ ©9»¢%¹º¡
Ï1 ®~" ~1°2|£®°21~=$ ¡¼R ®°=5£®~35¢¨LÏ¡wµ3®
W1(M) = [0→ T ] ⊂ W2(M) = [0→ G] ⊂ W3(M) = M .çß¡L ~¡Ï¬ ~®3 °=4³¨®1Ï¡
GrW1 (M) = [0→ T ], GrW2 (M) = [0→ A] et GrW3 (M) = [Y → 0] .çß¡¤¯=®~3¢¨¤°2$ 9=µµ¡¨¤®¡µ¡£L 1³2¡©o¡$ u m ljhöÌhjp~v3ljh ˧ßvthVi-k§/t-lBv3i-§»¡L W2
¡ 3µµ¡q1¨ µ=®C ~1¡nci-uh+vthVi-kki¢¡
Mb
1
"!$#%'&)(*,+-&/.0,1,*324&'5657( 890:%'#;<&'.#(=&)0>2-@? ACBEDFHGIKJBLBMNLBJOIP,QDSRTVUDWYXHZD\[)QNDG^]X_IZL`RI-RZBabWbXT
M 7→M t c ZWdTVXIDRKa I9[)QNDGVXIZL A 7→ At = Ext1(A,Gm)RTVUDWNF_ZL`a IFeFGOTfgBF
Bh:TabW IDFPdabI['QDGVX_IZLT 7→ Hom(T,Gm)
RTVUDWFZL-abIF-X_QLIF-IVXia Ij['QNDGVXIZLX 7→ Hom(X,Gm)RTVUDW`FZLja IFLTFIBEZlkm
nporqseqt<uwvxuqyz>|/`|~|z^ 53%+3`,rm4 <)M → S l 1 w < `$ H: l
τ≤0 RHomS(M,Gm[1])
DbS ¡^@ _ ¢_£¤ ¥E¢_£¡ −1^
0^§¦ ^ l ¢_ ¢_£¡= w¦ ¦ ¨ ¦ ¡ CbS ¨ i_¡^
l 1 w < © D DQXI
M t GI 1]fªQXWb[wP c ZW>RTJ:IDR ['QDGVX_QLWbIa abIfgID7X@RI M m«§QNfªfgI¬abIF@FGOTfBEFBh:TabW IDFR®¯ZDI°JBELXIVX±a IFjXQNLIF±IVXa IF
Z]fgQ7RZa IFR®²BEZX_LI°JBLxX³dTLWbUIDdX±aB´JLQJLWbTVX_T°RI°hW RZBEa WYX_TPQND
J:IZX9³dQW L c ZIabIF 1]fgQEX_WY['F
MIVX
(M t)tFQNDdX-GBDQDW c ZIfªID7X-WbFQNfgQLJOIFm © DªDQXI A P T P G P Ya IF§RWbµ>TLID7XF@GQNDFXWbXZBD7XF@RI
MP3QND DQXI
X = Hom(T,Gm)IVXa/®¯QD¶DQXI
T t = Hom(Y,Gm)m
© D\B[0→ A]t = [0→ At] , [0→ T ]t = [X → 0],
[Y → 0]t = [0→ T t] et [0→ G]t = [Xc→ At]
QN·cIFX@a®¸W fgBMI¬RI¬a®¸Ta TfªID7X@RI
Ext1(A, T )GQLLIFJ>QDRBDdXj¹¤aB¤GaBEFFI¬R®¯WbF_QfgQLJOWbFfgI¬RI
GJBLja®¸W FQfgQLJOWbF_fªIExt1(A, T )
∼→ Hom(X,At)m
º QWbXλ : M1 → M2
ZD\fªQNLJOW FfªIKID7XLI°RIZlk1]fgQEX_WY['F±FZL
Sm © D»J>QF_I
λm = GrW1 (λ)PQND
J:QNFIλab = GrW2 (λ)
IVX±a/®¯QD\J:QFIλet = GrW3 (λ)
m¼¾½¿CyÀuwvxuqyz>|'e|~|' m4 © DÁRWbX c ZI λ IFXpZDI W FQMNTDW I FW λm
IVXλab
FQD7XLIFJ>IGVXWb³NIfgIDdXRIFW FQMNTDWbIF±RICXQNLIFIVX±RIC³EBLW TVXTF±Bh:Ta WbIDDIFPIVX±FW
λetIFX±ZDI°WbDrÂIGVX_WbQND\¹´GQNDQ=Ã7BEZ"UDWm
Ä ®¸I^kWbFXIDGIR®¸ZDIWbFQNMTDWbIIVD7X_LIM1
IVXM2
WbfgJabW c ZI c ZIGIFRILDWbILFQND7X´fªÅfgI¶L_BEDMNFX_QLW c ZIPBEh>TabW ID"IVXTVXBEa ImÆHQEX_QDF c ZICa ICRZBaR®¯ZDICW FQMNTDWbI°IFXZDI°WbF_QMNTDWbIm º QNWYX λ : M →M t ZDI°W FQNMTDW Im¼¾½¿CyÀuwvxuqyz>|'e|~|~ m4 © DÇRWYX c ZI λ IFXCZDI´J>QaBELWbF_BEXW QDÇFW λab
IFXCZDI´J:QNa BLW F_BEXW QDÇRZMNLBRZTKBEh:Ta WbID»IXFW
λtm = λetm
È DIJ>QaBELWbF_BEXW QD¾IFX ¢ ¦ F_W4G®¯IFXÉZD¾W FQNfªQNLJOW FfªImÊÀBL¬RTVUDWbXW QDPabIRIMNLTR®¯ZDIJ:QNa BLW F_B<X_WbQND»IFX±a/®¯ID7XW ILdeg(λ) = deg(λab) deg(λm)2.
º WMIFX±J:QNa BLW FTPFQD»L_BEDMªXQNLW c ZI°IVXFQND»LBDMgTVX_BabI°FQND7XTMdBEZlkm
! "$##%!&'()%+*#,-.0/ 1 23!1/ -4* 5$
6787:9;7=<?>!@ABDCFE0BHG;I?JLK>MNGHJ+I7PO QR1S:Tf : M −→M ′ UVW SYXR1Z\[ V S W]WV T_^ W 1 `ba R\TS0c+XPX U ^ S de V3f R\S0T f g R aha WiUV a R\^jkS0X a WilW CbS m W TonqprR V?V R2T W Ker(f)
n W H−1 lU gs VWilWM
f // M ′ .t n W XTo^ W j^[X WV T_u\vn W ju\^ UV Z1^R U j WiwV S W Tojn u"T]X U ^ S u2jj W nY[hxy"z2| lW f m W T W XT a UV S l p UVWXT_^ U g T U ^ Wil p W~ T WV XSYR V ju V u g k[ Wil R V T4n W X4Z\^_u lU [XoXR V TGrW0 (Ker(f)) = Ker(fm) ,
GrW1 (Ker(f)) = Ker(fab) ,
GrW2 (Ker(f)) = Coker(fet) . W3lU u2n lW u\^T_S W ^ lW Ker(f) W XT Ker(f t) d QS f : M → M W XTnYu a U n0TSYjn0S g u"T_S0R V ju\^ UVWV TS W ^ n m R V jR\X W M [n] = Ker(f) d QS T m A W T Y XR V Tn W X g R V XTS0T U u V TX lW M m n W XZ1^u lU [XlW M [n]
XR V T T [n] m A[n] W T Y/nY d R aha W M [n] W T M t[n]X_R V T lU u U!~?lW u\^TS W ^ m SYn W~ S0XT WUVW u\jjn0S g u"T_S0R V vSYn0S V [u2SY^ W g u V R V SY UW M [n]⊗M [n]→ µn d! VW jR1nYu\^SYX_u2TSYR V j^S V g SYju2n W]lW MS VlU S:T UV u gg R U jn W a WV Tou\n:T W ^ V [ V R V?l [Z\[ V [^[ M [n]×M t[n]→ µn dQR\S0T
M UV 1 `ba R\TS0c W T n UVWV T_S W ^ d e V R\vT_S WV T UVW g R1^^ W XjR Vl u V g W vS W g T_S fWWV T_^ W np WV `X W a vn WolW XS0X_R\Z1[ V S W X f : M →M ′ T W nYn W X UW Ker(f) ⊂M [n] W Tnqp WV X W a vn WolW XFXR U X ` X g k[ a u\XWV Z\^R U j W X H ⊂ M [n] WV u2XX_R g S u V T f n W XR U X ` Z\^R U j W Ker(f) W T HnqprSYXR1Z\[ V S WV u2T U ^ W n0n W
lW M f1W ^X4n W gs VWilW H[0]→M m U S W XT4vS WV?UV 1 `ba R2T_S:c d6787 73"BH2B;I!KCIGJ+I$ABAA>"JL%BIKKB +I!K
1 ¡£¢ >¤JL¥¦K\7O §u V Xhn W ^ W XT W?lWg W TT W Tk¨X W m R Vw~W©UVV R a v^ W j^ W a S W ^ p m lWU!~WV T_S W ^X s ≤ g W T UVWV X W a vn W R1^ l R VV [ V R Vf S lW
D = d1 < d2 < · · · < ds ⊂ 1, · · · , g .QR1S WV T S UV g ku a ju\n0Z1[v^SY UWiW T M UV 1 `ba R\TS0cHj^S V g SYju2n W a WV TojR1nYu\^SYX[ lW Z WV ^ W g X U ^ S dª'«¬¯®¦°®±³²´Lµ;´b¶´b² dH· VW XT^ U g T U ^ WlWV S fW u U ju2^_u2kR\^S0 UW X U ^ M lW T¦¸j W D WV p W XTn u l R VV [ Wil p UV?l ^_u2j W u U?lW XR U X ` Z1^R U j W X wV SYX W T4jn u"T_X4X U ^ S
H1 ⊂ H2 ⊂ · · · ⊂ Hs ⊂M [p]T W n UW Hi
XR1S0T lW ^_u V Z pdijR U ^ i ≤ s W T UW Hs
XR1S:T;T_R2T£u2n W a WV TSYXR2T_^R1j W jR U ^;nqp¹u gg R U jn W a WV TX¸ a jn W g T_S0 UW X U ^ M [p]j^R f1WV u V T lW n ujR\n u2^S0X_u2TSYR V d
e Vºl [ a R V T_^ W nYu»j^R1jR\XS0TSYR V X U S f u V T W.WVºU T_S0nYS0X£u V T¼n u g R\^^ W XjR Vl u V g WPWV T_^ W XR U X ` Z\^R U j W X W TSYXR\Z1[ V S W X W T WV ^ W g R1jSYu V TPnYuj^ WUfW]lW nqpr[ V R V g [»u V u\n0R1Z UW jR U ^Fn W XX g k[ a u\XPu2v[n0S WV X½¿¾rÀ>%ÁÂ!à mj^R\jR1XS:T_S0R VÄ dÆÅ1Ç£dÈɱ\Ê;±Ë"®¦°®±³²´µÌ´b¶´Lµ dH· ÍÎ_ϺÐ$Î |\ÑÒ_"Ó¿ÔbÕ$y"ÖØ× Î£ÏÏ |!×+Ù"2xÓ Î£ÏÏ y"xÓ4Ô_ÚÛ|!×Ù22Ü Î xÓ Î£ÏºÝÞ ÜYhÒ_2Ó¿ÔbÕy"Ö× ÎiÐ$Î_Ï 1 ßbà y"Óq× á Ï M â ÖØ×+xÒ× â 2Ü Îà=Î xÓ â y"ÜY2ÖØ× Ï Ô ÏÐ$Î Õ Î xÖ Î g à |!x× ÏÐ;ã |!x ÎoÏ ÓqÖØ|ÒÓq|!Ö Î
Ð$Î x×+Ù Î "| â 2Öäy"Ö×ÚÛ| κÐÎ Óqz âÎ D Î x p å
1
"!$#&%(')+*-,&*/.-, )01# ' 2324*/.3/%5232768950) :;.3,=< ).1%;#>!?@)+*/.3*?@0';* 1 A 24%(0) :;.3,>*B#&*?@'/* gM = M0
// M1//
· · · // Ms−1//
Ms
λs
M t = M t
0
λ′0
M t1
oo
· · ·oo M ts−1
oo M ts
oo
M = M0
// M1// · · · // Ms−1
// Ms
%5CD EF) #5?G*IH@%5') J(%(?@9 *K2K!;, )0 ?L**/.M, 6@5*N,&*K EF) # ?L*IH@%5') J(%5?9 *I.61O@!')+*68';*/PQ EF) #5?G*H@%5') J(%(?@9 *4)R?S:S!')+*68';*T*/.U!$#> *VW WF)X# ?G*YH@%(') J1%5?@95*Y.6(O@!')+*68'/*;PZ [/%52ZO%1.1!/* Mi →M t
i →Mi*/.\!$#> *-VK K23680)]O^_)+; 0)+%(?4O ' p O%568' i ≤ s P`< ).S%;#&!?@)0* λs *;.a68?L*\O@%5 5'). A
0)+%5?8P λ′0 */.< )b?@c*'/.S*4,d< 6e?G*BO@%( ').S 0)0%5?fOg')b?L)]O5*h**4,>*$# ';!4,&* M → Mi*/.B!$#> iV
pdi O@%(68' i ≤ s jk@lRmGlnGlKoZp^qSrbs1rbpLtvu(wexzys1rb^wgl| ~e
H• ⊂ M [p] 4 ; 4K X SD@ 49 ^ D @ 3X ; wH•@ I ; Q ; X W H• W[@ ;X W ¡@ @/¢ M [p] ~eX V = ⊕2g
j=1 Fp · xj Fp £ g ; ¤Y Y 2g 3 ¥Y¦@ (xj) § ¢ ¨Y Y X 2g × 2g ; M ¢M@ ¦ g × g ©
J =
(0 J ′
−J ′ 0
),
ªJ ′ hY 7 ; £ @X3 B ¬«Y B eXB ¢8 W 1 =®Z¢¬ £
X GSp2g YIM M/@¢YIT ^¯ Spec(Fp) R £
GSp2g(R) = (γ, ν) ∈ GL2g(R)×Gm(R) | γt J γ = νJ .° I; r ≤ g ±¢¬@ ± £ ;@ ; I K ^ W1 V @ ² - I©
W1 =r⊕
j=1
Fp · xj .
®Z³ W2 = W⊥1 W3 = V ¦ ³ LS ´ 7 7 ¢ W• µ ³X
-X- £ ^-@ ¦ X P = StabGSp2g(W•) GSp2g LZUY¶I E¢¬g
^S¸·Q @ 4¹ L¨ s V StdD
• @ ©
V StdD
i =
di⊕
j=1
Fp · xj
1 ≤ i ≤ s ^®ZT ¨XM £ LM@ ¦ X PD = StabGSp2g
(V StdD
• )
! "$##%!&'()%+*#,-./10 -2 34!10 .5* 6$
78GSp2g 9
:<;>=?2@ 8
W = P (Fp) \ GSp2g(Fp) / PD(Fp) .
A 8B7 ?2CDE 8GF C?IHJ 8 ;$H=KILKIMGNOHPL 8 E 8 @=?I@PJQHPJ1?2;@L 8 E K"HPJQR 8 @ 78S78 C!T 7 LKI= 8 KIC!T 78V 9
UJL8 HLD@?2;H 78 @V@?2C@XWZY2L[?2C= 8 @ 78/\]8 R!J 78
P8 HPD ^
8 H_@PJWGSp2g ^ WL
8 HWLD
7` @JaYI; 8 ;$H_E 8 @VY2L[?2C= 8 @ 78b 8Oc E 78
GSp2g ^78L8 H 78
LD ^E K 7`d ?IMe=?2@JQHPJ1?2; 78gf LChK"HiMe?I;$HPL 8gF C 8 W 8 @[HiC; 8 ;@ 8 MjDE 8
k;J 8 H F C 8
W = WL \ WGSp2g/ WLD
.
l ?2CL/HP?ICHw ∈ W 8 H
1 ≤ i ≤ s ^?2;>=?2@ 8
mi = dimFp
((w · V StdD
i ) ∩ Gr1(W•))
ai = dimFp
(Im(w · V StdD
i ∩ W2 → Gr2(W•)))
ei = dimFp
(Im(w · V StdD
i → Gr3(W•)))
.
A 8 @5;?IMBDL 8 @ 7` H 8 L[MeJ1; 8 ;$Hw78 MeKI;J1NL 8 C;J F C 8
9U?IJ 8 ;HSC; d hKIMG=mKIE1Y ` DL[J F C 8nd ?I;; 8 T 8
^ MC;
1WZMe?3HPJQo]@CL
S ^=L[Ja; d J1=KIE 8 M 8 ;$Hp=?IE K3LJ1@ `
^78
LPK3;Y>H?2L[J F C 8r ^78 Y 8 ;L 8
g8 HH•
C; 8 @HLC d HCL 8 =K3LPK3h?2L[J F C 8e78 H c = 8 D 8 ;p 9q ?IC@rKsR2?2;@
R!C F C 8 EaK@[HPL[C d HPCL 8m7%tu8 TH 8 ;@Ja?I;v=K3;K d h `8w7` k;J1HBC; 7 LPK3= 8 K3Cv@CLM [p] ^
7 ?I;$HBE 8 @jY2LK 7 C ` @
GrW1 ^ GrW28 H
GrW3@P?I;$HrL 8 @= 8d HPJQR 8 M 8 ;$HjKI== 8 E ` @rMjCEQHPJ1=EaJ d K"HPJQo
^KID ` EaJ 8 ; 8 H ` HK3E 8
9l ?ICLxHP?2CH
1 ≤ i ≤ s ^?I;y=?I@ 8
mi = logp rg(Hi ∩GrW1
)
ai = logp rg(Im(Hi ∩W2(M)→ GrW2 )
)
ei = logp rg(Im(Hi → GrW3 )
)
:<;zKmi + ai + ei = di
8 HGE 8 @S@PCJQH 8 @(mi)i ^ (ai)i
8 H(ei)i
@?2;H d L?IJa@@PKI;H 8 @9:<; 7` k;JQHGC;
7 LPK3= 8 KIC>J1@?IHPL[?2= 8|sd LKI;@
V H•
•
78V8 ;>=?2@PKI;H
V H•
i =
(mi⊕
j=1
Fp · xj)⊕(
r+ai⊕
j=r+1
Fp · xi)⊕(
2g−r+ei⊕
j=2g−r+1
Fp · xj)
=?2CL1 ≤ i ≤ s 9
l KIL d ?2;R 8 ;$HJa?I;^@Jmi = 0
EaKm=L 8 MeJ1NL 8 @?2MeM 8S7 J1L 8d H 8B8 @H;CE1E 8_ J1E 8 ; 8 @[H78 MG~M 8 KR 8d
ai8 Hei 9
E 8 TJa@[H 8 C; ` E ` M 8 ;$He;?I;'C;J F C 8γ ∈ GSp2g(Fp)
F CJ d ?I;CY2C 8 E 8 @7 LPK3= 8 KIC!T
V H•
•
8 HV StdD
• 9U?2;J1MK3Y 8S7 K3;@ W 8 @[HDJ 8 ; 7` H 8 LMGJa; `8B8 Hx?I;EaK;?3H 8
wH•
dIt8 @[HE t Ja;RIK3LJaKI;H 7 J1@ d L 8 HmK3@P@? d J ` E K,@[HLC d HPCL 878 ;J1R 8 K3C=K3LPK3h?2L[J F C 8
H• ⊂ M [p]78 H c = 8 D 9 ;$R 8 L[@ 8 M 8 ;H
^@JM → S
8 @[HekT ` ?2;zR2?2JQHwKIJ1@ ` M 8 ;$H F C 8 =?2CLGH?2CHw ∈ W ^
JaE 8 T!Ja@[H 8 C; 8@[HPL[C d HPCL 8x78 ;J1R 8 KIC
H• ⊂M [p]H 8 E1E 8xF C 8
wH•= w 9
!"!$#&%')(+*,.-)/0*12%'4365879-;:=<>%?,.3@BADCEGFIHKJADLMADNLA0JOH0PQCR0SEUTV+WCXYEERCEUZYA[CR0\XYR0C]TH
1 ^`_ba.ced f8gihjKkl_mfnapoMg PrqHILEXYR0C]JZYSEEtslCi\OV+\MXvuwSHxTHIE giapyyzh|bghbgiz>~izyzoMh|hyh TSFOA0JXv\LtH$TH #x6 CCR0\H Ouv
ZYAGFAp\MVIN0RLtXYHTRCi\ZvHIERiH+\MEEMR0Ci\ZYHIETXYA0N0LMADHIE
(U → S)R0
SHIEt\SCEFOVIACR0LA0ZH+\
USC
ERSE>EFOVIAR0SqHIL\THICEH0PH+\TRCi\ZYHIEIFOHIEER0Ci\ZvHIETXADNLA0HIEFIRS\MAD\XvE
U //
S
U ′ // S ′ .
RX¡\(U → S)
SCRiH+\THOuv
¢¤£¥§¦©¨`ª¨«¦¬®e¯®`¬ ;° ± C
1>R0\Xv²THx³Sx2RLtT
MESL
(U → S)HIEt\´SC\LXvJZYH+\
(G, Y,MU)Rµ G HIEt\SCEFOVIAEHIX¶`A0VIZYXvHICTHLMADCN\RLtXYuwSHFIR0CEt\ADCi\ESL
SP
µ Y HIEt\SCLtVIEHnA0SESLSP
µ MU = [YU → GU ]HIEt\mSC
1>RD\MX¡6ESL
UPlR0
YUH+\GU
TVIEXYN0CHICi\ZYHIELtHIE\LXvF+\MXvRCETHYH+\
THG · U
¸b¹0º¼»½¾¿©¹[¬®e¯®ÁÀ ;° @ÃÂÄA0JLtIE§SGA3 ÅÇÆ vÈÉ. PlZYAÃJA0L\MXvH´\RLtXYuwSH´TH G HIEt\mXvER0\LX¡qlXADZYH´ESLZYH
EFOVIbAxCRLtbADZS
CÊCRD\MHMumU,S
ZA¤FnAD\VINR0LXvHTHIE1>RD\MX¡2ETHK³SRLtT ÌË A¤LHIEt\MLtXYF+\XYR0C · U XYCTSXv\SC
2R0CF+\MHISLMumU,S → MotU
uwSXHIEt\xJZvHIXYCHIHIC\WTIZYHGTÂÍADJLIEZYHU\MOVIR0LIH !Á !Î R0Xv\λ :
M → M ′ SCR0LJOXvEHÏTH 1>R0\Xv2ETHóSx2R0LT CTX¡\ÐuiSH
λHIEt\SCH d|aM~zydh EX λU HIEt\
SCHÐXYERN0VICXYHÐTH1>R0\MX¡2E CUTV+WCXv\ZvH g.kÑDÒ M t TH M FIR0H´ZÁÂÄSCXvuwSH
1>R0\XvTH§³Sx2RLtT
uwSXwV+\MHICTZvH1>R0\Xv
(MU)t Ó ER0CHÔXYEt\HICFIHLVIESZ¡\MHTHZYAÐFIRCEt\MLtSF+\MXvRCTH M t D± CHrÕ apÒYÑDod|nÑDcedea.yλ : M → M t HIEt\ÐSCH&XvEMR0NVICXvHÏ\HIZYZvH&uwSH λU ERXv\ÐSCHÏJR0ZADLXvEMAD\XYR0CKTS
1>R0\Xv
MU CKCRD\MH
MumU,S, polZYAÐFnAD\VINR0LXvHTHIE
1>R0\XvE©TH³SRLtTJRZYA0LtXYEVIE©ESL
(U → S)0Ë HIE;IFOHIE©THFIH+\t\MH
FnAD\VINR0LXvH§ER0Ci\ZvHIER0LJOXvEHIEmTH1>RD\MX¡2ETHϳSx2R0LT$uwSXLtHnEJHIF+\HICi\ZYHIEmJR0ZADLXvEMAD\XYR0CE Ó FIHERC\TR0CFTHIEXvERN0VnCXvHIE
R0Xv\M
SC1>R0\XvÃTH[³Sx2RLtTÖESL
(U → S)× RD\MR0CE
YPTPAPG
ZYHIEEFOVIbADEHICNLtRSJHIEÏuwSX²ZYSX²ERC\ÃADEMERiFnXvVIEÏH+\
X = Hom(T,Gm)6× RD\MR0CE Y ZYHJZvSEÏNLA0CT=ER0SEÁA0XvEFIHnA0S
FIRCEt\LSF+\XYZvHTHYÃSCXTÂÄSCHIFOH Y → G
JLtRZvRCNHnADCi\YU → GU
XGt TVIEXYN0CHZA§JADLt\XYHEHIX¶`A0VIZYXvHICCHKTH
M t PQR0C]CR0\H X ⊂ XZYHKJZvSEUNLA0CT]ER0SEt2A0XvEMFIHnADS]FIRCEt\LSF+\XYZvHKTH
XÃSCXØTÂÄSCH?IFOH X → Gt JLR0ZYR0CNHnADCi\ XU → GtU
) CÙJR0EMHX = X/X H+\
Y = Y/Y Ó FIHERC\TH+SÔbÁADXYEFIHnADSlÔGFIR0CEt\MLtSF+\MXYZvHIEmESLS× RD\MR0CE
DivSZvH´ÁADXYEFIHnADS$THIEmTX¡qlXYEHISLtEmTH
S× RSE
FIRCEt\LSXvERCE · JLtVIEHICi\SCH§2R0LHXYZvXYCVnADXYLtHFnA0CR0CXYuwSHB : Y ⊗X −→ DivS .
! "$##%!&'()%+*#,-.0/ 1 23!1/ -4* 5$
678891:9<;:>=%?A@<B91CD=FE7GS:H9<I0J>7;,JHCH@2I0J>=GK89<IL;JNM1O;OCDILE$7G
U = Spec(K)GPJ>E7GKQLG:SRT@2IL:VU
W GX@<7!Y X GPJY:91I0G;$J W 91;:DJZ@2;$J:X[\]9<^_^`G A G:DJaQLG^`9=bQLG=GcdOC9<;3=G:H@feBCDGM1O;OCDILE7G<gQL@
h @<Q07@2JIL9<;i=G K I0;=7I0J4=G:4IL:91^`91CD8jIL:^`G:
G(K)/G(R) ' T (K)/T (R) ' Hom(X,Z) .k Gl8Q091;M<G^_G;$J
Y → G(K)IL;=7ImJn7;GoRp9<C^`GlBILQ0IL;OX@2ILCDG
Y ⊗ X → ZE7I]:GoRp@ W JH9<CI0:GlG;
Y ⊗ X → ZGPJ>;G_=O8G;=,E7G_=G_Q @ W Q @2::G_=%?AI0:H9<^_9<C8jI0:^_G_=G M [ k @ W 9<;:DJHCD7 W JHI091;,=G
B=@<;:qQLG W @<:rM<O;OCH@2Q:X?sG;t=O=7ImJr8@2CqQL9 W @<Q0IL:H@2JIL9<;tQ0GdQ091;M>=G W j@2E7Gu=I h I0:HG7CqGPJr=G: W G;$JGdOPJH@<QLG<[vKw<xzy|1~wF% [ k @Rp9<C^`G
B8CDG;=:G: h @<Q0G7C:K=@2;:KQLGi:917:VURT@2IL: W GX@<7=G:K=I h I0:G7C:
=GS=91;JrQ0Gu:H7889<CDJqG:DJr=I0:pD91I0;$Jr=G
U[!6I
MG:DJq891Q @2CI0:O<g
B h OCImeGuQ @ W 91;=ImJHI091;K=Gu:D!^`OPJHCDILG:7I h @<;JHG h IL:VU"U h IL:>=GKQL@8@2CDJILGKOPJH@<Q0G φ : Y → X
=GKQ @i891QL@<CDIL:H@"JHI091;89<7CJH917JG:>:G W JHI091;:QL9 W @<Q0G: y GPJ y′ =G Y g91;@
B(y, φ(y′)) = B(y′, φ(y)) .6IqQL@89<Q @2CI0:H@2JIL91;FG:DJ>8CI0; W IL8@2QLG<g B IL;=7ImJN7;G_Rp9<C^`GKBILQ0IL;OX@2ILCDGK:D!^_OPJCI0E7GK:7CYQL91CD:E7G
Q?s91;iI0=G;$JI0eGX
GJYnQ?A@<I0=G=G
φ[
m3>PX"<+ ¢¡_£¤¥,¦§"¨©¢©$«ª¬Tp©$¡r ® @2;: W GPJDJHG.8@<CVJHI0G<g<;917:JHC@<=7I0:91;:=G:aCO:7QmJZ@"JH:a=7 W j@28I0JCG¯n=G`°s± n²³´ =@2;:aQ0GQ @2;M@2M1G=G: 1
U§^`9<JI0R+:4=G>µi7^nRp91CD=%[¶;$JC9$=7I·U:91;:¸Q @ W @"JHOM<91CDILG OuvComp
=9<;$J¹QLG:¸91B$GPJH:¸:91;$J¹Q0G:¹=IL@<M<CH@2^_^`G:(U → S ← S0)
g191ºSG:DJ¹7;
: W jO^K@K@"»K;GS;91CD^K@2Q¢GY W GQ0QLG;JXg91º S0
G:DJ¼7;:H9<7:VU§: W jO^K@`RpGCD^_OS=OPe;I8@2Cu7;IL=OX@2QCO=7ImJJHGQ0:½E$7G
S:91I0J
S0
U W 9<^_8Q0GPJXgGPJ½9<º U G:J¼7;':917:VU§: W jO^_@t917 h GCDJ½=G;:Gf=G S =IL:+9<IL;$J½=GS0
[k G:a¾b W jG:4=G W GPJJG W @2JHOM<91CDILG:9<;$JuQ0G:4=I @2M1C@<^`^_G: W 91^`^N7JZ@"JHI0R+:
U //
S
S0oo
U ′ // S ′ S ′0
oo
¶;$JC9$=7IL:91;:r^K@2IL;$JG;@<;$J]7;Gd;9<JIL9<;o=G¼891:ImJHI h ImJHO¼:7CrQ0G: 1U§^`9<JI0Rp:r=Gµi7^`Rp9<C=t89<Q @2CI0:O:X[
691I0G;$JM
7;1U§^`9<JI0R=Gdµ7^nRp91CD=`89<Q @2CI0:O4:7C
(U → S)GPJφ : Y → X
QT?AI0;«G W JHI091;_ W 91;9@<7e;I@2::9 W ILOG<[¿'ÀÁÂÃÄVÃÅÂÆ% Çȧ [
MG:DJ891:I0JI0RrCDGQ @"JHI h G^`G;$JS S0
:IÈ89<7C½J917JG`:G W JIL9<; Q09 W @<Q0Gn;91;;7Q0QLG
y=G
YgQ0G=I h I0:G7C B(y, φ(y))
G:DJuGPÉG W JHImRGPJu:9<;i:7889<CDJ W 91;$JILG;J S0
[k @K89<:HImJHI h ImJHO>=G M IL^`8QLI0E7G>E7G YS0
= XS0= 0
GPJ¼E$7GMU
G:J¼7;1U§^`9<JI0R W 9<j91^`91Q091M<ImU
E7G^`G;$J W 9<; W G;$JHCDOG;i=GM1CDO 0g%ÊËpÌXË
H−1(MU) = 0GPJYU
:?AI0;«G W JHG=@2;: GU
[ͼ;i;9<JG
MumU,S,S0, pol,+Q @ W @"JHOM<91CDILG½=GP: 1U§^`9<JI0Rp:.=Gµ7^nRp9<C=o89<Q @2CI0:HO:]:7C
(U → S)891:ImJHImRp:.CGQL@2JI h G^_G;$Ja S0
[ k @8C9<891:ImJHI091; mT ^_9<;$JHCDGCH@iE7%?sGQLQ0GKG:DJ>OE7I h @2QLG;JHGKQL@ W @2JOM19<CI0G DDpol
=OPe;ILG_=@<;:>Q0G8@<C@<M<CH@28jG`¶D¶¶Z[ÏÎo=G3°A± n²³´ [¢Í¼;;9<JG
DegU,S,S0, pol
QL@ W @2JOM191CDILGn=G:: W jO^_@<::G^`ImU@2BOQ0ILG;:
!"# S $&%&'(*),+.-*/ !102 ).% # + ! ( !1!"# U - 354- # %6/783 26# +,9 " -: 2 / ! 3;%&/<3 !"# S0 => /?4-*@A%6/4--*/ 26" 3 # -9 " -)B% 0 % # 3+.-C3 26# +,9 " -D4- GS0
!2 +,3+ !E2 3 # +,FG+B%&).- !E"# S0 =H - !JIK :ML- !J!2 /<3),- ! @ 2#N0 L+ ! @A- ! 4- ! :ML(*@A% !! -*@A+POQ%6'(*).+,-*/ ! 9 " + # - !0 -*: 3-*/<3R)B% 02 )B% # + ! %&3+ 2 /TS:*- !E2 /<3U4 2 /:T4- ! + !2 7(*/+.- ! =VXW6Y[Z\]^1W`_acbaBd5afe =hg > //-i3 #N2" F- 0 % ! )B%j: 2 /4+k3E+ 2 /4lm+ !2 3 # +kFn+.%6).+k3E(T4%&/ !porqsjtu $:ML =vvxwy=z 2 @A@-|),- # -*@A% # 9 " - H %&/ or~u $R = = =,*w $-*).),-|- ! 3 026"# 3E%6/<3/(*:*- !! %6+ # - 02"# %&).76(*' # + ! - #)clm-n3-*/ ! + 2 /4-iU%n/% " 44- G f*% "!! + o
SGA7 . =G=,*w - 3 oSGA3 [ =Pwf;= - 0 ) "! $:*- 3N3E-
: 2 /4+k3E+ 2 / 0 - # @A- 3U4lm(*).+,@A+,/- # )lrL 02 3L K! - omqsjtu $ vv;= = % wc=H -T' " 3R4-T)B%j: 2 / ! 3 #N" : 3E+ 2 /4- " @ 2# 4- ! 3U4lm- nL+.'- #R" / 2 /: 3E- "#
Mpol : MumU,S,S0, pol,+ −→ DegU,S,S0, pol
3E-*)9 " - ! + S - ! 33 # %6+k3J: 2 @ 0 ).- 3J4- 02 +./3f- # @( S0 - 3J4- 02 +,/<3J76(*/( # +,9 " - U = Spec(K) $ 2 /%6+,3-*/ ! -*@?'),+ ! 3-*@A-*/<3
Mpol(M)(K) = H0 (M(K)) .
H -i3EL( 26#K @- !" +,F&%6/<3R- ! 3U4 X " @j 26# 4$-*3U¡%6)k3E+,/7 ! - 3 z L%&+ =¢¤£¥§¦ \§¨6YW`_acbaBd5acbª© omqsjtu wy«5¬®§°¯¤±¯B²«¤±¯B³´µ±¯B¶·§=hg ¸;¹hºN»¼.½¾¿ºTÀGÁ * Á ¾¿ºMÀnà EÄ Á  Á¼cÅÀººN» ƹP¼ ¼Ç¾¿º
Mpol : MumU,S,S0, pol,+ −→ DegU,S,S0, pol .
¸;¹iÃEÈ Ä ¹¼B½º|ÀGÁºÈÅ Àn¼ÇÉ Ä ¹BºÁ º`Ê<º EÄ ¾¿ÈxË Â Ã¼cº;½ pÌ ¼Ç¾ M = (G, Y,MU = [YU → GU ]) TÌ ¼j¹Í  ÁÁ  ¾Qº G = Mpol(M) º¾8½¼ X(G) Ê<È;½¼,Ë&ÁºÎ¹.º *Ä ¼B½ º Ä À ; Á½¾ÃÀ ¾c¼cÏ ¹Bº Ä ½;½  ¼yÈÐ G Æ Ä ÃX¹BºÎ¾fÑÈ ÂÓÒÃEÔÕκ _aQ_aybÖ º¾ X ¹Bº *Ä ¼.½ º Ä À ; Án½¾cÃMÀ ¾c¼Ï ¹Bº Ä ½;½  ¼yÈ?Ð M Ö Â Á|Ê&¼B½QÆ Â ½ºÊ°Í ¼B½ Â Õ Â ÃxÆÑG¼.½Õκ;½  Á Ò¾  ÃM¼yº¹r½ EÄ Á  Á¼cÅ Àº;½
GS0
∼−→ GS0,
X(G)∼−→ X,
GU [p]∼−→ MU [p] .× ÁÎÆ Ä Ã¾c¼ Àn¹P¼yºMÃ Ö GU [p] º;½¾¤ÕjÀnÁ¼¤Ê°Í ÀnÁºi½¾cÃMÀ ¾cÀnÃEº?Ê°Íغٻ¾¿ºÁ½M¼  ÁÎÆ Ä Á Ä< ÑÈ;º
Ú ¯³¯B¶¯ÛÜÇÝ ~ ÜÞfß ~ µ§ÞÇ áà ´âÜ ~äã ®Óåæ´çÝhÞÜfÞ è<~ ÞÇ®§´çß é°å è µ®ÓÞfêë´¯ì í 2 +.-*/3 S " /î:ML%6@ 0%6),7(*' # +.9 " -j/ 2# @X%&)5-:*-*),).-*/3$ U " / 26" F- # 3i4-*/ ! -j4- S - 3 G " / ! :;L(*@X% ! -*@+kOQ%&'(*).+,-*/ !E"# S%6'(*),+.-*/ 02 )B% # + ! ( !"# U =ï 2 3 2 / ! KU ).-ð/ 2 <% " 4-ð)B% 02 ).% # + ! %Ó3E+ 2 /ñ4- GU - 3 K !2 /ò%&4L( # -*/:*-4%6/ ! G = z - 33-%64L( # -*/:*-ð- ! 3 " / ! :ML(*@A%X-*/ó7 #26"0 - ! 9 " % ! +POyô/+- 3 0 )B%Ó3 !"# S = %&/ !?omqsjtu $vNvM=§w $¡%&),3+./7 ! - 3 z L%&+8@ 2 /<3 # -*/<3õ9 " -ó),-ñ9 "2 3+.-*/<3 G/K - ! 3 # - 0# ( ! -*/<3E%6'),- 0 % #Î" / ! :ML(*@X%! -*@A+POQ%6'(*).+,-*/ª/ 2 3E( Gt → S = z 2 @A@- Gt
U - ! 3A+./4( 0 -*/4%6/3X4 " :ML 2 +Pö4-)B% 02 ).% # + ! %&3+ 2 /$÷).-3EL( 26#K @- Ú ¯ Ú ¯B² @ 2 /3 # -T9 " - Gt /-T4( 0 -*/4 0 % ! 4 " :ML 2 +P4-T).% 02 ).% # + ! %Ó3E+ 2 /4- GU =
í + (U → S ← S0)" / 2 'GøN- 3?4-Î)B% :%&37 26# +,- OuvComp $h)B% : 2 / ! 3 #" : 3+ 2 /[4- " @j 2# 4[- ! 3
: 2 @ 0 %&3+.'),- |)B%4 " %6),+,3( 4%6/ ! ).- ! -*/ !A2ùª! + M - ! 3 2 'Gø- 3A4- MumU,S,S0, pol,+ 3-*)R9 " - G =
Mpol(M) $ 2 /ó% Gt ∼→ Mpol(Mt) = %&/ ! ).% 0#N202! +k3E+ 2 / !" +,F&%&/<3E-6$ 2 / ! - 0 )B%6:*-A/ 2" F-% " 4%6/ !
).-:% !2ù S - ! 3 " /ú:;L%&@ 0 %&).76(' # +.9 " -|/ 26# @A%6)i-n:*-*).),-*/<39 " -*),: 2 /9 " -- 3 U - ! 3 " / 26" F- # 3
! "$##%!&'()%+*#,-.0/ 1 23!1/ -4* 5$
67S 89 7;:<:=7?>@AB>ABCD0:DEABF>7@GH7;:I6%JLKBCCAMD07@IN G O FD0F$P2KB@<D K2F$:IKBQ0R1ST@<DEU O 7?U O DVC=7@K O :=D0QE7W>A O @7X>@<DEGH7@4Q07CY>@<A1>@<DES;:SC4GZA6 O QEKBD0@7CYNHQEK\[]@ABF$:=D0^@7_67C4MABGZ>K2M;:=Da`MbK2:DEA1FC 8
cedgfBhVfi"jlkmjfnpoqsrqutquowvxLy.z|~b"$ 8 m1 _!?;b"e ;+¡+¢=£2+=B(G) : X(Gt)×X(G) −→ DivS
\;¤ £¥=¦?l"!4§=¨£2!©$«ª$\ =£g«¥£2_!,£"=£"¬ b§=£2+¦u©2=|§gI¥ S ′ → S "£2+B(G) = Mpol
(B(M−1
pol(G)).
® JLKB>@<^bCIQ07W:=¯SAB@^GH7 °±² QEK>A1QEKB@<DEC=K":=D0A1FGη → Gt
η
CJ³S;:=7F6´7F O F´GZAB@>¯D0C=GH7 G → GtU O D%DEF6 O Da: O F7µ^M¯7 φ : X(Gt)→ X(G) 8¶ K|[]AB@GH7_TDEQ0DEFSbK2DE@<7 B PS@<D0`7_QEKH@<7Q K":=D0A1FB (y, φ(y′)) = B (y′, φ(y))>A O @Z:=A O :=7CC=7M:=DEABFCQEAMbK2QE7C y 7;:y′67
X(Gt)7;:·Q K,>A1CD0:D0PD0:S67
M−1pol(G)
DEGH>QED0U O 7U O 7B(y, φ(y))
7C<: O F¸6D0P!D0C7 O @47;¹7M;:=Da[6ABF$:YQ07_C O >>AB@<:47C<:YQ07?Q0DE7 O 67_FABF¬KBFF O Q K":=D0A1F¸67 y 8º »¼½½¾]¿°ÀBÁ¿+ÂÃÅÄVÆÈÇÂÄÉÁ¿]ÊgÁ!Ë;Ì%½ÍÆÃÀ1Æ$Ê$ÏÎ Ð ABDE7F$:
(U → S ← S0) O FÑA1T$Ò7;:67OuvComp
7;:G O FÓABTÒ7;:367
DegU,S,S0, pol 8«Ô A2:=ABFC Gt Q07Õ6 O K2QÖ67 G 6S;`FDÖ6K2FC3Q07'>KB@K"×R1@KB>¯74>@SMS67F$: ²
λ : G→ Gt Q°JL7X!:7FCDEABFØC O @ S 67YQEK_>ABQ K2@D0CK":=D0A1FZC O @ U 8$Ô AB:A1FCSR1KBQ07GZ7F:eG : S → G
7;:eGt : S → Gt QE7CC7M;:=D0A1FCF7 O :=@<7C ² ΩG = e∗G ΩG/S
Q07\[°KBD0CM7bK O 67C 1×s[+A1@<GZ7C
6D0¹S@<7F$:=D07QEQ07CHD0F$P2KB@<D K2F$:7CHC O @ G ² 7;: ΩGt = e∗Gt ΩGt/S 8Ù FwC O >>ABC=7ÚU O 7 S → Spec(Z)7C<:
[]AB@GH7QEQ07GZ7F$:«Q0DECC7Ö7;:«U O 7Q07MABGZ>Q0SGZ7F:K2DE@<7 ∂S 67U6KBFC
S7C<: O FÚ6D0P!D0C7 O @«N\M@ABDEC7GZ7F:=CFA1@<GØK O X 8Ͷ K'>@<A1>ABC=Da:=D0A1FÛC O DaPBK2F$:=7ÚGHA1F$:@7ÚQ°JL7XD0C<:=7FM7Ú6%J O FÛGHA1@>¯D0CGZ7ÚU O DÏC=7@K'Q K´MQESÚ67PABÜ:=7_67_Q°JLS;:=KB>7_67_@7MABQEQ07GZ7F:Y6K2FCYQEKHMABFC<:=@ O M:DEABF¸67C4MA1GH>KBM;:D0`MbK":=D0A1FC 8
cedgfBhVfi"jlkmjfnpoqsrqÝquowvxLy.z|~Þb"½ßàá% 8 <1 ?!âY¦§=¨|§=£""°;KM/S : ΩG ⊗ Ωt
G −→ Ω1S/Z(log ∂S)
£¥¥ ¢ÍâY¦§¨\ª$\ã A6K2DE@K"× Ð >7FM7@gäVå + Ö;£$§" \KM/S : Coker
(ΩGt ⊗ ΩGt −→ ΩG ⊗ Ωt
G
)−→ Ω1
S/Z(log ∂S) ,
"æ¸E£YâY¦=§¨ ΩGt ⊗ ΩGt −→ ΩG ⊗ ΩtG çB"s x⊗ y ! λ∗(x)⊗ y − λ∗(y)⊗ x ä
¶ K?µ^M¯7è67 ã A6K2DE@K"× Ð >7FM7@ß7C<:.MABFC<:=@ O D0:7ÖNWQJLKBD067I67C 1×uGZA2:=Da[]CÏA1T:7F O CÏ>K2@ßQE7è[+A1FM;:7 O @U O K2CDa×uD0F$P7@<C=7_NHQEKHMA1FC<:@ O M;:DEABF¸67eé O G|[]AB@6 8
ºê3ÌbÊ"ëÀ2gëVÊgÆìÄÉÆìÿíÆÁëî½ÁÊgÁÉÂÊ"¿°ï%ëVÆßÎ ð A O @:A O :ÚABTÒ<7;: (U → S ← S0)67
OuvComp² FAB:=ABFC
DegU,S,S0, pol, 0
Q KØMbK2:SR1AB@D07\67CICM¯SGZKBCIC7GZD¡×lKBTSQED07FCGC O @ S 67e>K2@<:DE7
:=AB@D0U O 7èDECA2:=@<D0P!DEKBQ07èC O @ S0
6ABF$:ÏQ K?@7C<:=@<DEM;:DEABF·NU7C: O F·CM¯SGZKeKBTSQ0DE7F>A1QEKB@<DECSèG O FD6%J O F7C<:=@ O M;: O @<7¸67¸FDaP7bK O >K2@=K2¯A1@<DEU O 73673:lñ!>7 D 7Fp 8 ® 7¸GHòGZ7 ² FAB:A1FC MumU,S,S0, pol,+, 0
QEKMbK2:SR1AB@D07'67C
1×uGZA2:=Da[]CÚ67,é O GH[+A1@<6Ó>ABQ K2@D0CSCÚ>A1CD0:D0[+CÚ6A1F$:ÚQ07
1×uGHAB:D0[?C=A O Cm×aÒ=KBM7F:3C O @ U7C<:_G O FDÞ6%J O F7|C<:=@ O M;: O @<7H67|FD0P17bK O >KB@KB¯AB@D0U O 7H67\:lñ!>7 D 7Fp 8¶ KÚ>@<A1>A1CDa:=D0A1FÕC O D0P2KBF:=76SMA O Q07_6 O :=¯SAB@^GH7 ºaáÉ º >KB@.[]A1FM;:A1@<D K2QEDa:=S_67_QEKHMABFC:@ O M;:DEABF¸67eé O G|[]AB@6 8
1 "!
#%$&'(&*),+.-/+&*02143653/7 381:9<; =?>A@B,C4D"EF>"GHMpol I C4JKG I ELGC>NMPOG IRQ,SKT >"CDP>NJ>ND S EFM8UVB,H I >XW
Mpol : MumU,S,S0, pol,+, 0 −→ DegU,S,S0, pol, 0 .
Y[Z\\Z^]*_a`b]c`bdef\Peg`bh%ij*da\egkZmlonqpr\j*s*Z^s*ZrdZrd]turZrds*Zmvw`xs*j*lgZrds*Z1 y vw`\egzRd]4`ltK_egdprdv%jh*egds*Zda\P_aj*u\Pj*_aZrds*Z^h*egVZtKj|]t_t*`_eg~xj*Z 9
g(wmr? w(?<Rw(1 .
tKh*durZ%]t_PtKb_t]**Zc`hvw`bhV\P_aZ%~xj*Z%lgZrdutv]*ds*Z%vw`xs*j*lgZrds*Z1 y v`\egzRd]*_aeh*ureg]tlgZrvwZrhV\]c`blt_aedprd%~xj*e[d`bhV\NvNj*h*egdNs*Zda\P_aj*u\Pj*_aZrdNs*Zh*efVZtj]tK_PtK*`b_ae~xj*Zd`bh\Ntlgbpr*_ae~xj*Zrdr<Z\N`hlZrd
s*prur_ef\ Z"¡]*legureg\ZrvwZrh\ 9 Y[`bvvwZrh*¢r`h*d ]tK_£\_PtKeg\Zr_lgZ^utds¤n¥j*h*Z^da\_j*u\j*_Z^s*Z^h*efVZtKj|\_efetlgZ 9g(¦§¨xb©,R?cª«¬(V©,¦®¤(¯±°?ª
1 RX[² ³ `j*_´\P`j:\¨du*prvwtµtKcplgeZrhA[`bh
h*`\Z PA lgZmk*_p^s*Z ³ `begh*ut_ap^dj*_A× At 9
#%$&'(&*),+.-/+&*02143¶·3.14381¹¸"º §¨x6»:¼o½·<© [¾¿(ÀoV¤Á 9<; = S D S EFM8UB,H I >ÂJ>XW 1 Ã.Ä B,E I @PW>XW"EÅMPOG IRQ,S ÃT >"C*EF>NÆ TS D S E.M8UB,H I >NJVB,CE T >XWÇBbÈÉ>"EWWB,CE T >XW±@ S Ä IRTRT >PW±@B,H Ä MP>XWÊ J(Ë GCÌWDXÍ*M Ä SÎS ÈM TÏI >C A ÐÊ J>NJV>"GÑHXMPW> S GÑ X >E
Y ÐÊ J>NJV>"GÑ Ä B,H8ÒÍ I W Ä >XW c : X → At>"Ect : Y → A ÐÊ J(Ë GC4>ÓEoH IRQIoS,TÏI W S E I B,CJV> TS Gm Ã È I >aÑEF>"C:W I B,C (ct × c)∗PA
W"GxHY ×X Ô= S D S EFM8UVB,H I >J>XW 1 Ã8Ä B,E I @PWÓÒcH I CD I Ò S,T > Ä >CE[ÒB TS H I WMXW|>XW"EÕMPOG IRQKSKT >CEF>Æ TS D S EFM6UB,H I >JB,C*E T >XWBbȦÉr>"EWmWB,CE T >PW±@ S Ä IRTRT >XWA@B,H Ä MX>PWÊ J(Ë GCÌWDXÍ*M Ä SÎS ÈM TÏI >C A ÒcH I CD I Ò S,T > Ä >CE(ÒB TS H I WM£Ò S H λA ÐÊ J(Ë GCHPMXW> S G X ÐÊ J(Ë GC Ä BH.ÒÍ I W Ä > c : X → A ÐÊ J(Ë GC4>EoH IÖQIoSKTÏI W S E I B,C¨W"× Ä M"EoH I OG>ÓJV> TS Gm Ã È I >aÑbE.>"C:W I BCW× Ä M"E¦H I O"G > (c× c)∗PA Ð BØ PA >PW"E
Q GÙDPB ÄÇÄ >ÓGC4> Gm Ã È I >aÑbEF>"CW I B,CÌW"GH A× A QIoS λA Ôg(À´r<bx (N<?xÚ² Û `begZrh\
r ≤ gj*hÕZrh\eZr_?Z\
Xj*h
Z y vw`xs*j*lgZLleg*_ZLs*ZL_th* r 9ÜhÙ]4`dZT = Hom(X,Gm)
9 ÜhÙh*`\Z Ag−r lgZÇutv]Ùs*Zdmdu*prvwtd^tcprlegZrh*d^]*_aeh*ureg]tlgZrvwZrh\]c`blt_aedprds*Z^bZrh*_aZ
g − r Z\ Ag−r → Ag−rlgZ^du*prv´ttcprlegZrh|j*h*egbZr_dZrl 9
ÝÞßm0·+.-/+&*02143¶Å3659<; à h1 y vw`K\PefzÕ]_aeh*ureg]tlgZrvwZrhV\]4`ltK_egdpÙ_aees*efkpÙ]tK_
XZrda\lts*`bh*h*prZ
s¤n¥j*h*Z±z¦tKvwegllgZ(M,λ,GrW1 (M)
∼→ X,GrW3 (M)∼→ T)
`áM
Zrda\<j*h1 y v`\egz. λ j*h*Z[]c`blt_aedPt,\Peg`bh]*_egh*ure]tKlZws*Z
MZ\Õ`báltur`bv]4`dprZ´s*Z
GrW1 (λ)tbZru´lgZrdÕegd`bv`b_a]*egdvwZrd
GrW1 (M)∼→ X
Z\GrW1 (M t) = Hom(GrW3 (M),Gm)
∼→ XZrda\ l¦n¥egs*Zrh\Pef\Pp^s*Z
X9
Y[`bvvwZ lgZÂ_`j*]4Z s¤n¥tj:\P`vw`_]**egdvwZrdLs*ZÂltÓ]t_/\PegZÂtcprlegZrh*h*ZÂ]c`ltK_egdPprZGrW2 (M)
Zrda\Lkh*eoVlZb_a`bj*]cZs¤n¥tj:\`bv`b_a]**edvwZrd¾s¤nqj*h
1 y v`\egz(]*_egh*ure]tlgZrvwZrhV\£]c`ltK_edp_aees*efkpZrda\£prbtlgZrvwZrh\Úkh*e 9
!#""$ %&'($*)"+,-.0/ ,123 1/ -4) 5#
68797:2;=< M > <@?BACED@F9G<H1 I DJ:2;=KMLNHOFPQKR7?K0FC > <DJ<7#;OF: > CEPQKRHSHOG<UT1<7PQ<
g VW KXH:E7#;OPK0T1K0GK0YSHFCEPX Z
[]\_^E`a^b!cedfc^gihj klj k Znm oapUq=rs2t8u M p=v,wxs2yz|~pp,wv~p#|,~ vBvpUpM−→ B = Hom(X,Ag−r) −→ Spec(Z) ,
! M→ B pBvw w =vp, v #vy pOw =p E = Hom(Sym2(X),Gm
) : W H4G:E77<PQ:17H > C@? > CEHH<OG$K0H:1D@:1PQFAKRHDJ<OG<O?<;:1PQH=< W P4C W ?: W PH4G< > C@GSDJ:E7HQ;=PC2;KR:E7 Z |,t v,ws2w > H W ;G<D@:17#;P<P > CG< W KRD@<C2HH=<Pf;=K0:17 Z 687 W ;K > KRH< > C FP:EF:EH=K I;=K0:17¢¡£0¤a£¥¡£¡¦<7&PSFPQ<7CE7; > <HD@§DJ<H7:E;=C2;KR:E7H Z$¨ C©?A< M → B <~7ª:EKR<
MH W P c : X →
Ag−r
<; M → B <HQ; > <HFCE?<«G<«DJ:G W> <HUPQ< > C!;=KML¬G<H;PKMª K C > K0H=C2;KR:E7HUG< > CK0< ;<~7HK0:17HQ® IDJS;PK VW < (c × c)∗PAg−rZ¨ C? > C2H=H<]G$K0H:1D@:1PQFAK0HDJ<]G<?<;Q;=<]KR< ;<7HKR:E7&GSY7KM; W 7&S > SDJ<7;
G<Ext1
B
(Sym2(X),Gm
)= Ext1
B
(Z,Hom
(Sym2(X),Gm
)),G:17? W 7<O? > CEHH<OG$K0H:1D@:1PQFAKRHDJ<OG<
E I ;=:EPH< W PH4H W P B ¯VW K$<H;4?< >R> <OG< M Z¡£0¤a£ °£±³²´~µ1¶!·*¸º¹·N»¼½´¾a¿+À¹¶!·NÁ$º²ÄÃÆÅ0Ân¼º²Ä´¹~¶!ÂnµE¹Âº¶_²Äú²Ä¼n·*Dz#È£ÊÉ Ë8:177:E7H W 7<¦ª2C IPKRCE7;=<ÌG< > CÍFPQ:1F:1HK0;KR:E7&¡£M¤º£¥¡£¥¡UF: W P > <H 1 I DJ:2;=KMLNHÊD W 7KRHÊG$ W 7<8HQ;=P W ?; W PQ<8G<87KMª<~C W FC2P=C2A: IPK VW <ÍG<Í;e® F< D <7
p Z Ë:E77:17H I 7: W H w ∈ W <;87:E;:17Hmi ¯ ai <; ei > <HÌ<7#;KR<PH VW K >0W KºH:17;CEHH:?K0SH4F: W P 1 ≤ i ≤ s Z
[]\_^E`a^b!cedfc^gihj kljÏÎjÏh Znm oºpÐqBrs!t8u#pBv 1 Ñet w Òv8u*qÓus2y ptp,wlu yRs!,Rv|BvJ#pÍzp,=p g Ô¬#ps2 zw p r Ô$t v4aÕ pv,w¥,q,w =p¬p.*2p=s2Ìus2s_r ,p¬p.wÖup D p,wºaÕ *!s2,¥s2w w p, ppBv,wx|= 2s2y p,wxs2&qBrs2tÌu× us2s2tÐØ,w=py pBvÙÒs2tUy*y pBvÒ ,tÐ|BpBv]aÕ 92sz!=s2ttpq tUt wes2w¥ Ò
Y0
// Y1
// · · ·
// Ys
A0
// A1
// · · ·
// As
X0
X1oo
· · ·
oo Xs
oo
At0
At1oo
· · ·
oo Ats
oo
Y0//
Y1
// · · ·
// Ys
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// A1// · · · // As
1 "!
#%$'&#($*),+.-/+*0213+402$*+65784 τi &9#1:0 Gm ;=< +6#?>@$A#%784%+657 (Yi × Xi → Ai × Ati)∗ Pi 4%BC) Yi × Xi D 5"B)
0 ≤ i ≤ s E 5"F Pi #G4%$H1I0 < +6#=>@$J#%784,+65"7K&9#ML5"+.7N02)POM4%B) Ai × Ati Q $J#%1.1#G4'RB #'ST 1I0VU0NG#W0"-2027$X#%$Y1I0UZ0NG#'02),)%+6[,)G#'&2B\&"+*0/]2)02^W^_#M4/57$O`]@021#G4GaT 1#G4b5 <dc #%$e4&#_1I0'UZ0NG#f4%B D O%),+6#%B)P#f4/5"7$g&#P4h)POG4#P02B>i#%$1#G4kjV[PNGl#G4&#bNG#%$d$J#mU0NG#f4/5"7n$o&9#G4+.7 c #PN%$*+65"74qp X0 → Y0
#G4%$rB7s+I4/5"^_5) D lC+4%^t#qp D 5"B)o$J5"BC$ i ≤ s ar1#G4uNG5"^ D 5/4/OG4 Yi → Xi →Xi
4/5"7$_O`]@02B>wvx1I0x^yBC1z$*+ D 1z+*NP02$*+657 D 02) p #,$u1I#G4KNG5"75902B>w&#P4KNG5"^ D 5/4/OG4 Y0 → Yi#%$
Xi → X04/5"7$|&#yNP0")P&"+.702B>_)P#G4 D #GN,$*+ UP4 pei #%$ pmi aT 1#G45 <dc #%$e4m&#V1:0HUZ0NG#+.7UO,)%+6#,B)P#X4/5"7n$~&#G4|4/NGlO,^t04m0 < O%1z+*#%784&9#|]#%7n)G# g−r #%$1I#G4jV[PNGl#G4m&#NG#%$*$A#UZ09NG#r45"7$&#G4|+4/5P]9O%7+6#G4 p1#kNG5"^ D 1#?>C# At0 ← At1 ← · · · ← Ats
#G4,$1#V&2B0"1 &2B_NG5^ D 1#=>#A0 → A1 → · · · → As
p A0 → At0#%$ As → Ats
4/57$V&9#G4 D 5"1I02),+40"$*+65"784Pa At0 → A0#P4%$|BC7
+4/5"^t5") D lC+4,^_#qp D 5"BC)M$J5"B$ i ≤ s a~1I0hNG5"^ D 5/4/OG# Ai → Ati → Ai#G4,$H1I0u^yBC1z$*+ D 1z+6N02$*+65"7 D 0") p#%$Y1I#W&#`]")GO'&#g1 +I4/5P]O%7n+6# A0 → Ai
#G4%$ pai T D 5"BC)Y$J5"B$ i ≤ s a1I0m$*),+.-/+*021z+I402$*+65"7 τi +.7&"B+.$B7#Y$*)%+-/+*021z+4Z02$*+65"7W4%"^_O%$*),+*RB#&#Y1I0 < +6#?>$J#%784%+6574%2^tO%$*),+*RB # (Yi × Yi → Yi ×Xi → Ai × Ati)∗ Pi aT 1#G4$d)%+.-+*021z+40"$*+65"784 τi 4/5"7n$XNG5"^ D 02$*+ < 1#P4'02B>kj[GNGl#P4'&#$*)02784,+.$*+65"74 Yi → Yi+1a Xi ← Xi+1#%$ Ai → Ai+1
:[email protected]\qC"/q n qd qY¡ ¢m£x¤q¥@¦§Y = X
§¨_©dª«¥££n¥¨§ M0©:§¬,®¯°u¤±n§Z¦
1 ² °u¥¨³:´.¦_±8§µ@§Z£n¶?§ g · ¤n¶³:£n¬Z³I¤¯2©I§Z°u§Z£¨t¤¥©¯2¶³:¦¸Z¦t§¨¶³:µ@³:±n³:¹¸Z¦t¤¯¶Y · §¨_°'ºn£n³I¦t±ª«ºn£n§¦?¨P¶?ºn¬¨Pºn¶?§»¤¯2¶P¯2®n¥@¶?³I¼Cºn§±n§b¨J½C¤§ D §Z£
p ¾ ¢m£¿2¯K°u¥@£¨¶§¶¼Cºn§ M0
§Z¦?¨¯2©Iµ¸ZÀn¶³:¼Cºn§W§¨M§Z£\±n¥£n£§Z¶mºn£n§W±n§Z¦¬Z¶³:¤8¨P³I¥£\§%Á8¤n©:³I¬Z³Â¨P§ ¾ ¢m£\¤q§Zº8¨M©:§W±n¸Z¬Z¥@°u¤¥¦§Z¶§Z£©*ªÃºn£n³I¥£b±n³I¦.Ä¥³I£9¨P§M0 =
∐
w∈W
Mw0 ,
¥@Å Mw0
§Z¦?¨f©I§¬,®¯2°f¤Æ±n§Z¦1 ² °u¥¨³:´.¦f°'ºn£³:¦f±ªÃºn£n§b¦?¨¶ºn¬¨ºn¶§±n§b£n³Â¿9§¯º¤¯¶¯®n¥¶³:¼Cºn§±ª«³:£¿2¯ ²¶³I¯£9¨
w Ç ¬Z§h¬,®¯2°f¤x§Z¦?¨y¥º8¿9§Z¶=¨y§¨'´e§Z¶°u¸h±¯£n¦ M0 ¾ÉÈ ©Y¦Pº8Êu¨y±n§h°u¥@£¨¶§Z¶'¼Cºn§ Mw0
§Z¦?¨y¯©:µ@¸ ²Àn¶³:¼Cºn§g¤¥ºn¶¨P¥º8¨w ∈ W §¨k±n§g©I§o±n¸Z¬Z¶?³I¶?§o§%Á¤n©I³:¬Z³:¨§Z°f§Z£¨ ¾ ¢m£»£n¥2¨P§ Bw0 ©I§g¬,®¯2°f¤±n§o°f¥±nºn©I§Z¦
±n§Z¦±n³I¯µ¶P¯°u°f§Z¦¬Z¥°f°'º8¨P¯2¨P³Â´e¦¬Z¥°f°u§±¯£n¦©I¯y¤n¶¥¤¥¦³:¨³I¥£K:Ëdt¼Cºn³¦¥@£9¨V°'ºn£n³:¦V±ªÃºn£b³:¦¥ ²°f¥¶¤n®n³:¦°f§§Z£¨¶§©:§Z¦±n§ZºÁ±n³¯2µ@¶¯°u°f§Z¦Y0
// X0// Y0 et Y
× p // XId // Y .
¢m£'¶§Z°f¯¶?¼Cºn§Y¼Cºn§Y©:§Z¦É±n§Zµ¶¸Z¦±n§Z¦ÌÍZ¬,®n§Z¦Yi → Yi+1 · Xi+1 → Xi
§¨Ai → Ai+1
¦¥£¨É±n¸¨§Z¶°u³I£n¸Z¦¤¯¶
w ¾Î'ÏÐÑÐnÒ"ÓAÔ=ÓÐnÕÖ×ØÉ×JÙq×`Ö ¾Ú Û #NGln0"^ D 021Ã]O < ),+*RB# Bw0 #G4%$mB7x4/NPlO%^t0s#%7i]")G5B D #G4 D )G5 D )G#u4%B)B7#gB7+*5"7K&2+I4 c 5"+.7n$J#W&#-202),+6O%$JOP4W&#MÜ +6#`]#,1 Ag−r, 0 &5"7n$Y1#g$* D # D 02)P0/l5"),+*RB #'&O D #,7&(&9# w Ý O%^_5784%$*)02$*+657 Þ ß ¥³:¨ Aw0 ©I§k¬,®¯2°f¤_±n§m°u¥C±nºn©I§Z¦Y±n§Z¦r¬Z¥ºn¤n©I§Z¦Y±n§m±n³I¯µ¶P¯2°f°u§Z¦r¬Z¥°f°'º8¨G¯"¨P³Â´e¦¯½¯2£¨k©:§Z¦k°uàZ°f§Z¦V¤n¶?¥@¤n¶?³I¸¨¸Z¦k¼Cºn§g©:§g¬Z¥@ºn¤n©:§g´.¥@¶?°f¸g¤¯¶©¯y´d¯¬Z§g¦ºn¤q¸Z¶³:§Zºn¶§g§¨k©I¯u´e¯¬Z§g³I£8´.¸Z¶³:§Zºn¶§
!#""$ %& '($*)"+,-.0/,123 1/-4)5#
67869;:=<1>:=?@?BAC6AED:GF>H=FH=I90J9;H=KMLNPOQNRNSLTVUWDXAIJ@:=D0<1YZ>[9;\#7ACA]J@AI[J^7KAC7K9;H=K869;I*_H=9;K#JAC6A` :=>[9;Y]JYIa6Acb90A<1AD Ag−r, 0
6Ad<1AK>Ag − r A]Ja6AJWe FAdF:=>:=fH=>90\7A
a1 ≤ a2 ≤ · · · ≤ ag ≤ g − r .g Aihjf:=?@F Bw0 → Aw0
F:=>:=?@k]Jl>[AD0AI.mkhjfAIYi → Ai
A]JXi → Ati
\79 ` Y>9PnAK#Jo6AI.hH1K69PJl90H1KI6AphH1?@?c7Jq:!Jl9;H=Kr:=6Y\7:!JlA]IT=sXAphjf:2?BFcAI[JtY<1:=DuvD;:vD;90?B9PJlApF>[Hw_Ah]Jl9 ` Ap67c69;:=<=>l:2?B?BApI79 ` :2K#Jpx
yz |~|P.HomAw
0(X0, A0)
HomAw0(X0, A
t0)
88ppppppppppp
&&NNNNNNNNNNN
HomAw0(Y0, A0)
xxqqqqqqqqqqq
ffMMMMMMMMMMM
HomAw0(Y0, A
t0)
HomAw0(X1, A
t0) HomAw
0(Y0, A1)
HomAw0(X1, A1)
HomAw0(X1, A
t1)
OO
88ppppppppppp
&&NNNNNNNNNNNHomAw
0(Y1, A1)
OO
ffMMMMMMMMMMM
xxqqqqqqqqqqq
HomAw0(Y1, A
t1)
· · · · · ·
HomAw0(Xs, As)
HomAw0(Xs, A
ts)
88ppppppppppp
&&NNNNNNNNNNN
OO
HomAw0(Ys, As)
xxqqqqqqqqqqq
ffMMMMMMMMMMM
OO
HomAw0(Ys, A
ts)
1 !
"$#&%('*)+-,/.*01-2*+34 5#768*':9-2 Bw0 → Aw0 2*,;3(%2*<-%;.*,2*+3=#&>-5?2@<#&%(1-+A,'=9.*BC#D2*+AE%)1-<F2*,(<-%)<-%223G5H2I':9#&BC< Bw0 2*,;3(#&5?E.*>-%JH0K1-2LMN#A<-%)<O),JH3J?)+QP RTSNRVURWPXJHBC<-5HJ?01-2Y01-2Z5?2Z':9#&BC<[%2*5?#&3JH\ Mw
0 → Bw0<#%/#&BC83%2Z'*2*%]3=#&J?+-2*,
\W#&BCJH5?5H2*,^'*)BC<#_3/JH>-5?2*,$`-2a3/%;JHbJ?#5HJ?,/#_3/JH)c+-,$`-2Gm d >-J?2fe32*+-,J?)+-,Lcgh+i2*+j`-.*`-1-JT3^5?2k%;.*,/1-5T3=#_3^,1-JTb#&+3*L
lnmopNo-q_rts]ro-uwvOxy$xzFx|FL~ NZG==- Mw0 → Bw0
=f7K_/fKA_KQ&/_X= KF/& Ew
0
/fnE N -&::f/ ¡?i=- Mw
0
=:¢&£¤*¥:¦ M¨§£#+#&5H©K,/2I`-2*,`-2*1e`-2*%+-JH8*%2*,'*)c+-`-JT3/JH)c+-,`-2I5#i<-%)<O),JH3J?)+ªP RTSNRVURWPB7)c+«3/%;2¬01-2I5?2IEc%;)c1-<F2
`-2*,'#%#'38*%2*,`-2Ew
0
2*,;35#i5?JHBCJT3/2IJH+-`-1'3JHbc2I`-1`-J#&Ec%#B7BC2I,1-JHb&#&+3¬®¯ PRTSNRV°L²±I³
X0 ⊗X0
f
X1 ⊗X1
f
Xs ⊗Xs
f
Y0 ⊗X0 Y0 ⊗X1//oo Y1 ⊗X1 · · · //oo Ys ⊗Xs
Y0 ⊗ Y0
f
OO OO
Y1 ⊗ Y1
f
OO OO
Ys ⊗ Ys
f
OO OO
M2*,¢68*':9-2*,,)c+3J?+-`-1-JT3/2*,<#%¢'*2*5?5H2*,`-1`-J#&Ec%/#&BCB72¯ PRTSNRV°L£´³
Y0//
Y1//
· · · // Ys
X0
X1oo
· · ·oo Xs
oo
Y0// Y1
// · · · // Ys
23f //
,JHEc+-JHµ2A0K1-2¶5§²)c+Q#·'*)BC<F)c,.¶5?#·68*'=9-2¶'*)c%;%2*,<F)c+-`#&+3/2¶#bc2*'¶5§²J?,)BC)%<-9-JH,/B72¶`-2,;©B7.3/%JH2 ¯¸ a¹-³a0K1-J2*+«b«)J?2
a⊗ b ,1-% b⊗ a L
PRHSNRVºOR»a¼¾½À¿ÁÂÃFRaÄ ÅÆ#&+-,j'2<#%#Ec%#<-9-24+-)c1-,jB7)c+3%)+-,j0K1-2 Bw0 → Aw0 2*,3j1-+[,':9-.*BC##>F.*5HJ?2*+¶2301-2
Ew0
2*,;31-+A3/)%2LÈÇk2*,h%.*,1-5H3/#&3,h,/2*%;)c+«31É3J?5?JH,.*,`#&+-,h5?2n'*)c%;)c5H5#&J?%;27ÊÈRWPRVºRË4O5?1-J dBCÌ*B72¢+.*'*2*,,/#&J?%;2<F)c1-%¨`-.*B7)c+3%2*%^5#Æ<-%;)c<F)c,JT3/J?)+ÊÈRWPRVÍRWPLKgh+C,2¢`-)c+-+-2`-2,À,;©Bn>O)5?2*,εk<F)1-%
1 ≤ k ≤ r23a5W§£)++-)&3/2
[k, k′]5§²2*+-,2*Bj>-5H2G`-2*,a2*+«3/JH2*%,¡'*)cB7<-%JH,a2*+«3/%;2
k23k′LÉÎK)J?2*+«3
ei4mi
23ai5?2*,h2*+3J?2*%;,Æ#&,,/)'*J?.*,Ï
w ∈ W <O)1-%0 ≤ i ≤ s
Lgh+AµÉeÉ2j`-2*,hJH,/)BC)%<-9-JH,BC2*,Yi
∼→ ⊕rk=1Z · εk23Xi
∼→ ⊕rk=1Z · εk<F)c1-%
0 ≤ i ≤ s4É3/2*5H,0K1-2
Ð 5H2IBC)%<-9-J?,B72Y0 → Yi
2*+b«)J?2εk
,1-%p εk
,Jk ∈ [1, ei]
423,/1-%εk
,J?+-)+È4Ð 5H2IBC)%<-9-J?,B72
Xs → Xi
2*+b«)J?2εk
,1-%p εk
,Jk ∈ [r −ms + 1, r −mi]
4-23G,1-%εk
,/JH+-)c+ÈL
! "$##%!&'()%+*#,-./10 -2 34!10 .5* 6$
78:9;8=<8>7?A@CBEDGF$H Bw0 IKJ LNMPOQOSRTUM2TV5WX3Y5Z[OSM2T$\YRY^]_R`:]$_RVa`1ROQOSRV5WY[`1b:OSb:TX3b:YcRVdIeCf2ghgif 78 9;8j<8k78k7 IKJ lm"n H-op B q o4DErtsu-vQBwBxyu-z n H-o! H1
H2
H
H-pH ′ | q B"p~yHaD m q D-~3 m q HyD o n DH-p z:B3pDF$H B pjH-zD | q H (H1 +H2) ⊂ (H ∩H ′)
?BQz n v n pjH mj Hyr-p n+ HF q F n BE3B3vGvAHB/H1
//
B/H ′
B/H B/H2oo
OO
HyD-p^Hc2pjHoD nm ohF$H(H ∩H ′) / (H1 +H2) B3 B / (H1 ∩H2)
? u-v m oD-p~B3p nm o? LNR\c\tR`1b:OSb1\RWYMR\b12R P RVc\a[2XP`1R
(b1, b2) ∈ B ×B | b1 − b2 ∈ H ∩H ′ / (H1 ×H2) . XRa]$_bRT$2M2b1R (b1, b2) V_Y b1− b2 Z[Tb\_TRaV_Y+cR\tb1M2TwZR P V_Y (H ∩H ′)/ (H1 +H2) ZM2T\a`1RTM¡XP_¢RVc\
(b1, b2) ∈ B ×B | b1 − b2 ∈ H1 +H2 / (H1 ×H2) .£ T¢M2T`1_\aRT¢YcROwX3Y]$_XPT$\5]_R`¥¤¦b1OQOSRdYcVb:MPT¢Zb X32MPTXP`1RZR B ZXPTV B ×B b1TZ_b1\5_T¢b:VMPOQMPY=§Wb:VOSRZR B / (H1 ∩H2) Vt_Y^RTME¡$X3_¨IeCf2ghgif 78 9;8j<8k78¥© IKJ ¨Bwz n v n pjH mj Hyr-p n+ H P ′
i
F q F n BE3B3vGvAHHom(Yi, Ai) //
Hom(Xi, Ai)
Hom(Yi, Ati) Hom(Xi, A
ti)
OO
oo
HyD-p¨tH tuyDEHopkBxz:H B3 q oSDErtsu-vwBªBxyu-z n H-o! | q n HyD-p«u>B3zB q | q m p n H-op«FH Hom(Yi, Ai) B3/z HD m q D-3 m q H o n Hp z1B3p¬ m vAuSFHyDSu-z u-vwH-opD φ F$HHom(Yi, Ai)
pjH-zD | q H φ(εk) ∈ Ker(Ai → Ati)D n
k ∈ [ei + 1, r −mi]H-pφ(εk) ∈ Ai[p]
D n o m o? u-v m oD-p~B3p nm o? £ T¢WM2VR B = Hom(Yi, Ai) R\
H1 = φ ∈ B | φ(pYi) = 0 ,H2 = φ ∈ B | φ(Xi) ⊂ Ker(Ai → Ati) ,H = φ ∈ B | φ(pYi) ⊂ Ker(Ai → Ati) ,H ′ = φ ∈ B | φ(Xi) ⊂ Ai[p] .£ TX3WW`:b:]$_R`:R`:ROQOSRw®1¯®°®~®¥AW_b:Va`~¤±M2T¢M2²VRYc2R]_R H ∩H ′ = H1 +H2 R\³]$_R H1 ∩H2
RVc\`1RQV[OQX4RTh2YcM2_WRVXPTTM2T[ ]_b´RVc\²b1RT,VµYTb¥I £ T¶MPb1\]$_%¤¦b1`´RVc\W` X"\PYt·3Rw¸VtXZRVYb1W\tb1M2T¢R-¹W`1b:b\tRPI
1 !
"$#%'&)(*#%+(0 ≤ i ≤ s , #.-/-'#(0
Pi 1231547684 (*0:9'&#;0=<( 47> 0:?'%@? 452A & 2B686 0C#D+(*0=-%FE39 2 &G( 4 &?'0IHGJKMLNKPOQSRUTV0=-:-'0U<=#-'W*0=& >B2 -(XY%'0 1 0=WZ#DY;[0(*W?'#.-\( 1^]_4 -'? 4 <=0U0=W[( ≤ iQ
`ba.cdceagfihjkhml hnfohqpQ$r sNtvuIwyxz|Y~uItviBYxttHom(Yi, A0)
\y+~Pi
t~~xt =wywy.Bt~wyxuIN ut~
φ ∈ Hom(Yi, A0) x VB' φ(εk) ∈ Ker(A0 → Ati) pour k ∈[ei + 1, r −mi] . z'BxqYqtx* Pi t~b~|u8v\qtiuIw+~xBqwi@ ¡¢-£9'&[#Y<=¤=?'0¥9 2 &¦&[§<=%'&[&0=-'<=0¨W%'&
iQ3©0¥< 2 W i = 0
&[§=W*% 1 (*0¨?'%1 0 6)6 0FJ KMLNKªiK^JKP«QU¬Y%'9'9o#W*#-'WXY%'0 1 0=W 2 WW0=&[( 4 #-'W<=#.-'<=0=&- 2 -( Pi−1
W# 4 0=-( > §=& 4M® §0=W0(?'§¯6 #-(&#-'WZ<=0 171 0=W<=#-'<=0=&- 2 -( Pi Q©N0 A &#%'9o0 Pi 0=W[( 12)17456)4 (0U9'&#;[0=<( 4M> 0U?'0
Pi−1//
Hom(Yi−1, Ai)
Hom(Xi, Ati−1) P ′
ioo
OO
° 1 0=W[(XY%'#( 4 0=-(?'0 12±1547684 (*0U9'&[#;0=<( 4M> 0U?'%:? 452A & 2B686 0Hom(Yi−1, A0) //
Hom(Yi−1, Ai)
Hom(Xi, Ati−1) Hom(Yi, Ai)oo
OO
9 2 &)%'-/W*#%'WG¯ A &[#.%'9o0:?'#.-( 12 9'&#;0=<( 4 #- Qi
W%'&Hom(Yi−1, A0)
0=W[()²#.& 6 §=0:?'0 6 #&9'³ 4 W 6 0=W> §=& 47® 2 -( φ(εk) ∈ Ker(A0 → Ati−1)
9o#%'&k ∈ [ei−1 +1, r−mi−1]
QY"$#.W#-'WB = Hom(Yi, A0)
0(H1 = φ ∈ B | φ(Yi−1) = 0 ,H2 = φ ∈ B | φ(Yi) ⊂ Ker(A0 → Ai) ,H = φ ∈ B | φ(Yi−1) ⊂ Ker(A0 → Ai) ,H ′ = φ ∈ B | φ(Xi) ⊂ Ker(A0 → Ati−1) .¬ 4 Ri
?'§=W 45A -'0 1]S45682A 0 4 - > 0=&W0´?'0 Qi? 2 -'W B , #- > # 4 (µXY%'0 H ∩H ′ / (H1 +H2 + Ri) = 0
Qo©01 0 6)6 0J KMLNKªiK^JKJ 6 #.-(&0 21 #.&[W 12 W%'&;[0=<( 47>Y4 (§¶?'% 6 #.&[9'³ 4 W 6 0·- 2 (*%'&[0 1 ?'0 B > 0=&W Pi Q© ]_2 WW0=&[( 4 #-<=#.-'<=0=&[- 2 -\(ZW#.-:-'#¹¸ 2 %:&[§=W*% 1 (*0U?'%:< 2B1 <=% 1 ?'0 H1 ∩H2 , <=0UXY% 4 <=#-'< 1 %+( 12 &§=<=%'&[&0=-'<=0Q
º¼»'½»'¾n¾^¿'Àn½ agfihjkhml hnfih5jQ$r sNt¼~¹|'u Bw0 t~b~¹*|u\Áqt~x Aw0 fih5jkhGl hÂÃIÄb¹~UtEw
0
QÅr Ƶ#(#.-'WSw0 1 0 A &[#.%'9o0µ?'0=WÇ< 2 & 2 <(¤=&0=WÇ?'0 Ew
0
QY° 1 0=W[( 1547684 (*0 4 -'?'%'<=( 47> 0?'%:? 42BA & 26)6 0IHmJK7LNKOÃQSRUTQ`ba.cdceagfihjkhml h^Âhnf+Q$r sPu±mtUiB t S ′
i
BB¹ÈBxBuÉuItXi ⊗Xi
f //// Yi ⊗Xi Yi ⊗ Yifoooo
t~VZ Ê uIwyt^xt=$st*~8uIt~ εk ⊗ εl t εl ⊗ εk ∈ Yi ⊗Xi
wyËuÌuItÍu¹È\t)\B~S ′izwyYx
(k, l) ∈ [ei + 1, r −mi]2
! "$##%!&'()%+*#,-./10 -2 34!10 .5* 6$
798-:<;>=?-@BAC3@BDE;>=FHG IKJMLONQPRST1UVLWJYi ⊗ Yi → Yi ⊗Xi
N2XYJZ$[R\QP^]T1_`PaJZbJMJZ$cNQTdJεk ⊗ εlU[P
εk ⊗ εl − εl ⊗ εk si (k, l) ∈ [ei + 1, r −mi]2 ,
p εk ⊗ εl − εl ⊗ εk si (k, l) ∈ ([1, ei] ∪ [r −mi + 1, r])× [ei + 1, r −mi] ,
εk ⊗ εl − p εl ⊗ εk si (k, l) ∈ [ei + 1, r −mi]× ([1, ei] ∪ [r −mi + 1, r]) ,
p εk ⊗ εl − p εl ⊗ εk si (k, l) ∈ ([1, ei] ∪ [r −mi + 1, r])× ([1, ei] ∪ [r −mi + 1, r]) .
e9fg\3[YVPaJR\QPhYijdJLWN2PaRSTdULWJXi ⊗Xi → Yi ⊗Xi
JZcN2T1Jεk ⊗ εl
U[Pεk ⊗ εl − εl ⊗ εk
UTkJkYl\QRR\3PaYTdJZZJZYml
([1, ei] ∪ [r −mi + 1, r]) no Jjd\WU[pOY5RNQ[P5bNQZbjd[PaJ nqrNQ[PsYVNQ[Y
0 ≤ i ≤ sitNQZuZN3YVJ
Sijd\vjdT1LOTwYVJTdZ][bkYT1c2J][u]Td\QxQPV\3LOLWJyNQXYVJZ$[zl,R\QPhYVTdP
]J<h|w~ n JZZJbNQZUVJPhc3\QZ$Y5[Jj1JU^NQXaJkYVU5]N2ZY^jfT1Z]T1bJJUaY ≤ i nH2, K nr Z :<;3! Si ?-@DkA M @ ? 8- 8-: =@¡? εk⊗ εl @ εl⊗ εk ∈ Yi⊗Xi;"=@H:<¢-: D+:OC¤£ C3=? Si ¥ ;"A k @ l C"=? [ei + 1, r −mi]
F
798-:<;>=?-@BAC3@BDE;>=F¦G §¨Z©RPaNbª]J«R\3P¬Pa`b[PaPJZbJU[Pi n IKJ«b\3U
i = 0Pa`UV[jwYVJ][
jdJLWLOJ®|w~+¯ °B| n± [RRNQUN2ZUyj1JU'\QUUJPaYTdNQZUc`PaT1²`JUyRN2[P Si−1 n IKJ Z ³ LON$][jdJ Si JUYyj \jdT1LOTwYVJT1Z][bkYVTwcJ][]Td\QxQPV\3LOLWJ
Si−1 ←− Yi−1 ⊗Xi −→ S ′i .
I\´ªbµSJYi−1⊗Xi → Yi−1⊗Xi−1
JZcNQTdJεk⊗ εl
U[Pp εk⊗ εl
UVTl ∈ [r−mi+1, r−mi−1]
iJ-YU[P
εk⊗εlUT1ZN2Z n I\¨´ªbµSJ Yi−1⊗Xi → Yi⊗Xi
JZ$cNQTdJεk⊗εl
U[Ppεk⊗εl
UTk ∈ [ei−1 +1, ei]
iJkYmUV[P
εk ⊗ εlUVT1ZN2Z n §¨ZJZ]`][T1Ym[J Si JUaY¨j1TdXPaJQiJkYm[J εk ⊗ εl JkY εl ⊗ εk ∈ Yi ⊗Xi
N2Z$YLO¶LWJTdLO\QxQJ]\QZU
SiRNQ[P
kJkYl]\QZU
[ei + 1, r −mi]ibJ[TK\Qb·Sªkc2Jj \¸P`b[PaPJZbJ n
¹º»>º¼½¼¾¿½» , BÀ nr Z :s;3 Sw0 ?-@tDB-A Á @t £3AV;" ¥ Ew0 ?µ@t!=y@;"A F
|1~ Â4ÃOÄÆÅrÇÈÊÉrÄÌËrÈÍÎ$ÏÄHtÐ IKJÑLONQPRST1ULOJÑ]%fN2[XjdT][ÒZTwcJ\Q[ Mw0 →M
UVJ¨Ó¡\QbkYN2PaTdUJÑJZ[ZMLONQPRST1UVLWJ Bw0 → B JkYÆJZM[Z9LWN2PaRSTdULWJtJZYVPaJHYN2PaUJ[PUÆ[T$JUaYr`$[T1c3\QPaT \3Z$YrUNQ[UrjfT1UN2xQ`ZTdJEw
0 → E n IKJU´ªb·SJU Mw0 → M
JkY M0 → MUN2ZY RPNQRPJU nr± NQT1Y M jdJ
1 ³ LWNQYT1ÓÔ[ZTwcJPaUJjU[P M nÕ N3YVNQZU M [p]j1J¨x2PaN2[RJm²ZTJkY/Rjd\3Y/U[P M \QUUNbT1`l
M n IfJUR\QbJ¨]J¨LON][j1JU/PaJj \"YVTwÓM0 →M
R\QP\QLOªkYPJj1JU5]PV\3RJ\3[!Ö4T1UVN3YVPaN2RJU5]JM [p]
$[TKUN2ZY^]JY½×!RJ D n
1 "!
#$&%'$($*),+-/.0+132"4657498;:<4=?>@+AB465749C9D5@E78F4G=H+JIK4AL=2-/E@A2"E7-4M>@+N-+ O'DK-/.FPQE74 $SR TVUXWZYZ[w ∈ W [\]NU_^N`bacdeagf*fh[,i?jBfhfk]Z\eag\`&l@^N[
Z m f*j^N]Nnb[?op #$<%'$q(rts,u
Y0//
Y1//
· · · // Ys
X0
X1oo
· · ·oo Xs
oo
Y0// Y1
// · · · // Ys
v ]N`wJx?dy`&W[,n&[?ozi?jUN^N`&\`bjUNo^N[,nahNdyjBKjo`&\e`&jBU9#$<%'$q|$# rKTVU~UNj\[
MwX•,Y•, 0nb[ViafhH^N[Vf*jG^N]Nn&[?o^N[?o
1 m f*jg\e`<lFo^N[VcB[?UNdy[ g Nd`&UNi?`bagnb[?fh[?U\Kjnagd`&oxo f]NUN`bo^Q]NUN[Voy\d]Ni m\e]Ndy[^N[UN`<wB["ag]agdeagNjBdy` v ]N[^N[\;K[ D [\^Qt`bUwgagd`baU\w[?Up [\fk]NUN`&o^Q]NUk`bojf*jdNN`&oefh[^N]^N`agcBdafhf*[ p #$<%7$ru p jBZ\[?UG]cdegi?[nbaNdjBKjo`&\`bjUM#$&%'$|$#uo]Ndnb[^N`bacdeagf*fh[ p # $<%'$(Qrts,ur
j]Z\z`&oejf*jdNN`&of*[,[?U\d[,n&[?o^N`agcBdaf*fh[?o
Y0// X0
// Y0 et Y× p // X
Id // Y ,`bUN^N]N`<\z]NUN[,`&f*fh[?do`&jBU~j]ZwJ[?dy\[,[\l0[?dfhx?[
Mw0, X•,Y•
−→Mw0 . [?oiafhNo^N[*f*j^N]Nnb[?oa"wB[?i
X•[\
Y•WZYZx?oo[?dyw`&djU\n&jBdyo^N[*nba~^N[?oi?d`&Z\e`&jBU9Ndx?i?`&o[^N[*nba
oy\edag\`&Wia/\e`&jBU~^N[?oi?jBfhai\`&Wia/\e`&jBUNo p \eNx?jBdy?f*[$$03$#Zur
! "$#%&')(+*+ -,./-0!/-12&%-3(+')4%/-&%*5(6') !787) !12 !9( 3(;: 1<'8=!*5 (+/>*5'8?@0 78 !A$#%-1B&: 12/C:078 !D:
1 E 12/(6'GFH!IKJL/>0D 3MN !3(+0/>D0 O!/-5(6*503(6')/>P: RQS01TFH/>*5:P+0*;! !*5(6-')% 3D!/>1B&78U E(6')/>V: !V$#%12&V/>W.(+ !@0!,9&0')X&&*+/ZY.')12/-V78 ;*+U!+07G([\(]: ;! 3(+(+ ;!/>5(+*+03(+'8/-< !2&*5U!+ !*5^-9(78 "6/-0 E`_ *5/>0&a b: p E (6/-*++')/>cI%dV.4%c,%/-0;:U34%'8++/>;7) !DZ*5(+ !;78/@Z78 !D7 _ U!W*+')?@0 !!I
efhg%fjilkNmonqpKr-sutwvxr>p.sutHkayzTsZkK|k~[qp%zBnqp%|usutHz(6/-0.(78 *+ !5(6 : ! 3(5(6 (6#=!+ -,/>4.Y. 0
Z E 12/C:078 7)'8W*5 X : *6 _ g 3(7h/>
/-(+ B(X) = Hom(Sym2(X),Z)
7) Z E 1B/@:07) : !-&&7)'8Z\(6')/>W')78')UZ-')*+ !51BU3(6*5'8?@0 ! !C(+'8=!*5 !j+0*
X, 3(
C(X) ⊂ B(X) ⊗ R78 !- : !&&78')Z(+'8/-jW')78')UZ-')*+ !51BU!(+*+')?@0 !
+ !12' E :U34%'8 !&a/-6'G(6'G^9 !*+-:')Z-7*+(+'8/- !7hIb+ :/- 0 :U!!/-12&a/>+'G(6')/>&a/-7)>U!:*678 GL(X) E :12')++'8W78 Σ
: C(X)
?@0' !5(x7)'8++ &N/-0*785(6*503(60*5 A !9(6')=Z*5 B(X) &%*6 _ *+-&# `RI: l iT ¡H¢ I.£@/-')(
<, >:U!+' _ D7h-!!/>0&7) !12 !9(X: L:0%78'G(6UD !C(+*+ B(X⊗R)
3(Sym2(X⊗R)
IS&a/>+
σ∨ = x ∈ Sym2(X ⊗ R) tq < σ, x >≥ 0 σ⊥ = x ∈ Sym2(X ⊗ R) tq < σ, x >= 0
&a/>0*D(+/>0.(σ ∈ Σ
Ic¤X +/>9(*5 !+&N !3(+')^> !12 !9(0¥12/-/-¦§: l 3(0 _ *5/>0&a &a/>0*L7¨©::')(+'8/-cIª/-0*(6/-0.(σ ∈ Σ
,/>«:U!+' _ &%-* Xσ
78 &7)0j&N 3(+')(j?@0/-(+'8 !9(: X
(6 !7?@0 (6/-0.(6 !j7) !FH/>*512 !W'87)'8UZ'8*5 !
b ∈ σ + bF¨3(6/-*+')6 !9(L&%* Xσ ⊗Xσ
,% 3(L7¨/-P/-(+ r+/>P*+- _ Iaª-*; ¬Y. !12&7) O+' σ !5(:%-;7¨')C(6U!*5'8 !0*:
C(X),%/-P
Xσ = X 3(D+'
σ = 0 ,/>P Xσ = 0IaS&N/-+
Sσ = Sym2(Xσ) et Eσ = Spec(Z[Sσ ∩ σ∨]) .
¤ ": !*+') !*+$#U!1<T !5(;0S&7)/> _ !12 !C((+/>*5'8?C0 O2 -,10'®:c0 b5(+*6\(6'G4%Z(+'8/-S'8: ¬YU! b&%*78 !F¨! !R: σ ¯ 7°PF¨-! τ !/>*5*+ !+&a/>:785(6*+(+ Eτ
σ = Spec(Z[Sσ ∩ τ⊥])?@0'X !+(R0(6/>*5
?@0/(6') !C(l: EσI±ªx*O ¬Y. !1B&78 -,®7h/>*5' _ '8 0 !/-*+*5 Z+&a/-:7h0'8?C0 <5(+*6\(6 2/-0.^9 !*²(6 Eσ 3(O78 !>
σP7h0'8?C0 5(+*6\(6 <FH !*512U! -I³±¥&%-*+-1BU3(6*5'86\(6')/>: ´7°P5(6*+(+')4%Z\(6')/> !5(R:U!!*5/>')66C(6 -,
!#"$&%('*)+'-,.0/1"324"3$1. τ "$&%546"87:9 "3," τ ′ ; 9<$&%1/=9+%=">9+%1%?9 @A">) τ ′ "$1%B.C6 ; 4$="3,96$ ; !D9,@A/1"6 "," ; 9E$&%=/19%1"F9%&%?9 @A"F) τ GH 6I,AKJ6.L%M46N7:9.0$1 "O94I PQ6$&%=/&4 K%1.CR ; " Xσ$14/ Eσ S 24."$&%M46I24P%1.C"6T%M,4N7:9.0$1 "O94
PQ6$&%=96% XEσ S "6U,PQ6696%V$=9E/1"$&%1/1.C K%1.CP6W94YXZ$1%1/=9+%="$\[Xσ|Eτ
σ= Xτ .
] 9^/&"$1%1/1.0 K%=.0PQ6_," Xσ) ; 9^$&%=/19%1"3P4`T"/(%="8"$&%B64 ;0; "8"K%B$=9a/1"$&%=/&.C K%1.CP6b) ; 9a$1%1/=9+%=" Eτ
σ"$&%cAdT9 ; "
) Xσ$1. τ "$&%e.C6 ; 4$f,96$ ; !g.C6%1A/1.0"4/e," C(Xσ) Gh 9/e P6$&%=/&4 K%=.0PQ6 S Sym2(Xσ)
$1"5i ; PQ6d"5,96$; !#966"O943,"$j7kP6 K%=.0PQ6$j/1Ad4 ; .0l/1"$j$14/ Eσ S ,P6 m,96$ ; "n PQ/&i$o,"$p7k/=9 K%=.0PQ6$j/=9+%=.0PQ66" ;0; "$j," Eσ Gq 6> PQ6$1.0,AO/196T% ; "e,.L`Y.0$="4/r9$=$1P .0Af)B46"m7s/=9 K%=.0PQ6E/19%1.CP66" ;C; " S PQ6EPR%=.0"6%r46"m7kP/1tu"eR. ; .06AO9.0/1"$&vYtuAK%=/1.024" Bσ : Xσ ⊗Xσ → DivEσ
)>`9 ; "4/&$w,96$ ; "7k9.0$1 "O94U,"$w,.0`.C$1"4/1$w," Eσ Gx3yz|z~+*(1z** Gj VO^ Bσ *+ EbYO+<3Kss1s= ^K
Bσ : Xσ ⊗Xσ −→ DivEσ
K +u+Y* ?¡+¢=+ x £ Xσ <£s¤ Y B(x, x) +sk¥5= <¦ Q +\-§QU¡b£F¨YC:¡+¨£T x ©ª < 1¡©r« ¬ A/1.LJPQ6$ ; 987:9 K%1PQ/&.C$=9+%=.0PQ6$=4/m @924"B$1%1/=9+%=" Eτ
σ S PQ τ "$&%e46"B7:9 "5," σ G® P%=P6$ Xτ ; "6Pv94," ; 9E$14/¯1" K%=.0PQ6 X → Xτ G° ; $14±u%w,"tuPQ6T%=/&"/f24" ; 937kP6 %1.CP6U9$=$1P .CA")>%=PQ4%wA ; Atu"6% x⊗ y ∈ Xτ ⊗Xσ"$1%w$=96$w²A/&Pa6.i³ ; "\$14/ Eτ
σ G´h 9/w P6$1%1/14 K%1.CP6 S P6W9H0(Eτ
σ ,OEτσ) = Z[ s ∈ Xσ ⊗Xσ tq b(s) = 0 ∀b ∈ τ ] .
H /E$=. b ∈ τ P6M9 b(x ⊗ y) = 0 S ,P6 x ⊗ y .06,4.0%u46"Z7kP6 K%=.0PQ6µ/&AdQ4 ; .Cl/&"$14/ Eτσ G·¶ !g"$&%AdT9 ; "tu"6% ; "\ O9$V," ; 9E7sPQ6 K%=.0PQ6U.06`Q"/1$1"\24.¸"$1%V9$1$1P .0A"F) x⊗ (−y) S ,¸!#P ; "\/&A$=4 ; %?9+% G¹ P6%=/&PQ6$ ; 9b$1" P6,"9$=$1"/(%=.0PQ6 G ] "Z ³6",4´9 ; ," C(Xσ)
"$&% C(X∨σ ) ,¸!gPQº46"Z.06 ; 4$1.0PQ6
σ∨ ⊃ C(X∨σ ) G» . x "$&%·46"w$1" K%1.CPQ6 ; P O9 ; "w," Xσ S PQ6u`QPQ.L% x⊗x Ptatu"w46aA ; Atu"6%¼," C(X∨
σ ) G] 9½7kP6 K%=.0PQ6¾9$1$1P .CA"W) x ⊗ x "$1%^/&AdQ4 ; .Cl/&"W$14/ Eσ"K%^,¸!D9i/1l$ ; "$^i/&.C6 .0i"$adQA6A/194YXµ$=4/
; "$`9/&.CAK%1A$\%1PQ/&.C24"$ S ; " ; .C"4¿,"u$1"$F²A/&PQ$\"$&%F/&A46.CP6¨,"$\$&%=/19%1"$F9$1$1P .CA"$F94YXÀ7:9 "$ τ $14/; "$124" ;C; "O$ x⊗x i/1"6,3,"$p`9 ; "4/1$j$&%1/1.0 K%="tu"6%jiP$1.0%1.0`Q"$j PQtuta"·7sPQ/&ta" ; .C6AO9.C/&"n$14/ B(Xσ)⊗R G
H 6Á9ii" ;0; " Mσ; "U @9tuiM,"UtaP,4 ; "$^,"$ 1 '-tuP%1.07s$^,"Ud"6/1" g i/&.C6 .0i9 ; "ta"6%aiP ; 9/&.L'
$1A$E"K%u/&.Cd.C,.LJA$Ei9/ Xσ Go° ; 6"Z,Ai"6,Â24"Z," Xσ"K%Ei9$E," σ G H 6ÂiP$1"ZAdT9 ; "tu"6% Bσ =
Hom(Xσ,Ag−r) S PQ r "$&% ; "F/196da," Xσ G´Ã 96$ ; "F @9i.0%=/&"Fi/&A A,"6% S P6_9utaP6%=/&AF24" Mσ"$&%V46 Eσ '¡%1PQ/&$1"4/w$14/ Bσ G ® P%=P6$ Mσ; "\i/&P,4.0%w PQ6T%=/19 K%1A
Mσ =Mσ ×Eσ Eσ .] "E ?@9tui¿9 ; dQAR/&.C24" Mσ
"$&%F9+±^6"E$14/ Bσ "K%t>46.o,¸!#46">$1%1/=9+%=.LJ O9%1.CP6½.C6,"XA">i9/ ; "$7:9 "$c," σ G ® P%=P6$ Zτσ; 9a$&%=/19%1"39$=$1P .CA"3) ; 9a7k9 " τ "K% Zσ = Zσ
σ; !#46.024"8$&%=/19%1"F7k"/&taA" G ] "
! "$#%&'()#*"+"$#% ,.-/10$2!34650$7
Xσ
598;:!0</>=+?@A5CB2D:!2DEF0+2D?5Bσ
36GF98H56EI56E(8KJ Mσ×BσEσ
I5L@M0+E2DN?O5Eσ P 2!E(Q$0$?2!0+E(85C59835I5634656EI56E(8LJ Mσ RS ETE=$8H5&8=7U=+7?3 Xσ
:D5&/10$2!34650+7T46=+E3O8H?O7498H2DB:!5598Bσ : Xσ ⊗Xσ −→ DivMσ:)0</>=?O@M5CB2!:D2!EF0$2!?O5V=+B%8H56E(73K37? Mσ R%W 5CI2DQ*2D3567? Bσ(x⊗ x)
563O8K59X564982D/YI5C37ZZ=+?O8K:D5V:D2!567[I5E=EM0+EE7:!0$82!=+EMI5
x ∈ Xσ R\ 2 Xσ = X0+:D=?O3]=EM0
r = g ^ =+E_0 Mσ = Eσ598]:1G`=+E_0 Mσ = Eσ R
\ 2 σ = 0 ^ =Ea0 r = 0598 Mσ =Mσ = Ag R
b_c+dfegihjkcmlYnpon1l Rrq s =73tE5u46=E32DIF6?56?=+E3tI563YZ:D=Ev+56@M56Ew8H38H=+?2Dx7563tE=E<0$yME563tx75uI0+E3:zGF6E=E46FCI78|F6=?ON6@M5~z~)~) ^ Z'=+7?;=B%856E2!?7E5VI56346?O2!Z%82!=+Ev+:!=+B0+:D5CI7Qw=+2!32!E0$v5CI5C:1G`2DE%E2I56346=+@MZ0+4982D40.8H2D=E3 R( 0+E3:!046=E3O8H?O7498H2D=EMI53]46=@AZ0+498H240$82!=+E3598Z:!73Z?OF6462!3F6@A56E(8I0+E3:!563]Z?O=Z'=328H2D=E3;~z~)M598;~z~)~1E=73]7%8H2D:!2D356?=+E3:D5 /10+28x(75:D563]46E563]I5ΣE535?5646=+7%Q*?56Ew8
Z0+3 R~z~)~_9(%'trm>p'i> (~;¡ W 5C4|0$@MZ Mσ
EG¢0$I@M598KZ0$3KIG2!E P8H56?OZ?F98H0$82!=+EM@M=(I7:)0$2!?5 5¤£%Z:!2D462D85 R(¥ EM?59Q$0+E4¦|5 ^ :zG2!@A@M56?O32!=+EM=7%Q56?O85 (Mσ →Mσ)56E_0+I@M5¤8
7E5V56E[8¦0$E(8 x(75M§40$8F6v=+?2D5VB?OF65&J¨:)0/>?O=E(82!N6?O5©_I0+E3K:D5&356E3K372DQ$0+Ew8 R\ =2D56E(8 F 598GI567*£
4|0$@MZ337?Spec(Z)
598F → G
7E52!@A@M56?O3H2D=ET=+7%Qw56?ª8H5 R«m¬&®k¯±°ª¯²®³lYnponµ´npo Rrq W 0M40$8F6v=+?2D5<B?F65JM:)0A/>?O=E(82!N6?O5 (F → G)
563O8L:)0M40.8HF6v+=?O2!5B?F6556E¶v+?=+7Z=$·µI563_37?
Ouvx(72L0$33=462!5aJ¸8=7%8_=+BU598
(U → S):!0¸4i0.8HF6v+=?O2!5TI563_I2)0$v?0+@A@M563
2 P 46=+@M@¹7%8H0$82D/>3U //
F
S // G
S Eº0+ZZ'56:!:D5(U → S) »!¼½.¾¿ÀAÁÃÂ!Ä_ÅÆH½.¿Àz¾zÇ¤Æ¦È I5 (F → G)
:!040$8F6v=?O2!5aI563I2)0$v?0+@A@M563Z?F646F6I56Ew8H3tÉ=ET:!0AE=$8H5
(F → G)(U → S) Rb_c+dfegihjkcmlYnponµ´n>l Rrq W 0Ã40$8F6v=+?2D5
Ouv/>=?O@M57Eº32D8532C=+EÊ:!0f@¹7E28I5Ë:)08H=+Z'=:D=v+2!5
2!EI728H5<Z0$?Ì:!0A8=Z'=+:!=+v2!5 ŵ¼ÍÎ 37? U 598S ^ 598L:!0M40.8HF6v+=?O2!5B?OF65 (F → G)
563O8Ì7Ea4|0+@AZa37?46532D85 R =+EE=E3ÌJAZ?OF6356E(8Ì7E5I56346?2DZ%8H2D=ET5¤£*Z:!2D462D85I5<:!0A40.8HF6v+=?O2!5&B?OF65<JA:!0Ï/>?O=Ew8H2DN6?5
(Mσ →Mσ) .
\ =2D56E(8 (U → S)7EA=+BU598uI5
Ouv ^ 598 D 7EÏI2DQ*2D3567?u59X564982D/'2!?O?F6I74982!B:D5 ?F6I728uI5S R(S EAE=$8H5
pD : DivS → ZD
:!5C@A=?OZ|2!3@M5Cx720+33=462!5VJ8=7%8;I2DQ*2D3567?;3H0¹@¹7:D82!Z:D2!4628HFC:!5C:D=Ev¹I5D RS E?H0$ZZ'56:!:D5<x75<:1G`=+EË0M0+33=462!F7ET/10$2!34650+7a46=+E3O8H?O7498H2DB:!5
X598Ì7E5/>=+?@A5<B2!:D2!EF0$2!?O5<3OÐ*@MF98?2Dx75
B : X ⊗X → DivSJÏ8=7%8
1 P @M=$8H2/kI5¹ÑT7@A/=?OITZ?2!E462DZ0+:D56@M56Ew8ÌZ'=:!0+?O2!3F37? (U → S) R
"!$#%&#')(+*,(-#.0/&1+21435163798
(Mσ →Mσ)(U → S) :<;>=?:A@BDC)EFC)G6H:IH:<; 1 JLKMC)=6N OD;PH:RQSETKIOC)B-H;>ETB (U → S) H:U@:>VBD: g W9FB>NXVY>NZF[)?:>KM:V=AFC$?\[BN;]<;<WPB>N^@N6HN _I]<;`F[)B Xσ ;>EB U Wa=b:>?Z;dceEf:gFC)ETB=bC)E=b:h;:<Y>=6N4C)Vi?CY-[?: x Hj: X Wk?:HNmlN;:>ETB B(x, x) ;C)NX=":onk:DY>=6N O:>=IFC)ETBM=bC)E=pH)NmlN;:>EBq:onk:<Y=6N ONmBBD]DHEfY>=rN6se?:UBD]DHETNX= D H: S Wt?-u [FjF?vN6YD[=6N4C$VwseNm?vNmV]-[NmBD:;>xKh]>=6BN6ceEf: pD B : Xσ ⊗Xσ → Z ;C$Nm=NmVY?vE;:"Hj[V;y?-u [jH$z]>BD:>VY<:"H: σ | ]>KMC$V;>=6B-[=6N4C$V~ +^^U-y\yd-
S −→Mσ −→ Bσ-ya-hD+\^ G y-D-^<- S a\p-e--ph\¡-\d-\h Xσ e"pq,D^M\^h"\^^dj\\- g − r
7 +a^^q-i-^ej U → Mσ edd-^-d Xσ → G - U ^\Djq 1 LdD¢¤£q¢o-¥-^\\q\\-- (U → S)
7+I-¦Mj^§¨-^^-\D B ©\ª U → Mσ «¬eD
i S → Mσ
7~® ej-\^>¯TD-^id^-^-q U S 7° a-DaD\^y"±\-²r qy--a S = Spec(R) -h\R^-d^y\ Spec(K) ey\-D Mσ → Bσ y--T 7³ ´D^^DI¨< U → Eσ ©¨6«¬µeD´¨ S → Eσ
7 ³ jjDU\ H0(Eσ,O) = Z[Sym2(Xσ)]→ K j\ H0(Eσ,O) R 7f° h\D` S T\-gDD`^P\T¯¨d^-Ik- 1
7 +-¦dP`^IkD x⊗ y ∈ H0(Eσ,O) ⊂ Z[Sym2(Xσ)] e-gT\- D S fª pD B(x⊗ y) ≥ 0
7¶ «¬e\
H0(Eσ,O) = Z[ u⊗ v ∈ Xσ ⊗Xσ tq ∀b ∈ σ, b(u, v) ≥ 0 ]
e\¢mdy^\^\ pD B \^-U6«·y σ a¸T-- 7
¹hº»½¼!¾¿©ºÀ/&1L214351\Á798  S ^6 \< S →Mσ ^y^y^a¢od S D-- Zσ -et-\t- σ t\^hd\~Ã~ Σ ~6«·-\j¢o-y\^\\- pD B PDT\- D S 7ÄÅ6ÆÅÇ&ÅÈpÉÊË~Ì\ÍÏÎÏФÊeÑÒ)Í9ÓÑÍÏÒФËÏÐwÎ9ÔXÕÐÉÍ×Ö9ÉÒ$ÉfØ&ÙfÒ)ÔrÚ5Í&ÐÅÜÛ ³ S-\M>-D\ Σ ×D-Ý ¡^\--Ý´---e--ÝjD^- B(X)
7Â σ ∈ Σ
7 ³ ÞD Xσ \SD^ X --\Àß σ Ie r -ÞD 7 ³ ÞD Mσ, 0
\¨<i\-^¨ 1 L-^¢o¨ g I-j\\\-^- 9-^^^§ Xσ feg^g5«¬pDeDpU^ªD\pU+¸T D p 7 ³ ST¡\^--yj Mσ, 0 de<^yd-^-\qqf-\j,De¢m-
Mσ, 0 =∐
w∈Wσ
Mwσ, 0
! "$#%&'()#*"+"$#% ,,-!./021*34056+7 8!09:;$<=->04.(<?/0@+7;+:5'04A?/0&BC0ED*8
Wσ = WL \ WGSp2g/ WLD .
F ;:7G<=;+:<w ∈ Wσ H ;.I6J/34K47L->< /6+.A 8!6J57L;5';A-><-!;+.NMO>PQO)ROST:.T<=;+70
Ewσ, 0
0E< :.UKV6+WX5 Bwσ, 0<=048>AY9(:0 Mwσ, 0 → Bwσ, 0
A;-Z<Y:.U<;7LA04:7 A=;+:AEwσ, 0 [ \ 0 Z ] WX;/:8!0&8!->^70 Swσ, 0 = Hom(Ew
σ, 0,Gm)/3E_.-><Y:.I58!;+.@04WX04.(<Y<;7L-!9(:0`6$ab.0Ewσ, 0 → E
w
σ, 0 [ c .I/3E_.-Z<?8>0KV6$Wb5d6$8!@+34^7->9:0Mw
σ, 0 → Bwσ, 0K4;WXWb08>057;(/:-Z<YK;+.(<7=6$KE<=3Mw
σ, 0 =Mwσ, 0 ×E
wσ, 0 E
w
σ, 00E<?8>0KV6$Wb5d6$8!@+3^7L-!9(:0 Mσ, 0 → Spec(Z)K4;+WbWX08fe:.-!;+.I/-!Ag;+-!.(<0
Mσ, 0 =∐
w∈Wσ
Mw
σ, 0 .
\ 0KV6$Wb5 Mσ, 004AL<?Wh:.-i/e:.0AL<=76$<->_K6$<-!;+.I8!;(K6+8>04Wb04.(< jk047LWb340/0AL<=76$<04A
Zτσ, 0 =
∐
w∈Wσ
Zτ, wσ, 0
-!./021*3404A?56$7Y8>04A jl6+K404Aτ/0
σ [ c .I.;+<0 Zσ, 0 = Zσσ, 0
8lem:.->9:0AL<=76$<0&jk047WX340 [n 6$.AbK40U56$7=6$@76+5V0 H ;+.o/;..0U:.0U-!.(<04757L3E<p6q<=->;.o/0U8)6rK6$<34@;+7->0T_^7340Ist8!6rjk7L;.u<=->v470(Mσ, 0 →Mσ, 0)
6$.6+8>;@+:0wsxK4048>8!0h/0(Mσ →Mσ) [y ;-Z< (U → S)
:.z;^(g0E</0Ouv ['| ;+:%<
1 ] WX;+<->j&/0rI:WXjk;+7/M
A:7(U → S)
Wh:.-?/e:.0~AL<7:KE<:70d/0d.-Zu06$:56+76+V;7L-!9(:0H•/em->.($6+7L-)6$.(<
w ∈ W /3E_.-><w:.0jk6+WX-!8>8!0K4;WX56$<-!^8>0/0xjk;7LWb04Aw^-!8>-!.36$-!7L04AJ6+AA;K4-!3404AJ6+:*11 ]Wb;$<=-ZjkAY/0hI:WXjk;+7/
M/Hi H /;+.K:.I348!34WX04.(< B(M,H•)/0
Hom(Swσ, 0 ,DivS) [h+QqQ') [ (Mσ, 0 →Mσ, 0)(U → S) =2)$=qqf ( up 1 ¡¢xqf£ ¤ (¦¥z*¢w¡¤q ((2§= g 2* (U → S) ¨E*£©q§Q'£§ª2£«¬$)¢x2§Q!¬$£)©pp®¦2£>$£f $£ ¯Gp %¬$ Xσ 2°¢J*§£) Q± §2fª2l*pJ u²§£³+=¬qb¬$¬´q£f¨EJ (²fµ D 2§ p ®¶·2«J¨2Yq*`·q*©pª2f£¸§t)©ª=¬$! x ( Xσ )X $£³©£)2* B(M,H•)(x, x) ©q£¹kº»pª2f£ ¤b2¼q*h·q¹ $£³©£!©2*bkºpª2f£ ¤²£2== qª2f£f½E!w=2¡ $*£ D u S ®?!¬`¤q¢x½E£¾£§=¬q£pJ2µq¢x2f£f¨E pD B(M,H•) ©q£¹£§ª2¿(© u¬$§*À± ¬u ´2=2§ªp ( σ ÁSOfMO)ÂOUÃbÄÆÅÇlÈÉÆÄËÊÆÈÌ'Í(ÎĦOGÏ \ e-!.K48>:A-!;+.Ð/04A²AL<=7L:KE<=:7L04A²04.(<-!v47L04A
Sσ → Swσ, 0/;+..0À:.0
Ñ v4KV057L;57L0Ew
σ, 0 → Eσ0E<?8!6Jj;.KE<;7L-)6$8!-Z<=3/:I57;(/:-><YK4;.(<7=6$KE<=3/;+..0:.0 Ñ v4KV057;+570Mw
σ, 0 −→Mσ
!"#$&%(')+*-,/.!0*1")!2'"435%76! Mwσ, 0 →Mσ 89 %(:<;=.5>@?A"1:&;=%(:<"4')!"1:&%('$<!'2)"6B5>&6;=;C%ED76>@$&#GF:&"1%('
Mσ, 0//
Mσ, 0
Mσ
//Mσ).5'H$*1%(:JIK(DA%(:J3H%(>L$M"4D76*4%(::&.'$?>@.5?>@%(:%$N).5'$N*1%(:JIK(DA%(:NA.5>&"4O(.5'H$P6*1%(:N:&.5'$N)%(:!'"4.5':)"1:[email protected]"4'$M%(:S)%E?*4.5'B5%(;=%('$&:S$&.5>@.TU)6!WV 8XZY[]\^_5`aYcbedgfdghdgf 8ji 9 .;C;=%k*6l)#(D(.5;=?.:&"4$&"1.' Σ %(:@$m*4"1:&:&%k?.5!>n*16o:@$&>&!D$&!>&%k%('WF$M"4K(>&% B(X) p *4%EDA6;C? Mσ
%(:@$S*1"4:M:&%E:&!> Spec(Z) %$ Mσ →Mσ%(:@$<*4%ED(.5;=?*4#(;C%('H$P6"1>@%E)q,/!'
)"43"1:&%(!>Sr=D(>&."1:&%(;=%('$M:S'.>&;C6!WV 8s %E?>&.)!"t$Su0>@#Mσ, 0 ×Spec(Z) Spec (Z[1/p])
%(:@$v*4"1:&:&%w:&!> Spec(Z[1/p]) D76>2*1%(:+DPAx6;C?: Ag−r :M.'$+*1"4:&:M%(:+:&!> Spec(Z[1/p]) 8<y .5!$M%zR.5"4: pD(.5;=;C% Σ 'q,/%(:@$?6:*4"1:&:&%?.5!>|*4%(:)"t#(>&%('$&%(::@$M>@!D$M!>@%(:%('$&"1K(>@%(: Bwσ, 0 p *1%~DA6;C? Mw
σ, 0'q,%(:@$
?6:S*4"1:&:&%E:&!> Spec(Z[1/p]) %$Mw
σ, 0 →Mw
σ, 0'q,/%(:@$?6:*1%~D(.5;=?*1#(;C%('H$P6"1>@%)q,/!'N)"43W"4:&%(!>rD(>@.5"4:&%(;C%('$&:'.5>@;Z6!WV 8 %;C(;=% p *1%(::&"4'B5!*16>@"tF$M#(:S)%Mw
σ, 0 −→ Spec(Zp)'%E:&.5'H$<?6:S!'"4 !%(;C%('H$<D(%(*4*1%(:S)%(: Ag−r, 0 6735%(D r ≤ g 8N7CxR=g ']DA.5"4:&"4$!'%J?>&#(:&%('$M6$&"1.'#$P6*1%J:&!>Q&%(D$M"t3H% U → Bσ )!]DA6;=?6*4B5#(0>@"1 !% Bσ ?6>!'
:&DA#(;Z6=6C'%E'%E)#(?%(')6'$S !%E)!2>M6'BC)% Xσ 8 '2'.$M%Mσ = ( U ×Bσ
Mσ)U×BσZσ
*1%=D(.;C?*4#$M#=)!w:&DPA#(;Z626Z'% U ×Mσ*4%=*1.'B2)!:M.!:LFg:&DA#(;Z6NzR%(>&;=# U × Zσ 8j9 %=:&DA#(;C6)#(?%(')=)!=DA.5"V)% U p ;C(;=%:&".'=.5;C%G$D(%)%(>@'"1%(>)%(:'.$P6$M"4.5':?.5!> :M"4;C?*4"4u%(> 85¡ *D(.5'$&"1%('H$*1%:M.!:LFg:MDPA#(;Z6.5!35%(>@$
Mσ =Mσ ×MσMσ ,*1%:&.!:LFg:MDPA#(;Z6ZzR%(>&;=#)%D(.;C?*4#$M"4.5' U × Zσ !%*¢,.5'£'.$&%%('D(.5>@% Zσ %$~*4%)"43"1:&%(!>~rZD(>&."tF:&%(;C%('$&:S'.5>@;Z6!WV
∂Mσ = Mσ − Mσ . '6v!'%¤"1;=;C%(>@:&"1.']zR%(>&;=#(% Zσ → ∂Mσ 8a¥ .$M.': Mσ*1% 1 Fg;C.$M"tzS?>&"4'D("1?6*1%(;=%('$?.5*16>@"1:&#!'"435%(>&:&%(*q:&!> Mσ 8¡ *"1')!"t$<!' 1 Fg;C.$M"tza)%¦2!;=z.>&)2%('D(.>&%E'.$M# Mσ
:M!>(Mσ → Mσ
).
!#"%$&')(+* Mσ
!)"-,./0-(1*243*5+.!3$)/(+627!8,0%!)29*:,<;>=?!)(#!).,A@Zσ
3B!DC$)!)2E27!).,3B!8,6*0$)/."-,%0%F$DG,<;7/.B3
Mσ
!)"-,H6/"%;7,%;7I0-!)29*:,<;>=?!)(#!).,@Zσ J !)"K&?L6/,%&M)"<!)"HN!O2E*P$)/."%,%0%F$8,%;E/.QN!RSF(TIU/0-NQ"%/.?,=?')0-;7V')!)"WN/.$;E2B!DC;E"-,%!F.X"<$Y&')(1*T"<!)(#;>GZ*[')2E;E!).
Gσ −→ Mσ
\ F;B!)"-,]F.X/[^%!8,WN!29*T$*,%'_/0-;E!Deg
dMσ ,cMσ ,Zσ , pol
.
` .a2EF;b*"<"%/$;7!TF.cId*;E"%$)!*Fe$)/."%,%0%F$8,%;E[27!X(Gσ)
6*027!T,%&')/0-M)(+!gfh4fh9ia!8,F.!#*6627;E$*:,<;7/.[;E27;E.'*;E0-!
B(Gσ) : X(Gσ)⊗X(Gσ) −→ DivcMσ6*0j2E*k60-/6/"<;>,<;7/.lfh9ihnmhdf J ` .o0<*66!)272E! \ F!K24pq/.r*k')_*27!)(+!).?,s$)/."-,<0-F;>,tF.oId*;E"%$)!*Fr$)/."-,%0%F$DG,<;7[2E!Xσ
!8,sF.!HIu/0-(+!K[;72E;7.'*;E0-!Bσ
"%F0 Mσ J ` .r./,<!KN!K2E*k(#v)(+!K(+*.;7M)0%!K27!)"s/[w^-!8,<"t/[,<!).wF"6*0x;E(+*_!;E.?=?!)0-"<!"%F0 Mσ .` .y*66!)272E!Gσ
27!c"%$&')(+*z"%!)(+;GZ*[')27;E!).|"%F0 Mσ\ F;k60-/=w;E!).,lNFy"%$&')(+*z"%!)(+;GZ*[')2E;7!).
N!~0%*._,%/0-; \ F!~$)/."-,Y*.,TF.;7=!)0%"%!)2K"%F0 Bσ Js 2K!)"-,#')_*2@2E*c6*0-,%;E!l"<!)(#;>GZ*[')2E;E!)..!lN! Mσ J]*66!)27/."W2E!0-')"%F27,<*,W"<F;>=*.,)3NS@#*27,%;E._"W!8,kK&*; Jo:Z%dZ8qxTtjHht ¡chnihdf¢ Jj£ ¤Y¥§¦-¨©9ªD«¬¦+¦<ªT©Eª¯®:°Q®±Z²³w©9ªD°1¦Yª1´<µ¶®¶©4·8¸¦YªX(Gσ)
∼→ Xσ ¦D« B(Gσ)∼→ Bσ ¹º µP±Y¦Yª«4±D©4´D«4©®:¶S¦ Gσ » Mσ ¦YªD«t¸w¶lª¯´Y³¼°1µ#µ½¼D¥©¦D¶o²±©u¶´D©¾²µ¿
¥9¦D°1¦D¶«B²®:¥Eµ:±D©9ª¼o¦OÀ¦D¶±Y¦ g ¹º µ#±Y¦Yª«4±©´D«4©®:¶a¦ Gσ » Zσ ¦<ªD«x´%µ¶®:¶©4·8¸¦D°1¦D¶«§©9ª¯®:°1®:±Á²³¦P» Gσ ¹ºÂ µ²?²¥©´<µ«4©4®:¶Ã¦ÅÄT®?µ:©u±<µ:¿ÁƲ¦¶´Y¦D±±<¼<µ¥©9ª¦A¸¶©9ª¯®:°1®:±Á²³w©9ªD°1¦
KGσ/
cMσ: Sym2(ΩGσ
)∼−→ Ω1
cMσ/Z
(log ∂Mσ
).
!x6/;7.,§N')27;E$*:,§!)"-,§N!x(#/.,%0%!)0s29*Å[; ^%!)$8,%;7=;>,<'xN!x2E*ÅÇM)$&!xN!WÈO/wN*;E0%*:GÉw6!).$)!)0b/[,%!).wF!x6*029*Å60%/6/"%;7,%;E/.Sfh9iBh9Êh4f J H!8,%,%!xÇM)$Y&!W!)"-,§$)/."-,<0%F;>,WË ©dµ 27! 1
GÁ(#/,%;7IMσ
!8,§2E*[; ^-!)$8,<;>=;7,%'W60-/¯=;E!).?,N!24pqF.;7=!)0%"<*2E;>,<'N!
Mσ JÌ1ÍÎzϯÐÑtÍcUd Jj£ 2K!)"-,T$)0-F$);9*2KNBpÒF,%;E27;E"%!)0r29*27;E"%"%;7,%'lN! Mσ → Spec(Z)
6/F0r(#/.?,<0%!)0\ F!X29*aÇM)$Y&!XN!ÓÈ]/wN*;70<*GÉw6!).$)!)0+0%'*27;E"%!XF.Ô;7"%/(#/0-6&;E"%(+! J H!8,1;E"%/(+/0%6&;7"<(#!S./F"1"%!)0%*;E.N;7"%6!)."Y*[2E!6/F0x$)/.?,<0-Õ27!)0W27!60%/w$)!)"%"%F"WNBp¾*660%/C;E(+*,%;E/. J` .XN;7"%6/"%!6*0x$)/."%' \ F!).,WNBpÒF.!ÇM)$&! Mσ → Ag J ` .SN;>, \ F!2E!N;E*_0%*(#(+!
(Mσ Mσ
oo // Ag)
!)"-,]2E* ´<µ:±D«Z¦A¥9®¯´<µ:¥9¦§Ö®±D°1¦D¥u¥9¦Åªµ¶ª¶© Ë ¦<µ:¸ *"%"%/w$);7')!A@ σ ∈ Σ J
!" !#!%$'&)(+*,&.-0/21/435(6,7,809-:&);<*,&)(=.>?;@6?;@809BAC&)(D1-:E&);+!GF HJIKLNMOQPSRQTUIVVISWLNKWYX:IZ\[Y].^TU_`L\[YRa]W]TUIbSKRacedQRQT`K_MRQLfRQL\V]dQRQLGMRg[YIMKT`RQLhceTU]iX:ZjISWk\_UlQZjRaRQW<kmRQW]WkfdQI[YVk\RaMRgT]iW]k\KZ\RdQIS[nVTUlok\RpMRQL]WWR]Krqs_`WkmRoZjPSRW]WkaM]WLaTU]4dQIWLjkmZjKdokm_`ISWtMRYuvK[wX:ISZjMyxyHzISKLadRQZjdIWLaWI^k|][Y[nRQWkzcn_`[YVISL\RQZhMRQLzdQIWM_`k\_UIWLZ\RQT].km_~PSRLzcwKWvL\ISKL^Cd][YVB]TUbSOQZj_UKRX:RQZ\[nOx IKZvk\ISKkBL\dOQ[Y]
T ISWMOoW_`kBTU]d].kmOobSISZj_UROuvCompT
dQI[Y[nR5TU]LmIKL^Cd]k\OQbSIZ\_`R5MROuvComp
MISWkpTURQLpISjRokmLpL\R4ZjRQWkwL\KZT TURQLelQdRQLwL\IWkwdQRQTUT`RQLwK_G_UWMK_`LmRQW2kwT_UMRQW2km_~kmOL\KZ
TxgWvL\RMISWWRKWvM_U]bSZ\][n[YRMRd|][YVLL\KZ
T
F −→ G←− ZkmRQTKR
F → GL\IS_~kzKWR'_U[n[YRQZjL\_UIWBIKP2RQZjk\R Z → G
L\IS_~kzKWR'_`[Y[nRQZ\L\_UIWvX0RZj[YOQRRokF ×G ZL\IS_~kP_UMR dRQLjkj^c)^CM_UZjRKR F Rok
ZLmI_URQW2kzM_`L:jIS_`WkmLM]WL
Gx
t\\?:?yC x, ?]4d]k\OQbSIZ\_`RpZjOQRwVZjIdRpMRwTU]<X:Z\IWk\_UlQZjR(F → G ← Z)T
RQLjkT]d]k\OQbSIZ\_`RwZ\OQRnRQW¡bSZjISKVI¢MRQLL\KZ
OuvCompTK_£]L\L\IdQ_URYc4kmIKk'IS\Rok
(U → S ← S0)TU]
d]k\OQbSIZ\_`R(F −→ G←− Z)T (U −→ S ←− S0)MRQLM_]bSZ\][n[YRQL
2^CdQI[Y[eKkm]k\_`X:L
U //
F
S // G
S0//
OO
Z
OO
dQIS[nV]k\_UT`RQLz]KrqVZjI)\RQdok\_UISWLG→ T
RokS → T
x¤ I_`k
(U → S ← S0)KW5IS\Rok'MR
OuvCompTxgW5]VVRQTUT`R
(U → S ← S0)^CVI_UW2kiVZjIdR
MRiTU]nX:Z\IWk\_UlQZjRiMR(F → G ← Z)T
TU]%d].kmOQbISZj_UR'MRQLJM_]bSZ\][n[YRQLJVZjOQdQOQMRQWkmLQxgW¥MIWWRiKWRMRQL\dQZ\_`Vkm_`ISWvR¦qVT`_UdQ_~kmRMRTU]nd].kmOQbISZj_URZjOQR'VZjIdRMRTU]wX0Z\IWkm_`lQZ\R
( Mσ −→ Mσ ←− Zσ )U .gWYWIkmRdoIS[Y[nRVZjOQdQOQMRQ[Y[nRQWkGσ
T`RL\dOQ[Y]LmRQ[n_~^]OQT`_URQWnMRZ\]WbkmIZ\_`KRdQIWLjk|]Wk§KW_`PSRZ¨^L\RQTL\KZ Bσ x©T_`WMK_`khKW@L\dOQ[%]eLmRQ[n_~^]OQT`_URQW Gσ
MRaZm]WbpkmIZ\_`KRadQISWLjk|]WkhL\KZS0
V]Zf_`[%]bR_UW2P2RQZjLmRnª«¬
S0 −→ U −→ Bσ .¤ _GRQLjkiKW¡L\dOQ[%]L\RQ[Y_^]OQT`_URQW5L\KZ
S kmIKk'_UL\IS[nISZjV_UL\[nR GS0
∼→ GσLmKZ
S0_`WMK_`k'KWR
L\KZ0jRQdokm_UIWXσ → X(G)
MRX®]_`L\dQR]KrqdQIWLjkmZjKdokm_`TURQLL\KZSx
¯e°)±?².\?®y: x, ³µ´B¶·.¸¹¦º¨´¦»r¼¾½¦»r«t¬)¿m¿·¹¦«´¾Àf : S → Mσ Á ´¿¹mÂæÄ%¬¿´¦Äp«0Å
¬2Æà Á «´¦¸ f ∗Gσ¼mÃm¬ Á «¿´'»¸´wÃm½¦»r«0ª¬ Á ´¦¸¹|´eÇ´w¹m¬.º¨ÃCÈ·.¼«´m¿w´¦¸º¼m´
(Mσ −→ Mσ ←− Zσ)U(U −→ S ←− S0)
"!#!$%"&#'%!(!$% )+*
,-/.103240(-6587:9<;=>,"?,A@"2A9<BDC .1,A@ (G, G|S0
∼→ Gσ) E 9<F G ,4@-/B%GH9$IKJL,-M?:, DegU,S,S0, pol?,"?(=NO,G@=>9DG
;4,.10(-P=Q$, g C ;=G2=RCS0(.T,NU,G-VC9<.10(;=T@L5W@B%; U -6,.YXZB,[C9DB%;\-69<B%-]?(=Q+=T@L,B%;U,_^[,42-P= `W=;;454?(B2-P=PIZ.1,3;45a?(B%=- D ?, S E .cb 0C:C .d=P240(-P=>9<GeIZ=.d=G540(=;4, pD B(G) : Xσ⊗Xσ → Z @+9<=-f=G2.dB@+,]?:0([email protected] 0#?Dh5;4,G2A,?, σ E ,-Y-6,.fXB, σ @L9<=-iNW=G=Nj0(.CS0<;NW=k.T,A@W2Al<G,4@W?, Σ Q$5;= mn0(G-'2A,-P-6,C ;49AC ;=>5-65op 5NU9DG@-P;c0(-P=>9DGoq rtsuucvwSxyjz$u"ucsyjd|z$~ x1sSu"#~csSu"z("yOz(#s\1::su4s\Ss Gσ
:xs:1suS#S1xZcxucsSucsu':sucsyjs'uc$ G Su4vAwSxyOzu4sy3|z(~ xL1su4S S :xz(1suw% (4wSucsueSsTzSc$ #ucc1$grtscwSx#yjsTv#SuccSvZ4# Ss¢¡£Sy\¤_#¥1::sucsD¦ scy3sZUSs£S§z(ucu4v1sjS 1 |8y3$c¤"Ss¨¡eSy\¤_#©ucS (U → S) sZOTzSc$ #ucc1$ªPTS#SSs'S (U → S) |8$1:YSs Mσ
:rk«¬w:% $4wSucs'Ssy31SyOz$cxSs σ yjS1s'Ss'vLsZ4s®vwSss##s S0
z$Su Zσ n¯n#y3yjse1s£ucvwSxyjz°z<±OSs S suUv$yjSsZUsTz<4:sy3sU² S0MTz®vwSs
S →Mσucs"¤Kz(vZ4$cucs³sZ£SSs"®vAwSs
S −→ Mσ .
rze¤#SvZ4$c1z$cxOSsOTzev#Su4SvZ4#°Ss´¡eSy3¤_$c°y3#:4sUSs G suµK«¬yOz(#sO:sucsOSs Gσz$'vsZccs"®vwSs$
ªªU¶¸·_¹º»f¼<½_¾:½¿+º3ÀfÁ¹¼D ÁÄÃMÅ3ÀÄÅ p Æ ¿L ¼DÇL½_ ÈYiÉ rtsnu4vAwSxyOz]ucsyjd|z$~ x1s GσSsSsucvsS
z$u² Mσ%ÊM¸csZ(z$SvAws$1s]##S s]Ssu #cuSs p |>c#u4# Gσ[p]→ Mσ
Ssu4vsS´² Mσ%ÊM
sZËsZLS«Rz(Scu1s"cwSx#yjsjTS1sÌ1ucsSe1uc$yj#SwS1ucy3svLz$S$S1Ss
Gσ[p]∼−→Mσ[p]
ucS Mσ#ÍVz(Vv#Su4SvZ4#$s 1 |8y3$c¤ Mσ
u4sSsucvsSj² MσsZkuc#\##S s'Ss p |>4$cuc1$ Mσ[p]x:z(1sy3sÎ]ÏxucSyjsHvsv§sÏS1u4z(´Ss Gσ
su¸SSsHS#SSxsÐsÐz(cScsÐc4z(Su4vsSz(cs#z$#uSs Gσ[p]
su£z(1$x~ScSs$§¯isZccscsyjz$Ssu4scz¤_$Sz$y3s4z$s $Sv#SuccScsSSsv#y3z$vZcvLz<4#c#$Ñ>z(1sSs Ag, 0
ªO¶¸ÃfÃļDÂ%Ò½_ÓÕÔ¿D½Â ÈÏÀÄÅÇj¾#ÔS¼+¿LÅ%Ç3·_Ât¾$Ô·Å%Ç\Ö8 ¼<ÓÕÅ·_·_ÅÇÎ]1sSsS1sSs1sucvwSxyjz3ucsy3|z(~x1s
Gσ −→ Mσ
SsnSsucvLsSWz(uk² Mσ$ÊMWsyjSD:z$:VSsu/csvwSS1SsuV«Ø×]c1<#Ws/vssSz(VK«Rz(SScvAws
z$SucvwSxyOz GhσSsu'u4sy3|z(~ x1su4SSucvwSxyOzWxZAz(1s"ucS Mσ
rzWS##sLz$cx"TzWS1Suuc1$SvLz<4:siz$Ä4z( $V²µÙØÚ'Û\ÜSÝSÞ%su/SsiP«Ø#µsZËsvZcSs4zgvsZccsz(SScLÌyOz<4#µsWv$Sucs(z$:Suc#y3#cSwSucyjs Gh
σ[p]∼→ Mσ[p]
/vs´Snucs4zHSS1uc sSu4z$~S1s¸$SWS$ccs´v$Su4Svc1$Ssuv#y3z$vZcvLz<4#SugzL:svS:sLz(ß SsgSËxsvsyjSsScsgz(Ycz$S #ài²OÙØÚ'Û\ÜSÝÞxu4Ssgz$SuYK«¬sy3S1$SsgTzS$c1$OSsgv$Ss
wSsSucx1ssZ'S´cwSx#yjs"sÍf# sucv ¯isTzWxZ%4s[Ss"vLz$1ucs1suvLz$à4su'vLz(1su¤_$cy3s1su's4$Su1sSu$1cu'¤_syjxuµàÙØÚ'Û\ÜSÝÞz$SuTz\SxZSc1$eᵬâdã$¦
!#"!%$!&(')+*-,/.021+.34065,%7/.340!98 :<;==>6?A@CBDFEG;=HAI>KJ2>6BLFM6>KLGLG>NB@LGHO@BQPRTSGUV-WYXZWG[V]\_^a`Q^Yb^cbedgfihkjlnm oqpP4r s@CHtL
A u Bv;BB>; u >KL I u BwHAIx;?%PyzBwIHALF u >N?O>NM6@ u =?O>(Spec(A), Spec(A/I))
>6D|L->6BDGx6?AHO>6B~DGH u B>I>6D4M6@CBIHtLHA@CBD4x6 u Ht;?O>6BL>6D4D u HA;BL>6D4>6D|L4Cx6EGHtx6> =@ u EL@ u L f ∈ A[t]
>KL<LG@ u LG>EG;M6HOB>NDHAJ2=?A> a I> f I;BDA/I HO?>gHADGLG> u B>EG;M6HAB> a ∈ AI>
fEG>6?A>K;BL
a =@ u EL@ u L<IHO;E;J2J>NM6@CJ2J u L;LGHA-I>NDGMx6J2;D
Setale // Spec(A)
Spec(A/I)
p
ddIIIIIIIIII
88pppppppppp
?A>NJ2@EG=HADJ>pDG>NEG>6?AK>N>6B u BwJ@CE|=HODGJ2>NI> Spec(A) DGMx6J2;D Spec(A)→ S
P B=;EZLHAM u ?OHA>6E 4u BDGMx6J2;?A@M;?
SI>=@HOBL/>6EGJx
s>6D|L>6BDGx6?AHO>6BDGH>KLDG> u ?A>6J2>6BLDGH?O>M6@ u =?A> (S, s)
>6D|L<>6BDGx6?OHA>6BQPRTSGUV-WYXZWG[V]\_^a`Q^Yb^q\ P4r s@CHA>6BL
A u Bv;BB>; u >KL I u BwHAIx;?%Pn>6BDGx6?AHODGxNI>
(Spec(A) , Spec(A/I))>6D|L<?A>NM6@ u =?A> (
Spec(Ah) , Spec(Ah/Ih))
IxKBH=;ESpec(Ah) = lim←− S et Ih = IAh ,
@C?;?OHOJHALG>=EG@G>6MKLHt>>6D|L=EGHAD>D u Ez?;M;LGx6C@EGHA>I>6DIH;CEG;JJ2>6DM6@CJJ u L;LGHA/D=EGx6M6x6I>6BLGD6PyzBv;Ah/Ih ' A/I
PyzB=> u LI@CBMvIxKBHOE ?a¡>6BDGx?AHODGx ShS′
IQ u B¢DGMx6J2;£;¤ B>S?O>v?O@B¥IQ u B¢DG@ u D DGMx6J2;q>6E|J2x
S ′ P¦ §¨¥©ª«C¬-§T\_^a`Q^Yb^a` P4r _;NB@LGHO@B I>FM6@ u =?A><D|LE|HOMKLG>6J2>6BL>6BDx6?AHO>6B2BQ®;N=;DI>FD>6BD BHM6>6?A?O>IQ>6BDGx6?AHODGxD|LGEGHAMKL?A>?O@BIQ u B¯DG@ u D DGMx6J2;°q>6EGJxP4yzB±B>=> u LN=;DNIxKBHOE?a¡>6BDGx6?OHADxIQ u BDGMx6J ;B@Bw;²¤ B>?O>?O@BIQ u BDG@ u D DMx6J ;~q>6E|J2xM;E?A>6D?OHAJ2HtL>6D=EG@³G>6MKLGHAC>6DBQ>g´HOD|LG>6BL=;DL@ u G@ u EGDFI;BDF?O;M;²Lx6@CE|HO>NI>6DFDGMx6J ;D6P
!#"! !µ*4*4¶³'·_7%¸ j¹ 7q'3 06.,/'3 ºz'*n.0»)!8 ¼<B J@CE|=HODGJ2>½I>½DGMx6J ;D¾;²¤ B>6DSpec(B) → Spec(A)
>6D|LIHtL¿ÀcÁÂÃÅÄÇÆg¿2D6HA?4>6D|L=?O;LÈ ÉE|>6DzCx6@J2xKLE|HO u >6DE|x6 u ?AHO6E|>6D6PsH A >6D|L?O@M;?>KL>g´M6>6?O?A>6BL ?;Ê6M> Spec(A)→ Spec(A)
>6D|L<EGx6 u ?OHA6EG>PsH A >6D|L>gM6>6?A?O>6BL =@ u EL@ u LHOIx;?II>
A ?A>J@CE|=HODGJ>Spec(AI) −→ Spec(A)
I u D=>6MKLGEG>I u M6@J2=?OxKLGx I ;IHO u >9I> A I;BD-?A>D=>6MKLE|>I>A>6D|L-EGx6 u ?OHA>6E4=;E fÌË(h µÎÍÏ+ÐÒÑPÓPÔoaP
Õ @C=>6DGM u J2@BLE|>N?O>LGx6@CE|6J2>q@BI;J>6BL;?D u Ht;BL6P
"!#!$%"&#'%!(!$% )+*,.-0/214325$687:9<;>=;?9<;A@CBEDGFHJIKL%M?NPORQ S<TUWVYX[Z[\4U
Spec(B)→ Spec(A)UT]>^_U"`PZ[\4abdc2`ecgfhTU[`
Ti.]j\a^klUT%`mU[`]<^_aAn(o%pqk>X^_Ur`k0U]`PUopsUbtUT]`k0UppuUvc(k]xwUSpec(B)
o%TU'pqkbyk]zUR ^_|z+U[Z]?k~$UWpq]>^c(T4]zUwU`PZ[\abc2`eczfhTU[`pGks`[`PU[``o%^
Spec(A) $g_4u<
A+4$44+(uE
I44+$?
Spec(AI)u_d42
_ ¡¢4£¤4$d(¥4du_4Spec(A)
O.¦§ u¨Ed4h_¤4$©¢4«ªR# ¬£®4#44mr®4_#4r § $44+u(u$¯_4_°4¯u±°_4 ($4#ru4$_²_#³E_$J44®4$ ¦ 4´zu4 Oµ 7#6¶687·9<;l=;j9<;?9ORQ ¸|²klUT]
(A, I)o%T·Z[|²oP psU¹\UT%`PapqklUT:U]
no%T·UT]>klU^YTc(]>o^[Up º |2]z|²T` S =
Spec(A)U]Sn = Spec(A/In+1) ¸|²klUT4] S ′ o%T»`PZ[\abchpqks`_`PU·`o^
SU]t : Sn → S ′ oT
bt|2^¼4\ks`bdUewU`PZ[\abc2`"`o%^S ¾½ pxU¿$ks`]zUo%TU"`+U[Z]>kl|2T s : S → S ′ ]zUppsUeÀoUpsU[`bt|²^A\ks`btU_`Sn // S
s // S ′ et Snt // S ′
Z[|²ÁTZk>wUT]  abt|2T`]>^c(]>kl|2T °Ã Ä 4_tÅ£4u$
S ′ $4 ##_$Æ$t$ ¡gJ4d § u$Æ$t4Sn
<# Y44 #¹® § ¾_44"Ç?(E_$_(u$¢4syÈ©[¤4u_
S ′ → S
S ′ u // ArS
v // S ,#É
u°E[(uE
v°s42E_#«+(4#4®4 OÊ £ ¡Y_# #Y44È©¤4
s : S → S ′ ®4444s(Æ#$±4 $(Y$J_(ÇrË
S ′ u // ArS
v // S
s
DM
V_hq
z
Sn
??~~~~~~~~
OO
t
``AAAAAAAA
Ì°Íu©E #+±s′ : S → Ar
S
4Sn → Ar
S
_±_$2$4 O0¦ #44(S, S0)
¤44u4#4
s′"Ç?(E_$$' § ¤4+4u4
ArS
u#4Ʊ4 § u$Æ$4S0 → S ′ −→ Ar
S .Î $±u¯E[(u$ÏE¯¤44u·¯_$$4¤4·Å § ¤44_u«4
S ′ u«u$4Æ4S0O Î u
_4sy#44Eu#¢4sO
´¼44®4#4m_$44¡#±444ue § _4J § $44+u(u$44[²_m44 OÐ #
A4³(44+$tÑ"¤4¯%u+
I4¯4+$ E
n4t_Ï(4 O ªx$#4
S = Spec(A)S = Spec(AI)
ESn = Spec(A/In+1)
OR¦ d¤4u(Sh, Sh0 )
4(S, S0)
ut¬_$t®4«u±«4²E_ ¡«4«[¤4d(t4£zÒ¢ÓY4
SOrÔJ§ (4©t£¤4(
©±ÕÖ×ÖsÕÖ?ØS"u±e42E #JÓ_(_e4e¤4$
Sαu
ShOÙ #e4#44
"!$#&% "'()+*-,.S%.-/0&)1 2(!3%4.5)76849 :.<;2=2)7%.-/>, :@?A,( :/((;2 :*-,.
Sα%/B,.
α*-,CD*E46F68 2(!
9B.-4(@G+HI?J4%.-K2*ML1 NL7 26F68 POQR@QOQO@S)7L UT)1*<!E N,( N/((;2 N*-,.Sh
>,)V486FW268 N.< 2*<!E.<)7=X!-)7/(ZYSn>, [L74F/((;2 [)7()\!E)4L7 G]^(_4F/(=[.-;`4L7)1*-;N,( a4%%.-/`T)16D4b!E)1/B(_YcLd?A/B.<.-
nG
OQeRQRQgfghihijlkmVnporqsntkuwvyx&z|Bqj`sxz~kBqtxz|k+jborx&tx&z~z2quiz~uynxq+Q :/,*F4L7L1/B(*4%%.</= 2.L4c=`4.!E [L1/=`4L1 t/.-6F 2L7L1 N*E4(*"()10B `4,_4*E*-/=2)1;2 NY
σ ∈ ΣG+H^;X'()7*-*-/(*ML1 [*-=;26D4
Mh
σ = (U ×BσMσ)
hU×BσZσ ,
>,) 2*<!Lp? 2(*-;2L7)1*-;c,*-=;26D44bCD( U ×Mσ
L7 PL7/(9,|*-/,**E=;26D4gt 2.<68; U × Zσ GL 2*-! UT=2 2L1L7 2(!" X!M=2/(!-)7 2(!"L7 [*-/,*<*-=;2684F/,0 2.!
Mhσ =Mσ ×Mσ
Mh
σ ,L7 )\0&)1*- 2,.MYF=2.</B)1*- 268 2(!-*"(/B.<6D4,&T
∂Mh
σ =Mh
σ −Mhσ X!ML1 [*-/B,**-=;2684ct 2.<68;[@?A 2(*E;2L1)7*E4b!E)1/B(>, [Lp?/((/!- [ 2(=2/B.<
ZσG
IlVb-V ¢¡ ¢¡@ £ Gi¤ ¥¦3§<¨B©ªU«§F¬&rª®¯°U±D²ª§±c©³<²´°U¦µ©¢§UGhσ
ªU¬&¶ Mh
σ ·°U¶© ¸²«M¦7§ªº¹¶E»b³
¹¶©¢°U«$°EªªU¬&©·²«$§EªP¼
½ ©¦§ª«²´°¦µ©¢§UD¹¶U©e®U©J¹²b¦§U±g§«V¹»l¦7²¶©ª°P¾§º¿§U¶§gªU¬&¶ Mh
σ À½ ª`²D¶E§ª«d¶U©d®U«d©¢»b~ÁZσ
§ªU«®E²»l©dÂX¬+§±g§U«3©ª»b±D»b¶¹¯§IÁGσ À½ Gh
σ[p]§ªU«"®-²»b©dÂX¬+§U±D§U«5©ª»b±D»b¶¹¯§PÁ
Mσ[p]ªU¬&¶ Mh
σ À½ X(Ghσ)
§ªU«®E²»l©dÂX¬+§±g§U«5©ª»b±D»b¶¹¯§IÁXσ À½ B(Gh
σ)§EªU«"®E²b»b©pÂX¬+§U±D§U«3©7ª»b±g»l¶¹¯§PÁ
Bσ À½ ¦-à ²¹B¹¦µ©¢®E²b«d©¢»br¾§Äc»`¾B²©¶-²³ÅB¹§U®§¶P²ªª»®U©d°§PÁGhσ
©¾¬&©«5¬&©ª`»b±g»b¶¹¯©ªU±g§
Sym2(ΩGhσ)
∼−→ Ω1
Mhσ/Z
(log ∂Mh
σ) .
Æ °U±g»lªU«d¶-²«d©¢»lÇyÈ HI?J4%.-K2*5L7 "!-;2/B.-K26F ºO@QRQeOQpÉS Mσ
2*<!3L7)168)\!E M%.-/lÊ< 2=X!E)\0 M :*-=;2684* Mα
σL7)1*-*- 2*F*-,. Mh
σ
SË/Ìα%4.-=2/,.<!F,(Í 2(*- 26IÎL7 /B.</B((;'L\!E.-4(!`GÐÏ *E=;26D4*- 268)µ4Î;2L1)7 2(
Gσ%.-/0&)1 2(!"@?,(*-=;2684F*- 268)µ4Î;2L7)1 2(Gασ
*-,. Mα
σ
%/,.α*-,CD*E4686F 2(!M9.E4(@G+Ñi/B*-/(*
Mασ = Mα
σ ×Mhσ
Mhσ et ∂Mα
σ = Mα
σ ×Mhσ
∂Mh
σ
Ò /686F Mα
σ → Mh
σ
2*<!DL7)1*-*- S=2 _*-/(!D.- 2*-% 2=X!E)\0 26F 2(!D,(Ó/B,0B 2.<!D 2(*- X!D,(Ô)\0&)1*E 2,.DY=2.-/)7*- 26F 2(!E*"(/.-684,&T Mα
σ
G+Ï [*-=;26D4Gασ
2*<!M4Î;2L7)1 2(*-,. Mασ
X!">,)\!-!- NY84,9B68 2(!E 2.αS
/B(Õ% 2,!c*-,%%/B*- 2.I>,@?A)7L3 2*<!P%.-)1(=2)7%4L7 26F 2(!P%/L4.-)1*-;GÐ]^(Ö% 2,!c;29B4L1 268 2(!c*-,%%/B*- 2.I>, gL7 2*)7*-/68/.-%)1*-68 2*
Gσ[p]× Mσ∼−→Mσ[p] , Gσ × Zσ ∼−→ Gσ , X(Gσ)
∼−→ Xσ et B(Gσ)∼−→ Bσ%.-/0&)1 2(( 2(!"@?)1*E/68/.-%)1*-68 2*
Gασ [p]×Mα
σ∼−→Mσ[p] , G
ασ × Zσ
∼−→ Gσ , X(Gασ)
∼−→ Xσ et B(Gασ)
∼−→ Bσ .
"!#!$%"&#'%!(!$% )$*+,#-/.
n 021431 .5- 36718 . 026539;:=<>1?8$@2@29 -BA 023C9D3C9D3E7EF3HGIJIKGIKGL3 . 9NM , 1 ,$O.P- 31 . 02123CQ53R .P-D, 1 s :
Mh
σ →Mα
σ
A 0 - 6S3T9DUWVX39D3E , 65@2Y - Q5E73R , E7@ , Q5ZZσ, n → Mσ →Mα
σ ,,#[Zσ, n 3Q . 9B3\Q , 02Q^]_Q5RY2ZE`8bac36SE7ZFd23 Mh
σ d2ZWe1 - @8$6"9K8f@20 - Q5QP8(12R3 n ] - UEF3Fd23F9;M - dZT8(9gd23 Zσ :h 9iQ50j . 8(9 , 65Q>d23k@ , Q53T6 Ghσ = s∗ Gα
σ 3 . d23lVXZ6 - e36 A 023kQ - n 3Q . 8(Q5QP3mkn$6P8(12do 9NMp8$@2@29 - RT8 .5-B, 1qd23r , d8 - 658s] + @ 312R36t8$Q5Q , R - Z3\u Ghσ 6SZT8$9 - Q53v021 - Q , E , 65@2Y - QPEF3$oR3 A 0 - 6SZQ5029 . 3vd23bwpx'y\z2|o h5hSh :~%:poh_ :2:Dl3 . h_ :2:N:
3Q5RY2ZE78 Ghσ 1Mp3Q . @8(Q021 -BA 023E78 - Q9B3QR , EF@8$R .5- eRT8 .P-D, 12Q123d2Z@ 312d26 , 1 . @8(Qd20RY ,#- :
GIJI/I`%$icXc ? D¡2ii¢£c¡2¤I=¥ ¦ , 12Q - d2Z6 , 12Q9;M - E`8(n#3 - 1V#365Q53"d23 Ghσ @8(69K8v§URY23
Mσ −→Mh
σ .
<>1 ,#O.5- 31 . 021Q5RY2ZE78FQ53E - ] 8 O Z9 - 31Ghσ −→ Mσ .¨ M~8(@265UQ9K8\@26 , @ , Q -D.5-B, 1qGIGIKJIKGoR3Q5RY2ZE78Fd2ZWe1 -/. 02123"§URY23
f :(Mσ −→ Mσ ←− Zσ
)−→
(Mσ −→ Mσ ←− Zσ
)
d23H9K8©RT8 . Zn , 6 - 3 OuvCompU : M - E78$n$3 - 1VX36SQ53d23 Gσ @8(6 f 3Q .ª- Q , E , 65@2Y23u Ghσ 3 . f 3Q .
9NM - d231 .P-/. Zl8(0%]_d23QPQ502Qd23 Zσ :«¬1ac8 -/. o f 3Q . 9;M - d231 .5-D. Zd23(Mσ −→ Mσ ←− Zσ
)
®¯(°%±K® 02123@20 - QPQP8(12R3l8$6 O2-D. 658 - 653EF31 . n#658$12d23d239;M - dZT8(9|d23 Zσ :²k³´$µ´2¶s· ¸^·5´2¹»º¼¾½¼K¿g¼_À :iÁ Âà ®sÄ_ÅÆÇKÈ Ã f à ÈÉÊÉ_Ë(± Ã\Ã É Ã ÈÉÌ°%ÍË(°Éήs®Ä Å2ÆÇKÈ Ã ¯ à Mσ ÏÐ Ê®ÍÈÉNÄ5Ë(ÉNǾ®Í ÏiÑ 3Q - Q , E , 6S@2Y - Q5E73QÌd23 r , d8 - 658s] + @ 312R36Ò8$Q5Q , R - ZQ=u Gσ 3 . Gh
σ d2ZWe1 - QPQ531 .021 - Q , E , 65@2Y - Q5E73f ∗ Ω1
cMσ/Z(log ∂Mσ)
∼−→ Ω1cMσ/Z
(log ∂Mσ) .
¦ , E7EF3 f 3Q . 9NM - d231 .5-D. ZQ5026 Zσ o9;M - Q , E , 65@2Y - Q5E73@26SZRZd231 .Ó- 12d20 -D. 021 - Q , E , 6S@2Y - Q5EF3f ∗ Ω1
cMσ/Z
∼−→ Ω1cMσ/Z
.
Ô 0 - Q A 023 Mσ → Spec(Z) 3Q . a , 65EF39B9D3E731 . 9 - Q5Q53v9D3vR6 -D. U653'ÕP8(R ,$O2- 31qE , 1 . 653 A 023 f 3Q . Z . 8$9D3$: 3Q5RY2ZE`8Mσ
3Q . Y2312Q5Z9 - 3T1`6539B8 .5- V#3E731 . 8(0`ac36SE7Z Zσ : 8l6S3Q . 6 - R .P-D, 1`d23 f ulR3Óac36SE7ZÓ3Q . 021 - Q , E , 65@2Y - Q5E73d , 12R f 3Q . 021C8(0 .P, E , 65@2Y - Q5EF3$:
"!$#%'&($!$)%+*,!$)-/./013241657&88.9!:<;3=>&-/;3!)@?
f A %9-'#)&#-/!$=B!$.98C;3%/= AD &#*,#) A ;E)FG!./=>&-H;E!$)I)J57&2K-H2L8 A .9M# ANA - Ghσ A %9-O&#%/%/;PQ#);R( A .9% A 1TSVU# A Gσ W$X !$;R- (U → S ← S0)
#)Y!Z[ A -\M A OuvCompU W]^&B8.9!$8!$%/;R-H;E!$)_%/#;R(&)- A .92,%/#1E- A M A 1&`8./!8!%/;E-/;3!)baJcacedca Wfhgij^ikmlonpl/iqsr^tvutwxtGr Wzy |~m|<<Hv| f : S → Mh
σ |H|YT v| f ∗Gh
σ HH |V|`H< ||h|`Hmmv|H`|6H|(Mh
σ −→Mh
σ ←− Zσ)U(U −→ S ←− S0)
| Hmmv|VT| m¡ | (G, G|S0
∼→ Gσ) m¢ G |O<@$G£|L| DegU,S,S0, pol |V|6mH| 6e| g K<B|4¡¤¥¡ ||4¡ ¦ U § `<B^¨ ©m¡3| G|S0
∼→ Gσ
< S0 § | ¤|'¡m<ªm<'|<Y|G«4|6 ~ªHHm¤66 | HHm< D | S ¨ ¡T¡ vH66m¬K HH|pD B : Xσ ⊗Xσ → Z mO |V$ ¨ $H|H|V| σ § |O| K| σ mO` ¡Y |Ym|HY| Σ ®L'|6|+¡H¡v¯° 6/6v¯²± ³ !=>= A S A %9-"*,!=>81 A -´. A 13&-/;E( A = A )T-¶µ S0 D !$)&(Mσ −→ Mσ ←− Zσ)U(U −→ S ←− S0) = (Mσ −→Mh
σ ←− Zσ)U(U −→ S ←− S0) .·o1%/#¸>-'MJ5¥&8813;EU# A .+1&Y8.9!$8!%H;R-H;E!$)aceacdJca A -'M A *,!$=B8!% A .+1&h¹º,*C A S → Mσ
!Z- A )# A &( A *165»;3)T( A .9% A M A f W¼½¾À¿gÁ$Â\½r^t6uJtwxt6u Wzy ] A M;3&Ã.H&=>= A (Mh
σ −→ Mh
σ ←− Zσ)) A M2KÄ);E-Å8&%Å#)b!$Z[ A -ÅM A
OuvCompU
*&. Mh
σ
)J5 A %9-"8&%"*,!=>81 A - WacEÆ^cIÇ ÈÉÊËÌ>ÍGÎÏȤÍËÌBȤÍФÑÒÉmÓÕÔÖË<ÌacEÆ^cÕ×cÇ ÈÉÊËÌbÍÎÏ$ÈÍË<̦ÈÍФÑÒzÉÓÕÔJÖ^Ë<ÌbÌ,ÈØz̦ØzÓeÙËÈÖQcLÚ ] A %/*C2,=&&m¸) A Mh
σ A %9- 13;E=>;E- A8./![ A *K-H;R( A M A %/*C2,=&%Å2K-&1 A %Å%H#. U ×BσMσ A -Å1 A %/*C2,=>&% A =>;ÜÛo&Z2,13; A ) Gh
σ A %9-4M A -oÝ<8 A Ä);%/#. Mh
σ W ·o1 A :;E%9- A M!)*ª#)_%/*C2,=& Met
σ D #)_=>!./8C;3%/= A 2K-&1 AMet
σ −→ U ×BσMσU#; A %/-Å#)b;3%/!=>!./8C;E%H= A %/#. Zσ DA -4#)b%/*C2,=>&% A =B;RÛo&Z2,1E; A ) Get
σ
%/#. Met
σ
U#;zM A %/* A )M Ghσ WÞ )_8!$% A
Metσ =Mσ ×Mσ
Met
σ et ∂Met
σ =Met
σ −Metσ .] A %/*C2,=&% A =>;ÜÛo&Z2,13; A ) Get
σ
(T2,.9;EÄ A 1 A %B=>ß,= A %B8.9!$8.9;32K-/2,%BU# A * A 1E1 A %BM A Ghσ
M!))2 A %BM&)%1&B8./!8!$%/;E-/;3!)acedcdJcÕ× Wà &. A : A =B81 AD ;31 A %9-¶&Z2,13; A )8./;E)*,;38&1 A = A )-¶8!1&./;E%H2V%/#. Metσ A -"%/!$)Ã$.9!$#8 A M A p Ûv-/!$.9%/;3!) A %/-´*&)!);3U# A = A )-´;3%/!$=B!$.98C A µ
Mσ[p]×MσMet
σ .
X ; (U → S ← S0) A %9-´#)!Z[ A -"M A OuvCompU D -H!#- A ¹º,*C A(U −→ S ←− S0) −→ (Met
σ −→Met
σ ←− Zσ)
!#"
$&%('*),+-/.0&12$&%435)0 Mh
σ+),0 (S, S0)
%6$7-98%6:$&;6<21=%6:>@?A:CB;6BD12-EB%4<=)F30&.3.$&12-&1=.G:IHKJLKJNMOJPHQD:%B%6$&+60&123-/12.G:(B%6$R3.G12:-/$R307.+S8%6$RB%T<P)U'V07.G:-&1=W607%TB%
(Metσ −→M
et
σ ←− Zσ) .
X %R$&+S8;6YZ) Met
σ:[\%6$7-]35)$^D:1=_D%`YE),1=$]<2%6$]+6.YZ35),+6-&12a5+b)c-/12.G:$^:%RB;63%6:B0&.:-]35)$]BDd+S8.1NeK> X %
$&+S8;6YE)T)f;6<21=%6:Z30&12:+61=35),<=%6Yd%6:-]3.<P),0&12$/; Getσ →Met
σ1=:BD1N-gD:%`h5W6+S8% Met
σ → Ag . i 353%6<2.G:$j/k,lSmonTpPqj/k,p=nUk,p\rtsbulSvxwyn )$&$&.+61=;6%z σ ∈ Σ -&.GD-RB1P),|G0&)YdYZ%(Met
σ ←−Metσ −→ Ag
)
.Gf-&%6:D(35),0R<=)d+6.G:$7-&0&D+-&1=.:(30&;6+6;6B%6:-&%>
HKJ2MJPHKJ~G5 5#* t 5O5#5OcVOJ@ .-&.G:$ Met
σ, 0<=%
30&.BD12-`a5f0&;
Met
σ, 0//
Met
σ
Mσ, 0//Mσ .
<+6.:-/12%6:-R+6.GYdYZ%T$&.GD$$&+S8;6YZ)d.D t%60- Metσ, 0 ¡ _D1%6$7-B;a5:135)0`<=%T307.BD12-`a5f0&;
Metσ, 0
//
Metσ
Mσ, 0
//Mσ .
X )d307.G3.G$&1N-/12.G:($&D1N ),:-&%07;6$&D<2-&%TB%T<x[\)3307.e 1=YZ),-&1=.:4307;6$&%607 ,):t-<2%T|G07.GD3%TB% p ¢-/.0&$&12.G:>£¥¤#¦§¦¨c©«ª©&¦¬®¯P°±¯*¯² >³ ´ n¥µEqcl¶5·vP¸Sµ9n Met
σ, 0 →Metσ
n¹¸mºp&»¼n¹¸¶kj¹nd½n¥µEqb½,ypPn¹¸¥l/np=k,mxv ¾¿½tn¹¸¸mxlSyjSmxy l/n¹¸U½nTÀ5vÁn&k,yd¶kcl/k·5qclSvxwyn/¸U½nTmxÂS¶5n D nÀ p ¸yl Get
σ [p] ÃÄ sµ9q#À¸mxl&k,mxv¢q#À ÃÆÅ X %YZ.0&3812$/Yd% Mσ, 0 →Mσ
%6$7-A<x[Ç%6$&35)+6%B%Yd.BD<=%6$È0&%6<=),-&12'^B%6$È$7-&0&D+-/D07%6$B%E:12 t%b),DÉ35)0&)8.0&12_D%6$$/D0 Mσ[p] ¡ %-¥<2%E$&+S8;6YE)4)f;6<=12%6: Get
σ → Metσ%6$7-¥Y¥D:1]B[\D:
1=$&.YZ.0&3812$&YZ% Getσ [p]
∼→Mσ[p]×MσMet
σ>
.GD$)b t.:$R.Gf-&%b:D(D:(B1P),|G0&)YZYd%zd+b),0&07;6$R+b)0-/;6$&1=%6:$R%-`h5W6+S8%6$` t%60-/12+b)<2%6$307.G307%6$
Met
σ, 0
Metσ, 0
oo //
Ag, 0
Met
σMet
σoo // Ag .
! #"$$%'&(&)%+*'%-,."/102"3435%7698:;=<->?628><@8>BADCEF:GIHFJK<@LD<-M GON<28JQP8:28SR?T:UGVHFJK<@L<-;VWP < D <Mp"X2X2Y*'.)Z'%\[
σ ]_^ &`*'Y020a%'X2$Y ,cb %db .(eb%f*"0ag2%f&)Y*"&)%f"&)/1Z'h02.)eb%f"i1%'*d .)i1%"b7[+g9Ybg9%d*"0jg9%&(Y*"&)%k"&)/1Z'h0a.(eb%lX9" Xm .niD%"b ]oqprcs tu1vwpyxz(|zVxz~x ]_ YbXd "iDY Xd$ "Xdh%'X2Y1.) ,%,ZF .(0q,%'Xd*"0jg9%'Xd&(Y*"&)%'XdOY10a35%'&)&(%'X"iD%'*\ .niD%"b%Fg4,%\&(%'X4"$$02Y.)35%'0 ] "y35"bi".(X2%\02Z',b*Fg2.(Y %'35$'*%'02".ng4,%fg9Ybg9%f~"'Y &)%*'Y1 Dg90a1&)%l,Bb %lZFi1%' g9b%'&)&(%k"$$02Y.)3q"g2.(Y ]
! "#$
%'&)(+*-,/.0,213&)4+576989.;:)(+<>=+*?89./,/<;@A@7<>(+*?@7./*-,B&)8C69./*?@A<D,B&)@A./*-&;@7E>F/G+8C5AHI=+./*-E;8KJ),/.ML'=+(+.08C./@N&O6K57<>(QPSRTF/HI=+5VUW &;@7./(+,/.XFY6Z&)@A.X&;P+F/HD=3&O6K.)[]\^<>=+*_<;G`6K./(+<;(+*_=+(+.X,/<;ab43&),Y6K5Vc3,B&)6K57<>(d6K<;89<)efP3&;@7.XP+. Ag, 0 :+P+<;(I6_(+<>=+*P+F/,/895 W <;(+*'@A./*'4+8C./ab57g/89./*'4+8C<>4+8C5AFY69F/*/[S\h<;=+*',/<>(+*C6989=+57*K<;(+*'=+(+.i,/<>aj43&;,Y6957c3,B&O6K57<>(k'&)abF/@A57<>8CF/.ml:HD=+5X(SRn&o43&;*pP+.#*957(+E>=+@A&;8C5769F/*q6K<;895AHI=+./*/[h\h<;=+*pP+F/,/895 W <>(+*p.srt4+@A57,/5769./ab./(I6p@N&o*C6K89&)6957c3,B&O6K57<>(uP+./*,/<>aj43&;,Y6957c3,B&O6K57<>(+*Q<>G`69./(D=+./*/[-vw(`c3(S:?(+<;=+*jaj<>(I6989<;(+*j,/<;abaj./(I6b&)P3&;4`69./8Q@A./*Q89F/*9=+@V6Z&O6K*j&;=x,B&;*P+. Ag,n,0 [
yz|3zb~D>~D#~D!~tj>3B~Db>3~Dj3]IOS~D<;=+869<>=`6
σ ∈ Σ:<>(,21+<>57*9576i=+(+.,B&)8C69.@7<D,B&;@7.d&)@AE;F/G+8957HD=+.
(Met
σMet
σoo // Ag )
.Y6@|R<>(q,/<;(+*95AP+g/[email protected]*9,21+F/ab&
U =∐
σ∈Σ
Met
σ ,
HD=+5S./*C6^@A<I,B&;@7./ab./(I6_P+.6t4.c](+5S*9=+8Spec(Z)
[m(p4<;*9.U =
∐
σ∈Σ
Metσ , Z =
∐
σ∈Σ
Zσ et ∂U = U − U ,
HD=+5./*C6h=+(pP35 W 57*9./=+8hLb,/89<;5A*9./ab./(6K*h(+<;89ab&;=trqP+. U [m( (+<)6K. X =∐Xσ
:]HD=+5./*C6h=+(q¡¢&;57*9,/.B&;=,/<>(+*C6989=+,Y695AG+@7.*9=+8
U:.Y6
B =∐Bσ
:HD=+5£./*C6=+(+.¤¡<;89aj.G+5A@75A(+FB&)5A8C.*9=+8XL W &;@7./=38C*P3&;(+*@7.¡¢&)5A*9,/.B&)=
DivU[
m(p&j,/<;(+*C6K89=+5V6^=+(q*9,21+F/a¤&j*9./aj5VU&)GFB@75A./(G =
∐Gσ
*K=+8UHD=+5S./*96^&;GF/@A57./(p4+8C5A(+,/5743&;@7./ab./(I6
4<>@A&;8C5A*9F*9=+8U[!¥@.sr`5A*C69.iP+./*m57*K<;ab<;894+1+57*9ab./*
X(G)∼→ X
.Y6B(G)
∼→ B:.Y6'@|RT&;4+4+@75A,B&O6K57<>(¦P+.
§h<IP]&)5A89&OU©¨D4./(+,/./8M5A(+P+=+5V6^=+(q5A*9<;ab<;894+1+57*Kaj.KG/U : Sym2(ΩG)
∼−→ Ω1U/Z
(log ∂U
).
ª.X*9,21+F/a¤&j&;GF/@A57./(G*K=+8
U5A(+P+=+5V6^=+(qP+5A&;E;8K&)abaj.
(U Uoo // Ag
).
!"#$%&'()"*,+.-/10324
U → Ag4157685:9; <:410=6?>A@4B0+); M0 = Ag CDFEGIH,GJLKNMOK:GPRQSNT$SNT CVU W,XZY\[L]_^`abY\X U → Ag X3bcdcNe)fXgh dYi[jbck]:e)ckal[jgnm oqp 5?+I>A6sr94utv4w5?032x1y\+z+I$x1tv>|4 p 9 p >A@41;75?41t
Ag → Ag> p 9>A6Z9 p >v5:~Iyu~;
2>v5:yw4 4Z~ +)>v;:+LO 4 p 0141;KAg/Ag
: Sym2(ΩAg)
∼−→ Ω1Ag/Z4=68t|4B5:02x1y\+w+I$x1t|>v4 p G→ U
9 p >v5:~Iyu~I; 2>v5:yw4KG/U : Sym2(ΩG)
∼−→ Ω1U/Z .~)6?~ p 5
f : U → Agtv4.yu~I; 2>|5:yu4.> p 9>A6 +I;
G Coqp +G = f ∗Ag ~ p +
f ∗ Sym2(ΩAg) '
Sym2(ΩG)4=68t~ p +
f ∗KAg/Ag= KG/U ~ p 0f ∗ Ω1
Ag/Spec(Z)∼−→ Ω1
U/Spec(Z) . ~Iyuyw4#tv415032+Iy 58+)tvIx1;:>|r9415U
4=6 Ag 5:~ p 6nt|>v5:5:4155:9;Spec(Z) tv4#01;7>|6:/1;:4<?+)01~>|4 p yu~ p 6:;:4
r94f
415768x=6?+Itv4 C ~ p 5:> x1;:~ p 5ntv4B5:02x1y\+
R = U ×AgU = IsomU×U(pr∗1 G, pr∗2 G) .Ntn41576\y9 p > 4 419-/102415\x=6?+It|45w@41;75
U 4=6\t|402+)y +It|x1;7>vr94 Ag 41576\>v5:~Iyu~I; 24+I902+)y r9~)6?>|4 p 6
[U/R] C ~)6?~ p 5R
t|4 p ~;7y\+)tv>|5:x 4U ×Spec(Z) U
+ p 5R C 41576B9 p 5?032x1y\+yF9 p > 4 419-/1032415
R ⇒ U ;7~t|~ p I4+ p 6R ⇒ U C Nt4>v576:4w9 p>v5:~Iyu~I; 2>|5:yu4
pr∗1G∼→ pr∗2G
+)9 415:5?95 4R CL + ;:/15Vt|46?2x1~I;:/1yw4q >vt51x=6:4 p 4 p 9 p >|5?~L
yu~I; 2>|5:yu4+I9 415:5:95 4R +I; p ~I;:y\+)tv>A6?x 40145?032x1y\+ C *,+ x1yw~ p 576?;:+)6:>v~ pw 4t+ ;7~ ~5:>A6?>|~ p5:9>|@)+ p 6?4B415766?>|;:x14 4u ¡w¢£¤4=685:41;7@>|;?+ 4Byu~ /1tv4Z¥.014t|tv4 4Btv+ ;7~ ~I5?>A6?>|~ p§¦ ¨% CDFEGIH,GJLKNMOK:GPRQSNT$S©«ª=¬ ¡.¢£¤l¯®°±²%|³´ CVU W,X3bµX¶I·q¸n¹3º3`X3b R ⇒ U b»[LjcndcNeLfX3bgh dYi[jbck]:e)ckal[jgm ~I>v4 p 6z
9 p ~I> p 6 4R
4=6x
416y
5:415>|y\+)415 + p 5U C Nt5:9¼u6 48yw~ p 6?;741;
r94t|415½01~y tvx=6:x15 415½tv~0+)tv>|5:x15 4R
4 p z4=6 4
U4 p x
4=6y
5:~ p 6#>|5:~yw~; 2415¾ ake t|415 419-/102415 C ~ywyu4t|415-/1032415
R ⇒ U5:~ p 6x=6?+It|415 ~ p 41965:9 ~5:41;¿r94
x4=6y
5:~ p 6 415x1tvx1yw4 p 6:5 9i$~; ∂U C Nt$4>|5:6:4 415À01Á p 415
σx, σy ∈ Σ6:41tv5Âr94
x5:~I>|6À5:9;t+
σx_576:;?+L6?44=6
y5:9;
t+σy
_576?;:+)6:4 C ~I>v4 p 6 Ox Oy 4=6 Oz tv4158+ pp 4+I9 415n01~Iy t|x=6?x15 415ntv~0+It|>v5:x15n4 p x y 4=6z C *415
yu~I; 2>|5:yu415R ⇒ U
5?~ p 6 ~Iyu> p + p 6?5Â0+I;Ât|415Âyu~I; 2>|5?yw415R ⇒ U
5:~ p 6À5:9; <7410=6?>AÃÄ5 Coqpi >v5 ~5:4 > p <:410=6:>v~ p 5 + pp 4+)9
Oz
Ox.
>>||||||||Oy
P0
``BBBBBBBB
!"#$%#&'(*)+ ,.-
/02143#57698:140<;$/=?>@ 0AB/7A<C D$/FE7/75<14@HGI/7E014>"@$5<5I>@ 02;$/75<J7:K83L1M0J757NO @P@$>0/ Gx
/0 Gy34/75Q5IESR$J7=T85U5I/7=?1WVX8YJ7341L/7@$5Q5ID$A Spec(Ox) /0 Spec(Oy) >"Y0I/7@D$5QZ8AU14=T8:"/
1L@K[K/7AB5/\;$/ G N]^6_8Z$AI`75a3b8cZ$AI>Z>5I140I1L>@?degfebheihj.3L/ 1 Vk=?>01Ml ;$/mD$=nlo>"AB;n85I5> E71LJqp Gx14@$;$D$140UD$@$/
r `7EsR$/ Spec(Ox)→Mσx
/0<E7/734D$1t855I>E714JFp GyD$@$/ r `7EsR$/ Spec(Oy)→Mσy
N u#/E7v@$/ σx /75B0;$J0/7AB=?14@$J2Z8A B(Gx)
Ew8AxE6y/75B0\3z69D$@$14CD$/2E7v@$/2=?14@$1L=?83Z8AB=?1E7/7D?;$>@ 0x3g698;$R$J7AB/7@E7/<E7>"@ 0I1L/7@ 03L/75 pD B(Gx)
j>"| D /75I02D$@;$14[1L5I/7D$A<1LABAIJ7;$D$E0I1LY$34/F;$/ Spec(Ox) N~X3#/7@/75B02;$/=?7=/FZ>"D$A σy/0qE7>=?=/ B(Gx)|Oz' B(Gy)|Oz
, 3L/75E7v"@$/75 σx /0 σy 5I>@ 0qE7>@HGID$:D$J755>D$53g698E0I1L>@;$/ GL(X)/02143#/1L5B0/D$@145I>"=>"ABZ$R$1L5I=?/ Mσx
∼→Mσy
N O @>Y014/7@ 02D$@;$1L8:A8=?=/E7>"==PD080I14l
Spec(Ox) //Mσx
∼
Spec(Oz)
88qqqqqqqqqqq
&&MMMMMMMMMMM
Spec(Oy) //Mσy
u#/75F8@$@$/w8D Ox /0 Oy 5>@ 034/75E7>"=Z$3LJ0IJ75;$/7534>Ew8341L5IJ75;$/ Mσx
/0 Mσy
/7@34/751L=?8:/75;$DZ>"14@ 0<l*/7AI=J;$/ Spec(Oz) j$;$>"@$E143#/145B0/D$@1L5I>=?>AIZ$R$145I=?/
φ : Spec(Ox) ∼−→ Spec(Oy) .u6_8E014>"@;$/ GL(X) 14@$;$D$140D$@1L5I>"=>"ABZ$R$1L5I=//7@ 0AB/3L/75 1 Vk=>0I14l*5;$/mD$=l*>"AB;D$@$1M[K/7AB5I/73L55ID$A Mσx
/0 Mσy
j\/0143c/145B0/D$@145I>"=?>AIZ$R$145I=?/ Gx∼→ φ∗Gy
5D$A Spec(Ox) CD$1c;$/75IE7/7@$;3g6y1L5I>=?>AIZ$R$145I=?/ Gx
∼→ Gy5ID$A Spec(Oz) N >D$A;$J7=?>@ 0AB/7A3b8Z$AB>"Z>"5I1M014>"@j\>@E7>@$5I0IAID$1M0D$@$/ABJ0AI8E0I1L>@;$/3g6y1L@HGB/7E70I1L>@ Ox → Oz Nu69145I>"=>"ABZ$R$1L5I=?/ Gx
∼→ φ∗Gy5ID$A Spec(Ox) 1L@$;$D$1M02D$@;$1L8:A8=?=?/E7>=?=^D080I14l
R //
R
U × U // U × U
Spec(Frac(Ox))
@@
//
OO
Spec(Ox)(Id,φ)
OO
u6_8@$@$/w8D R /75B0@$>"AB=T83T/03L8 r `7EsR$/ Spec(Frac(Ox)) → R 5769J0I/7@$;/7@D$@=?>AIZ$R$145I=?/Spec(Ox) → R. /0B0/ r `7EsR$/5I/lz8E0I>"AB1L5I/Z8A?3L/E7>=?Z$34J0J;$/ R /7@ z /0T14@$;$D$140?Y$1L/7@D$@$/AIJ0IA8E014>"@;$/ Ox → Oz N
T"Uk ¡ Na¢ £2>"D$5F8w[K>@$5FD0I1L341L5IJZ>D$A3L8Z$AB/w=1L`7AB/nlo>1L53L/nlz8140CD$/3L/75E7v"@$/75;$/ Σ@$/5I/AI/7E7>D[AI/7@K02Z85wN
!"#$%&'()"
*,+.-/-0+2135437698;:=<>@?ABDCFE.GHGJIFKMLNI O%IO$PQSR TJUWVXZY[;\]M^`_badcfe]R ⇒ U
]dghiYkjml)ndXYoXZpqV]r h5UZ\]st ru XjghDnvU)hDwxXjsy z5k|v~FfvM
R×UR→ R k;fdR×UR→
R; .v|Z.| v¡)¢xQ£.|fd¤
R×UR|¥;f|v
U ¦ 7|v )%§¨¨.¥¤)7|f)v7©
R)|
R×U RQªJ«M
R×U R→ R¡¥7¬7¨¡v¡v;f
.v|v7®®7v7|f)v7¯;b|bZ.||fv|¡|ff®U
)|U
Q
ªJNMR → U × U
|¥° Ag|¥¤|v¡)v ¦ ;7¯.¥¡|v
R → U × U|¥
)7)U
|¥bk7Q
± ID²IO%I´³©µ¶E.HZ· ¸º¹»·¼G$½¿¾ºµ¶ÀE.G$Á¸SÃE¹»· ÄE.ù· G$½S¶À¶Ã½S¶0½S· ŵÃÆ,IÈÇ £È|fv|´7°M)¡ .É¥ v7
AΣ
g = [U/R] .z5|¥|f¡)v ¦ 7|v|f®|vSpec(Z) ¦ f Ag
.®Êb|v®;
∂AΣ
g = AΣ
g −Ag.k|v˧v7|vf|Ëvk%QJª|vM¯|vÍÌ5É$7
G → U|¥«¨
.®®|vvpr∗1G
∼→ pr∗2G|v
R.®b¯|vM|vÌ5)É7
G −→ AΣ
g
;f)¥7;vËÉ$7®¥|v7Qª¡¡7Zf7¯ËÎbfÌÊÏ¡$¯|v)Ìv¡|v
Sym2(ΩG)∼−→ Ω1
AΣ
g /Z(log ∂AΣ
g ) .
ªJ°|¥vfZf)v7.Ðv¤|v AΣ
g
¡)«|«Éf)M|«Ñk|f«§°¥.|vv|«.¥¤)/|¥¡v;f¥¡Èq|v¨É7Ò
Σ/GL(X)QªJ¨.¡7;v7iÓ¼v7Ò7. 7|¥vfZf#)|v|v)Ì
7§σ ∈ Σ
|¥¬|v.v¡ )f¬Ò¨¡v7´Ó¼.¥7Ò Mσ
. Zσ
¡)7# .v¡$#
Γσ = StabGL(Xσ)(σ) .ª|vM¤´|fÌ5)É$7G
.¥v|v¡$.N¡);#|v.v¡|v@) v7¤Ô¯@Ó¼.¥Gσ
1ÌÕvÓ
Mσ → Mσ ¦È vvN§Ñ.¡$.|vi¡)iDv..¥¡7|vNÖ¥.¡7;f.×Ö7¡v¡$.|vf.©ØI ± IÙºIq²QÚÒ¤v..b7;vfvbz5Û¨QÍÜQÍÝ#«:<¬>@?AB$¡$,¡7|,;d)7|QÏ.
(U → S)ÀÉÞv;
OuvQª¤) .©7v|v©
G;ÀÓ .;v@
(Ag →AΣ
g )(U → S).v|7Zf .¥7®||vM¤)||fÌ5)É$7|b|v
SQ
ߨàáâJáãZä5åÊäváæç1354$3Õè%3Õ4QSR é]êXZj%ch5]Ykn®n r U)\Íwg»]®Ykj%] rfë Ykw ìZU)\]jcd]V]cfU)h r lXZnMwx]dg@]MjhDnf](Ag → AΣ
g
)(U → S)
]hJ\7UcfU)h r lXZnMwx]V]dgÈg»cde ru UgÈg»] u w ^¥U.[ r \Íwx]jkgG′ → S
ë Ykw$g»XZjhFU.[ r \Íwx]Mjgo$nMw j%cwíoU)\] u ]jhoXZ\7UZnwg r gV]Òl]jnf]
ggYkn
U î ]h,h5]\íg ë Y]®\X»cfUZ\] u ]MjhoXYkn®\UhïXdoX\Xfl)wx] r h5U)\]«V]S î \]FêU)wg»cf]fU)Y
X(G′)]fgh
!"#$%#&'(*)+ ,.-
/0 1.243657298:5 X 52;3*<=5?>3A@2B5 σ ∈ Σ 8 1C72;<ED F:8.GHJIK57LK5MLK1C7$2436572;<NFPOQ1CIJRM5TS3*<U3*7HF3VIK5 pD B(G′)W1C0I2B1C02X83*YZ3A@Z5J0I3*IJIKH8C0 L2434S<A5[IH803*2 D 8 5 S \] [email protected] \=^ _a`[bcKd$egfThThiegjNkl`gm G′ n$o klm$cgk n hjN`gfp`gm q nr jAh o klbe[bs o U klm$t$s$klqus$m$`[vwgcJd$`U → Ag x s$`[j4y r mz"`gsq n$o?r j r m$|` og ~ mez"`gm qs$`gj n$oEr j r m$|"`gfp`gm q`gbEqs$m$k x s$`[cQh o Ag `gb?qbEe n h o e`qMb r m`kNb?qE`gm$cg`bE`qE`gb?q`j r cQhjl`Qfp`gm q X m n `gsqMt r m$cbEs n$nr bE` o x s$` S `gbEqTjN`b n `gcq o `tys$mhm$m$`Qhsj r cQhjc r f n jN`q!t$` nr klm qP` o fpe S0
mcKd r klbkq σ ∈ Σ fklm$kNfhj n h o fkjN`gb!cgm$`gb!t r m:qj4yht$d$e o `gm$cg`;c r m:qkN`gm:q=jl`gb B(G′)⊗pD r" D n h o c r s o qjN`gbt$kzkNbE`gs o b=t$` S "`q=jy r m o `gjNwz"`;jAhPvwgcJd$`U → Ag `gms$m$`!vwgcKd$` U →Met
σ x s$kbgyeqE`gm$t`gmS −→Met
σ
| oE cg`[jAh n$oErn$o kleqef r t$s$jAhk o ` n$oEr cJd$`t$`jAh oEr m qEkNw o `[t$` Met
σ
m`gmt$egt$s$kq x s$`jl`cKdhf n AΣ
g m$`t$e n `gm$tm$k#t$scJd r kt$`jNh n$o egbE`gm qhqEk r m U t$` Bσ m$kt$scJd r kt$`gbPh n$n$o?r kNfhqEk r m$b Get
σ j o `gb?q`Tf r m q o ` o jNh n$oErnr bEklqEk r mbEs$klzhm:q` nr s o z:e o k` o x s$`j4y r mhc r m$b?q o s$kqs$m$`c r f n hcgqEklcQhCqk r mt$` Ag
.a$C ¢¡£ E$¤¦¥§¢¨§ª©§« ¬ ® 5LKG$FCR9W¯FC<° HQSIJ34/05 AΣ
g5K@2aW IK1KW IK5P@0I Spec(Z) \
] [email protected] \^ _ah±vwgcJd$`²`gb?qbEe n h o eg`²t$`qB³ n `m$k9`qjy r m n `gsqh n$n jNk x s$` o jN`²c o klqEw o `zhjlshqEklt$` n$oErn$o `qEe!t$`p´µ·¶¸º¹4¹» ¼l½Z¾C¿6 Àr kl`gm q S s$mq o hklqt$` nr klm q|"egm$e o k x s$` η `q η → Ags$mf roEn d$klbEf` mz:`gsq;jN` n$oEr j r m$|"` o `gms$m$`!vwgcJd$`
S −→ AΣ
g
j r cQhjl`gf`gm q nr s o jNhMq rnr j r |"kl` O6W:W.O aÀr kq G′ jNhMzh o kleqehiegjNkl`gm$m$`bEs o η klm$t$s$klqE` n h o η → Ag _#`qd$e ro wgf`t$` o egt$s$cqEk r mÁbE`gfpkªb?qhi$jl` nr s o jN`gbPzh o kleqegbhiegjlkN`gm$m$`gbhbEbEs o `j4yÃ`klbEqE`gm$cg`t#yÃs$mbEcJd$egfTh!bE`gfpk¢hiegjlkN`gm G′ → S n$o?r j r m$|`Qhm:q G′ → η x s$klq?q`;`Ä`gcqEs$` o s$mpcJdhm$|`gf`gm q o hfpklet$`q o hklq ~ mcJdhm$|"`gfp`gm q!eqKhjN`t$`q o hkq n ` o f`q!t$`bs n$nr bE` o x s$` X(G) `gb?q x s r qkl`gm q!t$` X ~ mMq o hklqXht$fp`qÅsmMs$m$k x s$`9t$kzkNbE`gs o `Ä`gcqk o ets$kqXk oEo egt$s$cqkli$jN`jAh[vwgcJd$` B(G) : X ⊗X → Z`gb?qbE`gfkƪt$em$kN` nr bEklqEklz"`Ç o ht$kNcQhj o hCqk r m$m$`gj=tyh n$o wgbjNh n$oErnr bEklqEk r mÉÈÊAËÊVÌÊȲ`q Σ o `gc r sz o `jN`cgm$`t$`gbu r"o f`gb9bE`gfkƪt$em$kN`gb nr bEklqEklz"`gb! o ht$klcQhj o hqEk r m$m$`gj _ah n$o?r"n$o kNeqEef r t$s$jNhk o `MjNh o?r m qEkNw o `t$` (
Ag → AΣ
g
)
f r m q o `[hj ro b;j4yÃ`klbEqE`gm$cg`tys$m n$o?r j r m$|`gf`gm q
ÍTÎϯÐZÑ"ÒÎÁ¥§ª¨§ª©§4¥ ¬ _#`Tc r m q o jN`Tt$`TjyÓh nn$o?r klfThqEk r m¯heqEeMklm$t$kNb n `gm$bhi$jl` nr s o z"e o k` ojAh n$oErn$o `qEet$` AΣ
g
!"#$%&'()*#
+&,.-,/&,0213547698;:=<>3@?AB8DCE?F8DA<G4D<G:DH!I%<18J,LK MON*PQSRTOUN*VWN*XYVZNEU[TE\]EP_^`@aPb9cSdNeR]YfgUNERhcWbiaRN*cZbj]EPkNFlYTRmPbilTN*ngUNEVZNE[]YVobjpnTqN*QZQZ] Rbj`Tmr
Σ s t PkNEQZQZ] RbjTmr$Ru[NEp nTRNEVvcWTd9] RNEd9TqN*djXE`wVZb9p nTQWNEPQmPbilTN*n
Met
σMet
σoo // AgdN5RNEVvcWTxd9] RN*djTyN*djXE`wVobjpnTx^TxPb9lTN*nzUNEVZNE[]EVZb9p nTx^T$c|U%T D TP
p
Met
σ, 0
Metσ, 0
oo //
Ag, 0
Met
σMet
σoo // Ag
p nbdjnbR]YVoVZTQZU%]YP^ s%~ ]EPQWb9^`VZ]EPQmdjTQmQZR[`fN*Q
U0 =∐
σ∈Σ
Met
σ, 0 et U0 =∐
σ∈Σ
Metσ, 0 .
djQmb9P^nbjQZTPcmnPG^bjNEXEVWN*fkfgTxRNEVvcW`QZb9TPGr5R[TQlYTVocZbjRNEd9TQmUVZ]EUVZTQ
U0
U0oo //
Ag, 0
U Uoo // Ag ,UNEVR]YPQocZVZnR@cZbj]EPz^TQqRN*VocZTQmdj] RN*djTQqN*djXE`wVZb9p nTQqNFlYTRxPb9lYTNEn s MO`@aPb9QZQW]EPQmdjTxXEVZ]EnU]*^T
R0 ⇒ U0R]YfgfkTxd9TxUVZ]^nbicawVo`R0 = U0 ×Ag, 0
U0 . TQ^TnR[TQR0 ⇒ U0
QW]EPc`@cWNEdjTQqRNEVU → Ag
TQoc`@cuN*djTE\T@cd9T2Ru[N*fkUGp n]*cWb9TPc[U0/R0]TQocqwb9TPzQZVb9QW]Efk]EVZU[Tyr Ag, 0 s
EFY% s; ]YnV_^`fg]YPcZVZTVp nTU0 → Ag, 0
TQoc`@cuN*djTE\L]YPPN UN*QUVo] R`^`^bjVoTR@cWTfgTPc s MOn ¡¢N*b9cg^T_dN£fkNEnl*N*bjQZTVo`^nR@cWb9]YP\¤b9dSQZTf¥wd9T_^b9¦kRbjd9T^§NE^NEUcZTV5djN¨cWTR[Pb9p nT^Tk©§ªS05«¬.\wN*QW`TxQZnVd.®bjQZ]Yfg]YVoU[bjQZfgTx^Ty¯] ^NEb9VWNhv° U%TPRTV s+&,.-, ±,³²z´ABµ¶1·¸H ?F¹,LK MONEPQqRTxUN*VWN*XYVZNEU[TE\P]YnQm^`fk]EPcZVZ]EPQqdNgP]EVZfkNEd9b9cZ`y^T Mσ, 0
\p nbQZTVWNqncWb9djT7U]EnVDdNqUVZ]EU]EQZb9cZbj]EPg+&,.-,º&,.- sB» ]YnQ;ncZbjd9bjQZ]EPQ;djTQ;Vo`QZnd9cWN*cZQ;^T¼©¾½¿%AB¬+\QZTdj]EPed9TQWpnTdjQdjT$fk]YVoU[bjQZfgT Ag, 0 −→ Spec(Zp)
TQocmUdNBcqT@cqQWN¥awVZTxQZU%`RbN*djTxTQocmVZ`^nb9cZT sÀeÁEÂ=ÁÃBÄ|ÅvÄZÁÆÇ|%vÈ s; ÉDÊ¥ËuÌÍBμÏÍBЧÑÒÓ@ÔÕ.Ö@×Ê Ag, 0 ÊuØÙÛÚÜBÔÎÍ*ÐjÝÞ ÒγÜÚØÙ.ÔZÍ*Ù.ÕÜÚÝÛß M¥§NEUVoQ¼d9T2RVZbicWVZT2^T2P]YVofN*djbicW`2^T5° TVZVoTE\%b9dDQZn¦kc¼^T2fg]YPcZVZTVqp nT Ag, 0TQoc7Vo`XYnd9bjTVàTPkR] ^b9fkTPQZbj]EP
1T@cJp nTmQZTQJU%]Yb9PcWQJ^TR] ^bjfgTPQWb9]YP ≥ 2
QZ]YPcJ^TmUVo]E¡¸]EP^TnV ≥ 2 sMeN*UVZQ³d9TGcZ[`]YVofkTGUVZb9PRbjUN*dq^T©á½²G¬â.\7d9T>djb9TnãVo`XYnd9bjTV^T Ag, 0 R]EPRb9^T¨NlTRGd9T>d9bjTn
!"#$%#&'(*)+ ,.-
/"021$354635027879:;/4=<>/?A@$B5C>?D7>4 963E0;7F164$: Ag, 0 7>:291$7G<>/1$35?D7>4$:;35/4 ≥ 2 HI 7>B56J?A/4 9027FKL$7 Ag, 07>:29M02C>N"L$BE357>07>4O<>/1$35?D7>4$:;35/4 1 HP C>0;3RQ/"4$:=BS6T1$7>LU35V>?D7W<>/"4$1$3R93E/"4X1$LX<>023E9V>027W1$7W4$/"02?Y6B53R9CW1$7[Z7>020;7 H Z/357>4 9 A B]\^64$4$7_6LB5/ <_6B`1$7 Ag, 0 7>4aL$4a@/"3E4 9 x 1$7D@$02/bc/4$1$7>L$0 ≥ 2 79
m:;/4a351$C_6Bd?A6.U35?A6B HeI /"?D?A7 Ag, 0 →
Spec(Z[1/p]) 7>:29fBE35:;:;78/"4a@7>L9f:;L$@$@/":;7>0!KL$7 p ∈ m H#g \^64$4$7h6L A 7>:29@$BS6.9f:;L$0 Z(p)1$/4$< p4\i7>:29j@6:j1$3Ek3E:;7>L$0j1$7 0 164$: A Hl /9;/"4$: A = A/pA 79
m = m/m ∩ pA H m \^6@$0;V>:fnporqs.tuvw8 A7>:29x0;C>1$L$3R9y1$/4$<G:64$:z<>/"?D@/:64 9;7G3E?A?D7>0;N"C7 Hg 7>:x3E1$C_6LU6:;:;/<>3EC>:x1$7 A :;/"4 9x1$/4$<G?A3E4$35?A6LU H| 47>41$C>1$L$3E9~KL$7m
4\i7>:29O@6:O6:;:;/<>35C Hm \i6@$02V>:OB56@$02/"@/":;3R93E/"41$L@606N"0;6@$$7 P HE"H1$7!n`"v835B7U3E:297 x ∈ mKL$34\7>:;9`@6:`1$3Rk35:;7>L$01$7 0 164$: A Hg 6G:;L$3E9;7y1$7y@606?AV9;0;7>: (p, x)7>:29M$3E7>4O1$7B5/4$N"L$7>L$0 2 H
Z/3E9 σ ∈ Σ Hg 7<>/"02/"BEBS635027:;L$3Rk64 90;C>:;L$BR971$7BS6@$0;/@/":;3E9;35/4u]81$7>:<>/0;/BRBS635027>:fE]E79f R*Su8$79M1$7BS64$/"02?Y6B53R9C1$7>:j@$BE/"4$N7>?A7>4 9;:9/"0235K L$7>: H! $¡¢ $£¤£]¥$¦¤¡¢§¨©¤ª©¬« ©] He® ¯°±²$³´¶µT³·i¸º¹_»¼¾½¿À ° Mσ, 0 °ÂÁÃzÄÅ.¼¾´Y³·5ÆuSÇYÈÉÊ*ËÌ¢Í*ÎÐÏÒÑEÓÔeÍÕË Ê*ÉÎ×Ö"ÉË ÕÉÖYÎ×Í*ÕÉ Ë ÒxØ m 64$:Ò<>7M@60;6N06@$$78º/"4A<>/"4$:2902L$3E9ÙL$4$70;7>B569;35/4A1\C>KL$3Rk6B57>4$<>7C9Â6B57j:;L$0 U0 Hºl /9;/"4$: R0
B574$/"02?Y6B535:;C1$7 U0×U0164$: R0 H ¤B$7>:29Ò?JL$4$3
1$71$7>LU=ÚV><¾$7>:R0 ⇒ U0
KL$3C97>4$1$7>4 9R0 ⇒ U0 .
g 6!0;7>:29;0;3E<93E/"4A1$LA:;<¾$C>?A6:;7>?D3R¤6C>BE357>4 G Û U07>:29Ù6C>B53E7>48º?JL$4$3$1\iL$4A1$0;6@7_6L H• ⊂ G[p] H| 4O1$35:;@/":;7:;L$0 R0
1\L$4O35:;/"?D/"02@$$35:;?D7pr∗1 G
∼−→ pr∗2 G
KL$3e7>4 k"/"3E7 pr∗1 H•:;L$0 pr∗2 H• HÜ 60G4$/"0;?A6BE3E9;C8<>79F3E:/?A/0;@$$3E:;?A7:;7@$02/"BE/"4$N77>4WL$4W35:;/?A/0Ý
@$$35:;?D7:;L$0 R0 HÞ ¡¢ ß $à.¦¤áݦ; $â㨩¤ª©wä©åª He® ¯×°Áæ °¾ÀçGèMé±Â²°ÂÁ R0 ⇒ U0 ÁhÅ¢ÄÃj¹Ã¤³.·S°ÂÁ_Æê ¹´Å¢ÄÁü;³Ã½wÅ¢ÄÆÒë Z/3E9 z L$4O@/354º9M1$7 R0
79 x 8 y :;7>:j1$7>LU~3E?Y6N"7>:j164$: U0 Hì /"4 9;0;/4$:jKL$7B57>:Ù<>/"?D@$B5C9;C>: Oz 8 Ox 79 Oy 1$7>:zBE/<_6BE35:;C>:Ù1$7 R0
7>4 z 79Ù1$7 U07>4 x 79 y :;/"4º9z35:;/?A/0;@$$7>: H | 4@7>L9M:;L$@$@/":;7>0jK L$7 x 79 y :;/"4 9j1$7>:jC>B5C>?D7>4 9:j1$LO/"021 H ¤B#7U3E:2971$7>:j<>í"4$7>: σx 79 σy 1$7 Σ 9;7>B5:
KL$7 x 6@$@6029;357>4$4$7 Û B56 σx å:29;06.9779 y :;L$0jB56 σy å:290;69;7 H | 4O1$3E:;@/:;71\i3E4¢î;7><93E/"4$:j1\i64$4$7_6LUOz
Ox.
>>||||||||Oy
P0
``BBBBBBBB
!"#$%&'()"*,+.-*0/214357698:/./2*76<;>=?8)@57AB;C*7D/ Gx
*,+ Gy E @+2*7DF/.G8)H.;B6I8)JK*0;BDLK*7H2/2*0-* G /MFH Spec(Ox)*,+ Spec(Ox) NPO AC/I/ E D+IJK57D57H2;BQF*76<*7D+R8:@$57AB;C*7D/7SP6TFD;C/I-UVFD*W/X+MHXF1,+MFHX*Y-*WD;BLK*8:FZG8)HM8)3 E =H2;BQF* H•, x
*,+ H•, yS*,+[;CA*\;B/2+2*]/2FH Spec(Oz) FD^;C/ E 6 E H2G3;B/M6<*
pr∗1 Gx∼−→ pr∗2 Gy
QF;_H2*7/2G$*71,+M*`AB*7//X+MHXF1,+MFHX*7/-*`D;BLK*8:F Na *7/^8)DD*8:Fb\ Ox *,+ Oy / E D+D E HX6I8)Fb\c-UV8:GHXd7/AC*1 E H E ABA8);CHX*Re%fhgf if j Nlk UV8:GH2d7/mAC8^GH E G E /2;B+M; E DZgfe%f ifngS E D E @+M;B*7D+m-*7Fb\ 1 =o6 E +2;Bp /m-*RqWF6rp E HX-8L*71m/X+2H2F1,+MFHX*m-*mD;BLK*8:FY8)/2/ E 17;C57/ts Gx
*7+ GyS*,+uA8<GH E G E /2;B+2; E Dvj%fhgf>wlfngx6 E D+2H2*mAnUV*\;B/X+M*7D17*
-*#yd7143*7/Spec(Ox) −→Mσx, 0 et Spec(Oy) −→Mσx, 0G8:HzAC*7/2QF*ABAC*7/r-*7/217*7D-*7D+rAC*7/r-*7Fb\ 1 =o6 E +M;>p/<8L*71|/X+MHXF1,+MFHX*|-*|D;BLK*8:F N~ DHM8);C/ E DD*|*\8:1=
+M*76<*7D+1 E 6<69*I-8)D/A8^GH2*7FLK*I-*IAC8WGH E G E /2;>+M; E De%fngf jG E FHm1 E D/X+2H2F;BH2*IFD*9H25,+2HM8:1,+2; E D-*AhU;CDX*71,+M; E D Ox → Oz N
DL E ;>+u1 E 696<*]GH2571757-*76<69*7D+[QF* R0 ⇒ U0*7/X+[FDWJ:H E FG E: -*]5,+M8:AB* N GG$*7A E D/
AΣ
g, 0 = [ U0/R0 ]
AC*_138:6<G8:ABJK57@HX;CQF*_QF E +M;B*7D+ N \bGAC;BQF E D/H28:G;B-*769*7D+A8TGH E GH2;B5,+M5_6 E -FA8);CHX*#szAC8TpH E D+M;Bd7H2*L57HX;B57*]G8)H
(Ag, 0 → AΣ
g, 0
).
E ;B*7D+ (U → S) FD E @2*,+<-* Ouv *,+ (G′, H ′•)
FD E @2*,+<-* Ag, 0(U) NP D/MFGG E /M*|QF* G′
/7UV5,+2*7D-R*7DRFDR/214357698T/2*769;=?8:@$57AB;C*7DR/2FH S Na *QF E +2;C*7D+ G′/H ′i
*7/X+8)A E H2/HX*7GH257/2*7D+M8:@AB*G8:HFD/2143576I8x/2*769;=?8:@$57AC;B*7D`/2FH S G E FH 0 ≤ i ≤ s N FGG E / E D/#QF* X(G′) / E ;B+#FDvQF E +M;B*7D+#-* X Nk U8)GH2d7/AC8TGH E G E /2;B+2; E D`gfBwf%f jR*,+AB*_-;8)JKH28:6<69*<gf>wlf NV# SbA8pn8)69;BACAB*_-*7/p E HX69*7/@;BAC;BD58:;BH2*7/B(G′/H ′
i)-5,D;B+uFDW6 E HXG3;B/269* B(G′, H ′
•)-* Sym2(X ⊗Q) -8:D/ DivS ⊗Q N$a 8<GH E GH2;B5,+M5
6 E -FA8);CHX*rsRAC8xpH E D+M;Bd7H2*z/2F;BL)8)D+M*z/2*z6 E D+2H2*z1 E 696<*zA8xGH E G E /2;B+2; E Defngf %fngS%*7DvF+2;CAB;C/M8:D+#AC8GH E G E /2;>+M; E D.j%fhgf>wfhg NT:?2¡ ¢?£$¢¥¤¢¦ N¨§ ©lª«)¬M®¯°h±ª (Ag, 0 → AΣ
g, 0
)(U → S) ¯²)¬2²)³I´µh¬Mª·¶ª¸¹Kºn»ª4µ¸
(G′, H ′•) ±ª Ag, 0(U) µ¼ª4¶¸z½®ª G′ ¸¾À¿µ¼ª4Á±|ªÁ.®bÁW¸ÃÂÄ¿³9²r¸Ãª³rÅ ÆX²º4¿¶ÅhªÁW¸®b¬ S ǪµPµ¼ª¶¸T½,®ª]¶ÃÂ2²)Æ
¶ª³IªÁµl¯®b¬¶B²Rµ¼¯¶M«Å¥ª9¿µo²)¶ªr±ª S ǶªPÈ7²)ŸÃª2²)® X(G) ª¸µ[½,®µhÅ¥ªÁµ[±ª X ª4µÅ ¶ÉªXÊKŸ4µ¼ª σ ∈ Σ
±ÁµP¶2¾ ²±Ä¿¬MªÁ%ªzÂÁµhÅ¥ªÁµP¶C²[Ȭ4³xªTº,Å ¶ËÅ Á%¿M²)Å ¬ª pD B(G′, H ′•) ¯®b¬]µ¼®bµ±)Å ÌÃÅC¸Ãª®b¬#Å ¬4¬¿M±®ÂµhÅhº¶ª
¬M¿M±)®bÅ µ D ±ª S Ía *#+235 E HXd769*]/2F;BL)8)D+uHX57/2F69*]AB*7/uGHX*769;Bd7H2*7/[GH E GHX;C5,+257/u-*]D E +MHX*]1 E D/X+MHXF1,+M; E D N
ÎPÏlÐ Ñ:ÒÔÓÕ ¢o£¢¥¤¢h N¨§ ©ªTÂIJ)³t¯²)¶V«¿7º,¬Ån½,®ª AΣ
g, 0 ÁªT±¿?¯ªÁ$±R½®ª±ª Σ Í%Ö¶lª¸µÉ¸Ã¿?¯²)¬M¿Ç×Á%¬4ƳI²¶ªµØ¸7²¯$¬M¼»ªÂ4µhÅ¥Á^Ì:ª¬M¸ Spec(Z) ª¸4µ±ª_µnÙ4¯ªØÚ×ÁÅǯ¶C²)µ¼ªÇ~ÛÚº¬ª¸[«¿³I¿µh¬4Åh½,®ª4³xªÁµ×¬¿2±)®bÅ µ¼ªM¸Í
!"#$% &('
)+*-,/.0!1324,/56798:*67 Ag, 0,/5;1<1=*<5;>?@*A7BC*6%D*<*7+8EF*HG8D7I*J>K6LB;890M!AN0!1<1=*<,O0;A7IPOD8:*6
Ag, 0 //
AΣ
g, 0
Ag // AΣ
g
QSRUT ,O.*/DV?@*A798:,N0!E*/DW2AO5/2AO*/D(X#Y+6<20!A798:,>KEZ8:*AO[ AΣ
g, 0
*/D72\A/5/2\A/*UD>KASpec(Z)
*7+*/D7]>K6#*^,O5;1V20_,`798 aS,O0!798:56LBC* Ag, 0
Xb/EF*cG@8D7I*J>K6dD(,/.P1e0fD*1<8`c0gPEZ8:*6G
*7W>K6hB!AO0/2*O0!>iBC*VM!A/5>2*/Dkjl>0D8`man68D2Eo0;7pD
H• ⊂ G[p]D>KA AΣ
g, 0
jl>8SP7I*6#BC*67+E*D(,/.P1e0q0_gPEZ8:*6L*7+E*kB!AO0/2*O0!>r>K68?!*AOD*ED>A Ag, 0X
s P1t56%D79AN0!798:56#Xu vwyxnz!xNxOwyH|O~xxN~@C|wyx]y~xNywyywyxefyoz!|Nwyxwozy~@xc|Ocl|O~vz N~ ~xN|O~r\te~@C|OcwVw3owyxSy~@ z!l|O(z!|No~@xSwV wywyC|SwVw Σ
Kvz z;|O|Owwl|JozqNyl|No~@hwyxNwyxwyxJy~@ z!l|O(z!|No~@x c~owywyC|JwyxNyxN|/z;|Oxwrmq\;#%9vwOz w(z@Lxy~@%|OwyC| z@z@ yNwyyweLOz w(z!¡_wycxNwynxN Ag, 0
z@xowe¢c~ w=Cz@xNZ£:\ z;| G[p]
vw z! AΣ
g, 0
wyxc|3k¤w^xc|NOz;|Oz;|O~ z@Nz@fl|Ocyw z@U¥~!|OwyC|Uw Σwl|3oz
y~ ol|No~@d¦p~cewyowkw-~¢§wkoztxc|ONz!|Nw-z@xNxN~yoyw-¨σ ∈ Σ
wyxc|oxN~@e~@ w-z@i~!|OwyC|wkzy~ ol|No~@h¦~cewyowfw Mσ, 0
wfo~¢rwZσ, 0
z!JL¢@N~ w<9vwexN/ y=z©xNwyeZ£ªz@yowyG
wl|^ow=Oz w(z! H•
y~@Ncwyx ~@wyC| z!^ywl|koxN~@e~@ xOfwtz@¡~!|OwyC|kwq«¥<¦p~@N¡1£
e~!|O¦Mσ
k\wUxOzJxc|Ocl|OcwUwU_w(z!=(z@~@owUxN Mσ, 0
C¬z!xWow z!Oz!¢Nz wUxN@z!C|y~ cyyoxNwJozfy~@-z!|N~NwJr~crwJz<y~ z@l|N(z;|O~
q®^¯#°L±³²\´@µ¶·]´²%µ¶p¯¸º¹f²°L»¼¶¯\»C½K¹¬VwyK¾ c~¿yewyxÀxNxNoxc|Owy_|(vw cwyewyÀwyxc|ÀwJy(cocw NyyxNyewyC|ÀowJ~@Nrw
AΣ
g, 0
wl|e~@|Oz@efwyC|eowyxecwyz;|O~xeyowyywrwy_|OcwrowyxeÁycwyC|Owyxexc|NOz;|OwyxyWvwrxNwyy~@Âwyxc|=wrxNwyz@cOz!xNxNwy©wyxrxN¢oz@c|Nyx©|N~cowyx©o_|OwyH_wyz@_|rz@x©z4y~ z@l|N(z!|No~@ ~@©~@%|Owyà z! Äz(ÅCz!C|=w¥wyx=xNo¢@z!N|Oyx=wyx Ag−r, 0 ~@ r ≤ g
Æ+w¥/ z@ xNwyOzhoxNxOwrxNSpec(Z[1/p])
wl|ÀowJy~ oyfwyC|Oz@NwJw AgŧxOwyNzfrKoxNwyÀ¨fyc~xOwyfwyC|NxU~@Nez@K¾#
Æ+wyx c~¿(fwyx+xNw3cyxN~_wyC|+xNok|/z!yewyC|+wytC|NN~xOz@C|+Ç,/5;1V2\E*cGK*,O5;689jl>* C y~_|Ow£z@_|C(X)
!vzUyy~ ~xN|Oo~@ ΣwC(X)
o|FwWyy~ ~@xN|No~@SΣ
w C ywWCy~|¨<wJwyxNyc |No~@rw¾ oy|NwJr#~@Nrw
AΣ
g, 0 .
!"#$%&'()*#+-,/.%0213-4256*78029:02,3-;<029:.76*=202;>780?=A@BDCFEHG0
Σ.6*;>1,/[email protected],/GOP1,0?G42=2BD9:.BIL<CQ3<C8BI,
SG0 C R =20TS 1CD.%02;<9:03UG0T=2BI,LV3W;V1C8;V0TG02LX=2BD9:.6I=3<CQM=6Y3WCQBD,LU6*9Z4278CQBD;V4202L R LW6*,LXL<C8,5D1786I;VCQ3<4TGO[BD;VC85IC8,03WBI;<CQS 10I\]&^ _&^a`^bc%dfeg<hie2jfklDh enmocmkp!h qrhismutae2cmvcjTh eI^xw +-,/C8,3<;<BG1CQ3yG02L-,B*3z6Y3WCQBD,Ly6I,6*78BI5D102L =202787Q02LTG1:.6I;<6I5I;W6*.@0 `^ _&^| \ BDCJ3
V = ⊕2gj=1Z ·xj
1,Z ~ 9:B G178078CQ;<0G0;<6I,5
2g91,CGOP1,0
6IL<0(xj)
\+-,7Q091,CJ3NG0786!BD;V9Z06*7Q3W02;V,420GBD,,420.6*;786:9Z6*3<;<CQ=20
J =
(0 J ′
−J ′ 0
),
BDJ ′
02L<376-9n6Y3W;VC8=206I,3WC ~ GC6*5DBI,6I7Q0GBI,37802L=2B 0n=2CQ02,3<L,BI,, 178LL<BI,3425D6I1iE 1\+-,:;<6I..%02787Q0
S 10 D = d1 < · · · < ds\+-,:G4M,CQ31,0=A@6Y,0.6*;W6I@BI;<CQS 106*13WB G16*780
V•D
G0V
02,:.%BDLW6*,3 R.%BD1;1 ≤ i ≤ s R
ViD =
di⊕
j=1
Z · xj ⊕2g⊕
j=di+1
p Z · xj
et Vs+iD =
2g−ds+1−i⊕
j=1
Z · xj ⊕2g⊕
j=2g−ds+1−i+1
p Z .
B*3WBI,LΓ0
780-LV3z6*C87QC8LW6Y3W021;xG0V•
D
G6I,LGSp2g(Z) =IOP02LV37Q0-LWBI1L ~ 5D;VBD1.%0-G0y=2BI,5D;V102,=20-.6I;W6 ~@BD;VC8S106*LWL<B =2CQ4 D \* C V ′
02L<3U1,LWBI1L ~ 02L<.6I=20T3<BI3W6I7Q029Z02,3fCQL<BI3<;<BI.0a6*=3W021;XGC8;V02=3UG0V R ,BI3<BD,L
rL<BD,;W6*,5/03
V ′⊥L<BD,BD;3W@BI5DBI,6I7a\OP02,3WCQ02;
rG43W02;V9ZC8,07aOP02,L<029780G0.BIL<CQ3<C8BI,L;V0276Y3WCJ02L
Wr W!2 7Q0.6*;W6*5D;<6I.@0 `^_&^ | R BI=203$02,L<0297Q043W6ICJ3$,BI3<4 W \OPC89Z6I5I0G0V•
D
G6*,LV/V ′⊥G4M,CQ31,GC6*5D;<6I9:9Z0=2BI9Z913z6Y3WCJ
Y0//
Y1//
· · · // Ys
X0
X1oo
· · ·oo Xs
oo
Y0// Y1
// · · · // Ys ,.6*;V3<C8=217QC802; R V ′G4M,CJ31,427849Z02,3
w ∈ Wr
\T+-,02LV3G6*,L7Q0=6IG;V0GO 6*..78CQ=6*3WCQBD,G0786=2BD,LV3<;<1=3<C8BI,G1.6I;<6I5D;<6I.@0 `^J¡X^ ¢ R 03N7£O[BD,.%0213=2BD,L<C8G42;V02;7Q0=A@6I9:.G09ZBG17Q02L
MV ′, 0 =MwX•,Y•, 0 .¤ O 6*.;<¥2L/7Q02L/;<42L<17J3z6Y3WL/G1¦.6I;<6I5I;W6*.@0 `^Q¡X^| R CQ7?0EC8LV3<01,§=z@6I9:.¨6I7Q5D42;VC8S10 BV ′, 0
03/1,3WBI;<0
EV ′, 0
3<0278LTS10 MV ′, 0
L<BICQ3T1,EV ′, 0 ~ 3WBI;<L<021;L<1; BV ′, 0
\©fBILWBI,LSV ′, 0 = Hom(EV ′, 0,Gm)
\ª-O[02LV3N1,0LV3W;V1=3<1;<002,3<C8¥2;V0LW1;
Sym2(V/V ′⊥)⊗ R\
!"#$% &'&()*)*),+.-/1032465487-9;:=<>?4@)BA CED!FGDIHJ
C KMLON HJ NQPSRKTNVUN JWJXDIYJ[Z N JX\]!^ N J_FGD!F`] KTNQPaN HbFdceJGDfZFGg[D\ N Jdh=]I^iF N YgXJ U ceg N ^iFXJ UN V jk DIYgWFGDIY%F V ′ ∈ C l HDIFXDHJ C(V/V ′⊥) KTN ^QmIH NnUN JdhDg PaN JR c K cTHo']!cTg N JSJXp P oiFXgXceqY N JSJ NQP crZ U oisHc N Jt\DIJXceFXceu N JSJXYg (V/V ′⊥) ⊗ R U DH6F KTN gX] U ce^'] KvN J[F U oisHcJXYg
Q jk DIYgFXDY%F V ′′ ⊂ V ′ l c KNw cTJ[F N YH N ceH^ K YJXceDH UN C(V/V ′′⊥) U ]IHJ C(V/V ′⊥) j CED!FGDIHJ CKTN qYDIFXc N H6F UNxKyL YHcTDIH U cTJzXDIcTH6F N ∐
V ′∈C
C(V/V ′⊥)
\]Ig K ]ng NQK ]fFGceDH UL oQq6Yceu!] KeN H^ NN H| N H U gXo N \]!g KTN JcTH^ K YJXcTDIHJ UN C(V/V ′′⊥) U ]!HJ C(V/V ′⊥)\DYgFGDIYJxhM]I^iF N YgXJ U ceg N ^iFXJxFGD!F`] KTNQPN H6F.ceJXDIFXgXDI\ N J V ′′ ⊂ V ′ j;~LN J[F.YH1^QD P \ KeNw%N ^QDIHcTqY N q6YcN J[F3H]fFGYg NQKTKeNQPaN HbF P YHc UL YH N ]I^iFXcTDIH UN GSp2g(Z) j!N JXDIYJ[Z N JX\]!^ N ⊕2g
j=g+1Z·xjhM]!^iF N Yg U cTg N ^iF
UN V N J[FFXDIF`] KTNQPN H6FcTJXDIFXgXDI\ NxN F l JGc KyL DH,s w%N YHceJXD P Dg[\cTJ PaNX
∼−→g⊕
j=1
Z · xj = V/(⊕2gj=g+1Z · xj
)⊥,
DHD R FGc N H6FaYH N ceHz N ^iFGceDH UN C(X) U ]!HJ C j~BN F[F N cTHz N ^iFGceDH N J[FaoQqYcru!]Ig[c]!H6F N JXDYJ KML ]!^iFGceDHUN GL(X) l uY^QD PPaN YHJXDIYJZ|g[DY\ NxUN GSp2g(Z) j()*)()_d7-/10-'::-9;;>7-/103246547-9;:=<>?4@)BA H]I\\ NQKTKeN6``yff ¢¡!£G!¤X¥! ¦§¤X¥!yf¦ ¦¨6¦ C K ] U DHHo N§UL YH N§U oQ^QD P \DJXcrFGceDHd\D K pbo U gX] KeN gX]!FXcTDIHH NQKTKeN§UN C(V/V ′⊥)\DYgFXDY%F
V ′ ∈ C © DIH UNQP ]!H UNN H_DY%FXg N qY NK ] U oQ^QD P \DJXceFXcTDIH UN C(V/V ′′⊥)JXDcrF K ].g N J[FGg[cT^ZFGceDHdª
C(V/V ′′⊥) UNK ] U oQ^QD P \DJXcrFGceDH UN C(V/V ′⊥)JXcV ′′ ⊂ V ′ j« DIceF S
YH N§U oQ^QD P \DJXcrFGcTDIH\D K pbo U gX] KeN gG]fFGceDHH NQKeKTNxUN C j¬®X¯°3±³²±X´°¶µ·=¸¹·µ·º j;» H U ceFEq6Y N S N J[F] UP cTJXJXc RKTN g NQK ]!FXceu NQPaN H6FEª Γ0
JXc NQKeKTN¨N J[FEoQqYcruI]ZgXcT]IHbF N JXDYJ KyL ]I^iFXcTDIH UN Γ0 N FtJXc KeN qYDIFXc N HbF S/Γ0 N J[FSsHc j H U ceFtq6Y N S N J[F K ceJXJ N JXcv\DIYgFGDIY%FV ′ ∈ C lK ] U oQ^QD P \DJXcrFGceDH¼ceH U YceF N \]!g S
JGYgC(V/V ′⊥) N J[F K ceJXJ N g NQK ]!FXceu NQPaN H6FWª K ]J[FGg[Y^iFGYg NxN H6FGce½Qg NxU Y] KTNxUN SV ′, 0 j
¾ FXDY%F N¿U oQ^QD P \DJXcrFGceDH Σ UN C(X)qYc N J[F GL(X)
Z³] UP ceJGJXc RKTN DHa\ N Y%Fv]!JXJXD^Qc N gJXDIHÀDIg R ceF NJXDYJGSp2g(Z) U ]!HJ C l ^ N q6Yc U oiF N g P ceH N YH NEU oQ^QD P \DJXceFXcTDIH SΣ UN C qYc N J[F Γ0
Z³] UP ceJXJXc RKeNIjÁ ¤f¤l SΣH LON J[F\]!J K ceJXJ NIjb Hg N u!]IH^` Nl c KNw ceJ[F N YH NU oQ^QD P \DIJGcrFGceDH S UN C qYc%gG]fÃÀH N SΣN F¿q6Yc N J[F K cTJXJ N_ÄÆÅOÇWȶÉʨËÌXÌ`j¢ÍjOÎ DIY ÅÏ :=9ÑÐÒ Ë?ÓjÕÔfÖ6×`j
()*)rع) Ï 2-9?Ù@4/149#Ú-@Û-Ü`?Ý>¹5ÞÝ@'Q-7b:àßS)vA H¶s w%N YH NU oQ^QD P \DJXcrFGceDH Γ0
Z] UP cTJXJXc RKTN S UN C j@ÂKTKeN \DYgXgX]a\]!g Nw%NQP \ KTN¨á FGg N oQ|b] KeN ª SΣ l DIY R c N H á FGg N.K cTJXJ NIjk DYgFGDIY%Fσ ∈ S l c K@Nw cTJ[F N YHÀYHcTq6Y N JXDIYJ[Z N JX\]!^ N FXDIFG] KeNQPaN H6FvceJXDIFXgXDI\ N V ′
σ
hM]!^iF N Yg U ceg N ^iF UN V F NQK qY N σJXDcrFEcTH^ K YJ U ]IHJ KML ceH6FGoQg[c N Yg UN C(V/V ′σ⊥) j H U cTJX\DJ N ] K DIgXJ U Yâ^] P \V] K |o R g[cTq6Y N MV ′
σ , 0 N FU Y\ K DIH| NQPN H6FFGDIgXD!ã U ] KMV ′
σ , 0 →MV ′σ , σ, 0
!"#$%&'()"
*+,.-0/21430576/21982+,:1<;>=?)821@,A8CB+D30EF6- σ ⊂ Hom(Sym2(V/V ′σ⊥),R) -G14B-IH?J/21982+3G19+8@-I-06LK
19,:M08@- SV ′σ , 0 ⊂ Sym2(V/V ′
σ⊥) ⊗ R NPO -Q3SR?7TU=V?7H:WFX0Y82,A*+- MV ′
σ , σ, 0-0/21Z?)[\6-<B-<1^]=-4_6,
/@+8 BV ′σ , 0 NL` H-0/21aTb+6,Bcd+6-e/2198@?)1@,:_3?f19,:5F6gHA53?)HA-0TU-061ihj-082TkX0-e=?78@?7TUXG1982X0-l=?)8iHA-0/mhn?730-0/iB- σ No 6p6571@- ZV ′
σ , σ, 0Hncd+6,:*+-l/2198@?)1@-ehj-082TkX0- Nq 5F,r1 V ′ +6p/@5F+/sKt-0/@=?730-u195)1S?)HA-0Tk-061,:/@571@8@5F=$-ehj?73G1@-0+8
B,A82-03G1vB- V Nw 571@5F6/MV ′, 0 →MV ′, 0
HA-Q=HA576WF-0TU-06119578@57xyB?)H#HA53?)HA-0TU-061ZB-Q1z]L=$-Q_6,#/@+8 BV ′, 0?)/@/9530,AX;|HA?82-0/21982,A3G1@,A576VB-
S; C(V/V ′⊥) ~ -G1b;HA?/21@8@+3G1@+8@-\-061@,AM082- SV ′, 0 NO -\3R?)Tk=?)HAW7X0Y8@,:*+- MV ′, 0-0/21bTb+6,Bcd+6-
/2198@?)1@,:_3?f19,:5F6J=?78@?7TUXG1982X0-Z=?)8kHA?8@-0/21@8@,:3G19,:5F6JB-S
; C(V/V ′⊥) Nw 571@5F6/ ZV ′, 0Hc+6,A5F6JB-0/
/2198@?)1@-0/a=?)89?)TkXG1@8@X0-0/i=?78iB-0/i30EF6-0/i,A630H:+/aB?)6/aHncd,:61@X08@,:-0+8iB- C(V/V ′⊥) NO -e3R?)Tk= ZV ′, 0-0/21
?7H:5F82/P+6Z/@5F+/sKt3R?)Tk=hj-082TkXB- MV ′, 0 N\7Fjt N q ,
S-0/21HA,:/@/9- ~ H:-0/3SR?7TU=/ MV ′
σ , σ, 0-G1 MV ′, 0
/@5F61aHA,:/9/@-0/a/@+8 BV ′, 0 NO -0+8@/Z/@,:6W7+H?)8@,r19X0/Z/@5761B576330-0HAH:-0/ZB-0/Z3SR?7Tk=/ Ag−r, 0 =$5F+8 r ≥ 0 NP 6V=?78s19,:30+HA,:-08 ~ HA-0/3R?)Tk=/ MV ′
σ , σ, 0-G1 MV ′, 0
/@5761UH:,A/@/@-0//@+8 Spec(Z[1/p]) -G1UH:-p305FTU=HAX0TU-061S?),A82-pB-0/U/21@89?f19-0/5F+F-0821@-0/P-0/21v+6ZB,:,A/@-0+8v;308@57,A/@-0TU-0619/P6578@Tk?7+L N% ¡% ¢% £p¤¥$¦§¨©<ª«r¬¦¥®¥¤§¯°n©®¨©7 m± q 5F,r1 V ′ ∈ C NO -\/21S?)Y,AH:,A/9?)1@-0+8.B- V ′ B?76/ Γ0?7W7,:1/@+8 V/V ′⊥ N o 6k6571@- ΓV ′
/@576\,:T\?)WF-PB?76/ GL(V/V ′⊥) NO -PWF825F+=$- ΓV ′?)WF,r1/@+8 C(V/V ′⊥)-06k/@19?7Y,AHA,:/9?761WFH:5FY?)HA-0TU-061H?#8@-0/21982,A3G1@,A576\B-
S; C(V/V ′⊥) NL² -PW78@57+=-v?7W7,:1B5F63P/@+8 MV ′, 0 N
O -k*+571@,A-061 MV ′, 0/ΓV ′-0/21.B-U1z]L=$-k_6,³/9+8 Spec(Z) 3?)8
S/Γ0-0/21´_6, Nµ 57+81@5F+1´30EF6- σB-
S ~ 5F6C6571@- ΓσHA-./21S?)Y,AH:,A/9?f19-0+8uB- σ B?76/ ΓV ′
σN%² c-0/@1u+64/@57+/sKtWF8@57+=$-._6,B- GL(V/V ′
σ⊥)*+,?7W7,:1P/9+8 MV ′
σ , σ, 0 N% ¡% ¶% ·.¬¤¨©U¸t¥¤f®¨¹n¹ ¨L©¬©©0¥º° »¨L©¼
S m± µ 57+81@5F+1P30EF6- σ B-
S ~ 6571@5F6/
MV ′σ , σ, 0
HA-g305FTU=HAXG1@Xghj578@TU-0HiB- MV ′σ , σ, 0
HA-pH:5F6W<B- ZV ′σ , σ, 0 N³O -gWF8@57+=$-g_6, Γσ
?7W7,:1/@+8½30-p/@3RX0Tk?hj578@TU-0H ~ =+,A/@*+c,AH=8@X0/@-082F-H?/2198@?)1@- ZV ′
σ ,σ, 0 Nµ 5F+8a195F+1 V ′ ∈ C ~ 6571@5F6/
MV ′, 0
HA-P305FTU=HAXG1@XPhj578@Tk-0HB- MV ′, 0H:-vHA576W´B- ZV ′, 0 N` H-0/21T½+6,Bc+6-u?73G1@,A576\B- ΓV ′
-G1HA-P*+571@,A-061-0/21Ph 5F82Tk-0HAH:-0Tk-061vB-#1z]L=$-#_6,/@+8 Spec(Z) N% ¡% ¾ \¿lÀ¥$ÀºF»Zª¦|¯»¥¤Á®¨Z§¤f°jÀºF°j§¬¹z i± q 5F,:-061 Σ +6-lBX03057Tk=$5F/@,r19,:5F6pB- C(X) H:,A/@/@-=$5F+8H?/@1@8@+3G1@+8@--0619,:M08@- Sym2(X) ~ -G1
S+6-BX0357Tk=$57/@,:1@,A5F6ZB- C *+,8@?)[k6-
SΣ Nw 5)19576/AΣ
g
H?305FTU=?73G1@,:_3?f19,:5F61@5F8257xyB?)HA-B- Ag 30576/21982+,:1@-=?78PÂ?7Hr19,:6WF/P-G1 ² R?), N
!"#$% &'(*),+-./10325468796;:<6>=%?A@ BDCFE;GHIJKEMLONQPSRT!UWVXT!CZY\[^]`_Hba`L<E
AS
g, 0a`LOHWEDIJcV_SdDV_SEeIL_Spec(Z) f a`LOHWPDdgNJhHiENJ Ag, 0 PDdUeUjEkdgLOl!E_JmnENOIE f EJWJKEC*a`L<EoCqprdgNsT!HJWLN
m!HbT^Y!_qT!UeUtEePST!_JK[DIHiEN
Ag, 0 //
AS
g, 0
Ag // AΣ
g .
u EoPDR%T!UvV AS
g, 0
EDIJwUeLONHwm9p LON#EMIJb_qTgJbH xPST!JhHidgNyCdPST!CEUjENJ#z^E_Uj[DEVT!_qT!Ut[Jb_S[DEV%T!_S/Γ0 |A dNPDdgUWVC[J~[z^dg_UjECCECdgNOYm\ECTeIJb_qT!JKETIDIdPHb[DEtJ~dgLOJPSgNE
σmnE
SEDIJwHI^dgUjd_~VRE
MV ′σ , σ, 0 / Γσ .
dgLO_J~dgLOJV ′ ∈ C f CEtPDdgUWVC[JK[z^dg_UtECm\E AS
g, 0
CECdgNOYmnECqp LNHidgN3m\EDIIJb_qT!JKEDITIDI^dPHi[DESItTgL1GPDgN#EDImnE
Sa`LOHAIdgNJ*HNPCLnIm\TgN%ICqp HNJK[_HiEL_m\E
C(V/V ′⊥)EDIJ*HIdgUtdg_>VRE
MV ′, 0 / ΓV ′ .
u EPSRT!UvVT!CYn[^]`_Hba`LE AS
g, 0
EDIJNdg_UtT!CEJAI^TWV_SdK^ESPJbHidgNILO_Spec(Z)
EDIJVCT!JKEMx]`_SEDIYn[DdgUt[Jb_Hba`L<ESI_S[Sm!LOHJKEDI |A H S
ESIJCHISIE f I^EDIIHNOY!LOCTg_HJK[DIIdNJWPSECCEDIemnEDIoPDRT!UWVI Ag−r, 0 VdgLO_ r ≥ 0 | NV%T!_JhHiPLOCHbE_ f HCESIJvCHIDIEeILO_ Spec(Z[1/p])EJvCEPDdgUWVC[UjENJ~T!H_SEkmnE Ag, 0 EDIJvLONm!Hl^HIELO_
_SECT!JbH zP_SdgHIEUjENJ8IN#dg_UT!L1G |!*^1 ¡
S = SΣ ¢ t AΣ
g, 0 = AS
g, 0
`£*£q¤¥¦ §©¨¦ªv«¬¬®¬h¯°§©¥¦¦±q²M¦ ª³±!£q ±§q¥`´\ nµ¶¥¦1°§©¨¦£q¤¥¦ §q¨¦ªo«¬b¯¬·¬« ?¸ [UjdN%IJb_qT!JbHidN |¹ º »§q¦°§©¦¼»y¦ 1©£S§©¶`£S± »¼ AΣ
g, 0
¦»h½¾!¼1°%£S1n£e¥^¿1¨¦§q`ª¦n£ ?cÀ 1¶§£S 1¶%£
σ ∈ Σ ¢ 1£q Sσh½Z¦q¦ª³e¼¦¦Á1¦¼
Sµ¶±!°°1§;£S±¦^\£Âkh½Z 1§q³±Ã£Se¼
σq ¶
b½Z1`£q± 1ļGSp2g(Z)
?<À q 1
MSσ , 0 =∐
τ∈Sσ
MV ′τ ,τ, 0 .
Å ±Ã£ Met
σ
¶M^1§;£SM ^!!¿¥¦³§©±µn¶MD!±´^1¶ ? º Ä°# 1qMet
Sσ , 0 = MSσ , 0 ×MσMet
σ ,
º k¼¥`Ʊ£1 §q US, 0¦ ªoªvb½¶± 1k¼v£q ¶¦q¤¥¦ª1 Met
Sσ , 0
° ¶§σ ∈ Σ
? º k¼¥`Ʊ£¿§© ¶° !Çi¼RS, 0
¦ 1ªªoM¼1M°!§S!¿§q1°¤«¬h¯<¬·`£²h½Z 1´\¥¦§©±ÆMµ¶½±¦©£²¥`£S1 ? º Ä 1£q
AS
g, 0 = US, 0 / RS, 0
!"#$%&'()"
*,+.-0/1)24351)*,687:9;=<?>@+#>@A8B=<,+:CB:DEFCHGA8<?BI1)CJ*,+:JKI7:24A8CJ=B=;L1MBL<?ANCJI+:JK3;=A83A8J=<?B=<,A8CJO18C1)*,A86N@+:JPRQ%SUTSV +WB Q%SUTSX%SUT >@+*,+H6N;YAN@3$A)Z[I+
RS, 0
;Y+:JL3$+:-WB=+*,1\JYB=;L1MBL<^]-1)B=<,A8C_+WB`>@ab<,*K<?I+:CBL<^]+I+:JJYBL;=1)B=+:Jc-:A8Cd=@68@7+:JeJ=AN@Je*Uaf18-WB=<,A8CgI+
Γ0
DhEFCi+:CgI7:I@<^Bc*,+:Jc18J=J=+:;YB=<,A8CJc;Y+:*1MBL<^G+:J P *,1jJYBL;=1)B=<?]k-1)B=<,A8C5I+l*1`-:A82431)-WBL<^]-1)B=<,A8CDm+:Jn1)@BL;Y+:Jo18J=J=+:;YB=<,A8CJo;Y7:J=@*?B=+:CBoI+4prqHstMu Qwv +:BOI+l*1x3;=AN3$A8J=<^kBL<?ANC Q%SyTS zSyT D
Q%S V%SX%S| t~tM R tt$ t S AN<?+:CB(U → S)
@CRA89dY+WBeI+Ouv
+WB(G′, H ′
•)@CA89d=+WBoI+ Ag, 0(U)
B=+:*>@+G′ J:ab7WBL+:CIj+:C@CjJ=-0/7:2414JL+:2`<^k1)9$7:*,<?+:CJ=@;
SD$oA8B=ANCJ
G′i = G′/H ′
i
3$A8@;1 ≤ i ≤ s
D*+w<,JYBL+l@C+l-/1),C+lIaf<,J=A86N7:C<?+:J
G′ = G′0 −→ G′
1 −→ G′2 −→ · · · −→ G′
s+:CB=;=+HJ=-0/7:2|1)J`J=+:2`<^k189$7:*?<,+:CJxJL@;S
Dn1)33$+:*,A8CJ`>@abANC_1I7W]C<K*,+HI@18* Iaf@C¡JL-/7:2|1J=+:2`<^k189$7:*?<,+:C5I1)CJo*?+l318;=1868;L1)3/+ TSQ%S¢ D*+<?JYBL+lIA8C-#@C+l-/1),C+lI+l24AN;Y3/<,J=2`+:J£ Q%SV%S X Db¤l¥
X(G′0t) −→ X
(G′
1t)−→ · · · −→ X
(G′st)−→ X (G′
s ) −→ · · · → X(G′0) .
@33$ANJ=A8CJO>@af<?*+w<,JYBL+V ′ ∈ C
+WBn@C+lJ=@; d=+:-WB=<,A8CjI+V/V ′⊥ I1)CJ
X(G)>@<<?I+:CBL<^]+¦*yaf<?2|1)6N+
I+V•
D
+WB £ QSV%S X Db¤l¥0Dw§¨af183;Y©:J *1e3;=A83A8JL<^BL<?ANC TS^ªS «%SV¬ *1¦y182`<,*?*,+FI+:JA8;=2`+:JK9<?*,<?C718<?;=+:JB(G′
i)I7W]C<?BO@C24AN;Y3/<,J=2`+B(G′, H ′
•)I+SV ′, 0
I1)CJDivS
Dm1`3;YAN3;Y<,7WB=7l2`AI@*1)<,;Y+ P *1¨;YANCBL<?©:;=+J=@<?G)18CBL+l+:JYBO*1¨BL;=18I@-WBL<?ANC5I+l*,1`3;YAN3$ANJ=<^BL<?ANC Q%SyTSX%S V +:CB=+:;=2`+lI+¦-:A8243*?++l-:A8C<,>@+8D®c¯°8±°²M³´µ³=°¶¸·¹yº¹[»¹¼ D½ ¾À¿iÁ)ÂLÃMÄÅÃÆUÇ¿ (Ag, 0 → AS
g, 0
)(U → S)
ÅÈ)Â=È)É|ÊËUÂL¿_Ì¿ÍÎÃNÏyп0ËÍ
(G′, H ′•)
Ç¿ Ag, 0(U)ËÑ¿0ÌÒÍgÓWÄ¿
G′ ÍÔÖÕË¿×ÇØ¿×ÙÄw×ÚÍÛÜÕÉ|ÈÍ¿ÉxÝ ÞµÈÏ0ÕÌßÝU¿×àÍÄÂS á ¿Ë|Ë¿ÌÒÍgÓÄ¿
ÌÃÛ=È)Ì¿Éâ¿0×ËÅÃMÄÂ`Ì,ÈËÑÃÅÃÌÃLÁ)Ý[¿ãÕ0ËÈ)Ì¿âÇ¿S á Ý Ì¿YäNÝÍ0ËÑ¿
V ′ ∈ C á Äw׿¨ÍÄw пÛËUÝUÃM×V/V ′⊥ → X(G′)ÝUÇ¿0×ËUÝ å ÈM×ËâÌ=Ô Ý É|ÈÁ¿Ç¿
V•D
¿Ëçæ ·¹º¹[»èré ê á ¿Ëσ ∈ S
ÇÃM×ËâÌ=Ô ÈÇÜÕÂL¿×%Û¿¡ÛÃ×ËUÝ[¿×ËãÌ,ÈjëÃÂÉ|¿ÏWÝ ÌßÝ ×%ÕLÈ)Ý ÂL¿
pD B(G′, H ′•)
ÅÃÄwÂlËÑÃMÄwËKÇ)Ý ìÝÍ¿ÄwÂlÝ Â0ÂÕ=Ç)ÄÛËUÝyÏWÌ¿¦ÂLÕLÇ)ÄÝ ËD
Ç¿S
è
Q%S Q%SHí tMî8ÀtçÀçÀ ï$~tMî ~ çÎÎtâðy t A8<?B
n ≥ 3@C¦+:CBL<?+;CA8C¦I<^Gw<,J=<?9*,+318;
pDnA)BLA8CJ Ag,n → Spec (Z[1/n])
*Uab+:J=318-:+I+ñ24AI@*,+:JI+:JOJ=-0/7:2418Jo1)97:*?<,+:CJ3;=<?C-:<,31)*,+:24+:CBo3$A8*1);=<?JL7:JI+l6N+:C;Y+
g2c@C<,JIaf@C+lJYB=;=@-WBL@;Y+lI+#C<?GN+18@
Å$Â0Ý ×%Û0ÝÒÅÈ)Ì¿¨¿×n
D*31);L1)24©WB=;=+lIANC-l*?+:JOJ=-0/7:2|1)Jo189$7:*,<?+:CJA
2¨@C<?JOIaf@C5<?JLA824A8;=3/<?J=24+
V / nV∼−→ A[n]
>@<n;Y+:JL3$+:-WB=+ç*?+:JâA8;=2`+:JãJYòw2`3*,+:-WB=<,>@+:J P @CØJ=-1)*1)<,;=+3;Y©:JD §caÒ1)3;=©:Jçpfó¸ônõ ªv[¬ Ag,n +:JYBã@CJ=-0/7:2|1x>@1)J=<^k3;=AdY+:-WBL<^öJ=@;
Spec(Z[1/n])D÷öA8J=ANCJO7:61)*,+:2`+:CB
Ag,n,0 = Ag, 0 ×AgAg,n .
EFCgI7:-:;=<^B P 3;=7:J=+:CBc-:AN2`24+:CBc-:AN2c9<,C+:;e*1j-:ANCJYB=;=@-WB=<,ANCgI+:Jc-:A82431)-WBL<^]-1MBL<?ANCJcI+ Ag, 0 +WBI+ Ag,n 3$AN@;A89BL+:C<?;I+:JO-:A82431)-WBL<^]-1)B=<,A8CJKB=AN;YA8Z[I18*?+:JOI+ Ag,n,0 ¬ >@<J=+:;YANCB@B=<,*?+:JO*?AN;YJOI+#*,1-:AN2`318-WB=<?]-1MBL<?ANC24<?C<,2|1)*,+eI+ Ag,n,0 D%oAN@JF-0/1)C6N+:A8CJF*,1âI+:J=-:;Y<,3B=<,A8CçI+:JF-:AN2`318-WB=<?]-1MBL<?ANCJ
!"# $%&('*),+.-/+.0.12+.-43/+5+687:9;;6=<>9?)@<&-BAC69D>.&(EF;9>G),<=H>9#),<&-IEJ<=-<EJ96=+LKM-&/0N5O.>.P,<Q3R&-0N/-+;P,+.EF<S.PT+U0T)@P,9#),<=H>9?)@<=&(-V+.-N)@+.PTEJ+U5+
C1;/<0W/-+U0T)@P,9#),<=H>9?)@<=&(-V;6/02H-+U+.-N),+.P,EF+U5+
S XY &<=) Σ /-+M5O.>.&(EF;&0,<=),<&-F;&6=Z(O.5P@9#6+ GL(X) [ 9#5EJ<=0@0,<='6+M5+ C(X) X(\ &(0,&-0S = SΣ XR] 7^+.0T)
69_5O.>.&(EF;&(0,<=),<&-`;&(6QZRO.5P,96=+ GSp2g(Z) [ 9#5EJ<=0,0,<'6=+S
5+ C &('*),+.-/+a+.-`;P,+.-9#-!)b687^&(P,'<Q)@+c5+ Σ59-0 C 0,&/0d687:9>G)@<=&(-J5+ GSp2g(Z) XRe 7:9;PTS.0gf:h2ikjlm1(),n XRoqprXtsXtu +G)wv Xtux +G)w6=+2)@nO.&P,S.EJ+ayzz|z~1<6+*<=0T)@+U5+.0W>.&(EF;9>G),<=H>9?)@<=&(-0
Ag → Ag , Ag,n → Ag,n et Ag, 0 → Ag, 090,0,&>.<=O.+.0gA Σ +G)cA
S X\ &(0,&(-0Ag,n, 0 = Ag, 0 ×Ag
Ag,n .] 7^+.0T)g/-+U>.&EJ;9#>G)@<QH>9#),<&(-V5+ Ag,n,0 0,/P Spec(Z[1/n]) Xyz yz~zN* Ag,n, 0 z e 9-0U>.+J;9P,9P@9#;n+1m&(-5O.>.P,<=)5+./0T)@P,9#),<=H>9?)@<=&(-0-9#),/P,+.6=6+.0W5+ Ag,n, 0 X\ &(0,&-0
Γn = Ker(GSp2g(Z)→ GSp2g(Z/nZ)
).
e 79#;P,S.0f:hMiFjlm1*)@n XoqprXtsXtu +G)gv Xtux +G)g6=+)@nO.&P,S.EF+yzz|z~16=+.0g>n9EF;0c9#6O.'P,<=/+.0Ag , Ag,n et Ag, 0
0,&(-R)Er/-<=057^/-+0T)@P,9#),<=H>9?)@<=&(-_;9P,9EFOG)@PTO.+;9#PS/GSp2g(Z) 1
S/Γn+G)
S/Γ01?9#<-0,<!/+w57:/-+
57:/-+W0T),P@9?)@<QH>9#),<&(-J;6=/0d(PT&(0,0,<S.PT+W;9P,9EFOG)@P,O.+W;9#PC/GSp2g(Z) 1
C/Γn+G)
C/Γ0 X!e 9-0d6+WPT+.0T)@+5+>.+;9#P@9#(P,9;n+1&(-5O.>.PT<=)a;6/0a;PTO.>.<0,O.EJ+.-R)>.+.0a0T)@P,9#),<=H>9?)@<=&(-0 X Y &(<Q) V ′ ∈ C Xo 6+*<=0T)@+5+.03#9PT<OG),O.0M5+ Y <+.+.6 AV ′ , AV ′, n et AV ′, 0 5+a)Z;+;9#P@9#n&(PT</+67:<=E 9#(+5+ V•
D59#-0 V ′⊥/V ′ ¡ 1
5+.0W0,>¢nO.E 9#0g9'O.6<=+.-0BV ′ → AV ′ , BV ′, n → AV ′, n et BV ′, 0 → AV ′, 0 ,
5+.0 Z [ EF&5/6+.0W6<='P,+.0 SV ′ , SV ′, n et SV ′, 0 ,5+.02)@&P,0,+./PT0
MV ′ → BV ′ , MV ′, n → BV ′, n et MV ′, 0 → BV ′, 0
0,&(/0W6=+.0W),&(PT+.0EV ′ = Hom(SV ′ ,Gm) , EV ′, n = Hom(SV ′, n,Gm) et EV ′, 0 = Hom(SV ′, 0,Gm) ,
+G)g5+.02H'PTO.0g+.-V;6=&(-+.EJ+.-R)@02)@&P,<=/+.0W6&!>96=+EF+.-!)W5+)£Z;+H-<MV ′ → BV ′ , MV ′, n → BV ′, n et MV ′, 0 → BV ′, 0
/<0,&-!)ME/-<0M57^/-+0T),P@9?)@+c¤¥+.PTEJO.+a>9#-&(-<=/+ X*¦ &#)@&-0 ΓV ′1 ΓV ′, n
+G) ΓV ′, 06+.0b0,)@9'<=6<=0@9#),+./P,0
5+ V ′ 59-0 GSp2g(Z) 159#-0 Γn+G)W59-0 Γ0 X
¦ &),&(-0SV ′
67:+.-0,+.E'6+F5+.0 σ ∈ S<=->.6/0U59-0U687^<-!),O.P,<=+./PU5+ C(V/V ′⊥) X\ &(/P),&(/*) σ ∈
SV ′1&(-V5<0,;&(0,+U5+H'PTO.0g+.-V;6=&(-+.EJ+.-R)@02)@&P,<=/+.0g9#§J-+.0W5+)£Z;+H-<
Mσ → BV ′ , Mσ, n → BV ′, n et Mσ, 0 → BV ′, 0
!"#$%&'()*#+-,./10234,.5$07698;:*6;/=<>:?69/[email protected]:A8;:A+DC6;8;C5$B:*8E150GFH:*>5025 σ IJ :A.0G69@A,0KE150>:*0LE1:M07698;:*6;5$B:A8;:?N+OCP6;8;C5DB:*8 σ Q E: σ N7RSUT;V*SXWYZ5076[EH34,./=\ ,5D07698;:*6;5MF]58;+DC5 I_^ @A6;@`.0 Γσ
5P6 Γσ,0E=50a0;69:Ab/=E1/=09:*6;5,8;0
25 σ 2:*.0 GSp2g(Z) 5P6K2:A.0 Γ0 c >5[0;@A.6d250KeA8;@A,B50<./=0 IfhgAikjlnm`opgrqsUqsutsvt Ixw y @A+O+D5 n ≥ 3 LE=5a0;69:Ab/=E1/=09:*6;5,8K25 σ 2:A.0 Γn
5076K6987/=z /:*E23:*B8;|0dE=5E15+D+O5a25- 5878;5O~A IJ 34:AB87|0aE150a6;C@`87|+O50au II 5P6[v II 25~M5P6[E=5D6;C@`87|+O5&!HL_/=E5/=0;6;5O250
/10;@A+O@A8;B/=0;+O50Q &&U I Y MV ′ / ΓV ′
∼−→ AV ′
g , MV ′, n∼−→ A
V ′
g,n / ΓV ′, n , MV ′, o / ΓV ′, 0∼−→ A
V ′
g,o
Mσ / Γσ∼−→ A
σ
g , Mσ, n∼−→ A
σ
g,n et Mσ, o / Γσ, 0∼−→ A
σ
g,05.6;8;5>@`+DBE1CP69C0F]@A8;+D5E10E=5E=@`.e 250 V ′ Nv076;89:?69505P6250 σ Nv07698;:*6950 Iy @`.0;/=2C8;@`.0-E150-B8;@ 2,/¡690<b8;C0
AV ′, n,0 = AV ′, 0 ×AV ′AV ′, n , BV ′, n,0 = BV ′, 0 ×BV ′
BV ′, n ,
MV ′, n,0 = MV ′, 0 ×MV ′MV ′, n , Mσ, n,0 = Mσ, 0 ×Mσ
Mσ, n ,E:M0;@`+D+O5[:*+h:*E1e`:A+DC5SV ′, n,0 = SV ′, 0
⊕
SV ′
SV ′, n ,
E15Ge`87@`,B%5 ΓV ′, n,0 = ΓV ′, 0∩ΓV ′, n5P6E=5e`87@`,B%5 Γn,0 = Γn∩Γ0 I`¢ 500;>£C+h:*0 MV ′, n,0
5P6 Mσ,n,00;@`.6d250<b8;C0K5.¤BE=@`.e`5+D5.6;0K6;@`87/1\,50d0;,8 BV ′, n,0LC\ ,/¡z*:A87/:*.690K0;@`,0KE=5$69@A8;5
EV ′, n,0 = Hom(SV ′, n,0,Gm) .
¢ :DB87@`B%@`0;/¡69/=@`.¤0;,/¡zA:*.6;5[87C0;,E=6;5a/1+D+OC2/1:*6;5+O5.6K250K/10;@`+D@`87B/10;+D50 Q & &H I Y I¥-l¦A§_¦¨?©uª«©;¦¬qs®q&suts]¯ Ixw ° W²±´³V*µZ¶·V*¸4¹º»PT¼H½P¾W Ag,n, 0 W9RSZµ¾¿¼ÁÀ_ ¾¿WDRSUT;V*SH¼ ÃG±9V*SH¼®Ä?¿ ¶V*T;V*ŵº£SUT´º9W¶V*T
C/Γn,0 ÆÈÇ Ä?¿É±9Ä?µZ¶%¸1ºSXºËÊÄTµhW¸¸Wa¸Ä?¿¹ÀW[¸=V V ′ Å®RSUT;V*SXWW´RS̼1RnÄ?µÄTu¶³WV*¾ ½P¾ÄSU¼®W¿SMV ′, n,0 / ΓV ′, n,0
À*¾Í±´Ä?µZ¶¸ºSXºÊÄ?TµhW¸pÀW MV ′, n,0¸W¸1Ä?¿¹ÀW¸1VOR£SUT9V?SXWÊWT£µhº´WD±9V?¿&Ä?¿¼H½P¾W Æ ° WD±9³V*µZ¶ Ag,n, 0 W9RS
ºv¹V?¸WµW£¿SεM¾ ¿¼ÎÀ_ ¾ ¿&WaRSUT;V*SU¼ Ãα9V*SU¼®Ä¿¶%¸¾RÃË¿&WK¶V*T;V*µºSHT´º´W¶V*TS/Γn,0 ÆxÇ Ä?¿É±´Ä?µZ¶¸ºSXºÊÄTµhW¸
¸W[¸1Ä?¿¹²ÀW[¸1V σ Å®RSUT;V*SXWW´RS¼RnÄ?µÄTu¶³WV*¾Ï±´Ä?µÁ¶%¸ºSXºÊÄ?T£µW£¸Mσ,n,0
ÀW Mσ,n,0¸W[¸1Ä?¿¹²ÀW[¸1V σ Å®RSUT;V*SXW Æ
¢ 5a>@A8;@AE1E1:A/=8;5a0;,/=z*:*.6d87C0;,E=6;5a2,FH:A/¡6d\,5 Ag,n 5P6 Mσ,n,[email protected];>£C+O:A0KB%@`,8 σ ∈ S I
Ð ¦l¦ÑuÑHj©ulgÍqs®qsvts®q Ixw ° WM±9³V*µÁ¶kV*¸4¹º»PT£¼U½P¾W Ag,n, 0 W9RS̾¿ÒW´Ru¶V±´WV*¸¹º»PT£¼U½P¾W Æ
!"# $&%'( '()(+*-,./02143650872/191:&;2<=9>?1>@A<BC36/021(ED FHGIJGLKM
G0 NPO MJQSRTVUXWW#YT NPZO K\[K Z^]_OV` M ONMJ[ ` Ag,n,0 a O I H• NPOcb` Wd O W[e[K Z^]_OV` M OVNEbO G0[p] fhg GMJGLKMGi = G0/Hi
dG[ ` 1 ≤ i ≤ s fji NOklZ MmI O [K O QSRW&nK OobpqZ MJGLrTVK ZO M
G0 −→ G1 −→ · · · −→ Gs
O K!I `JO MJQSRTVU+W#McW#YT NZPO KM+MJ[ ` Ag,n,0 fts O IJRTVG `mu U O '( )( v(xw O I NO M ` TVMJ[ N IyW#IJM bO z|~ a imif4UXGK!I `JO K!I[ O GiM p TI O K bO K[KMyQRTVU+WM O U Z^ W#YT NZPO K
Gi −→ Ag,n, 0dGL[ ` 0 ≤ i ≤ s a O I[ OXN WQSRW&nK OXbp|Z MyGrLTVK ZPO MM p TI O K bO K[K O QSRW#nK O+bO UXG ` dR Z MJU O M O K!I `JOMJQSRTVU+W#M~M O U Z^ WYT NPZO KM flg G[ ` IyG[lI0 ≤ i ≤ s a GKKGI O ei N WM O QI Z GLKK O [lI `JObO Gi O I Nxp GKdGLM O
Ωi = e∗i ΩGi/Ag,n, 0et ωi = det( Ωi ) . O M bOV` K ZPOV` MMJGLK!I bO MxW Z MJQ O W[ kN GQW NPO U O K_I NZ Y `JO MMy[ ` Ag,n, 0 bO` WKr `mO MJd O QI Z 6M g O I 1 f2 K
Wdd OVNPNO ωi NPO i u U Oty! 4¡! fi N¢OklZ MmI O [K O MJ[ Z I OobO UXG ` dR Z MJU O Mωs −→ ωs−1 −→ · · · −→ ω0
b T£K Z M¢MJ[ ` Ag,n,0 a [ ZLZ K b [ Z M O K!I bO M Z MJGLUG ` dR Z MJU O M2W#d `Ju M¢QSRW#Kr O U O K!I bO YW#M O dW ` Spec(Z[1/np]) fg G[ ` IyG[M
V ′ ∈ C O I 0 ≤ i ≤ s a ZN¤OkZ MmI O [KMJQSRTVUXWM O U Z WYT NZPO K GV ′, i −→ BV ′, n,0![ Z
O MmIHr N GYW NPO U O K!I Ok I O KM Z GLK0 −→ TV ′, i −→ GV ′, i −→ AV ′, i −→ 0
bp [K¥MJQSRTVUXW¦W#YT NZPO K¥dW ` [K§IJG `mO¨y©6ªV« dW ` Wr ` W#dR O M¬ fqf¯® O I¬ f|lf±°_²Sfs O MJQSRTVU+W+WYT NZPO K AV ′, id ` G ]ZPO K!I bp [K³MyQRTVU+WWYT NPZO KMJ[ ` AV ′, n,0 f?s O r ` G[d OobO MQW ` WQI uV`JO M XV ′, i b [IyG `JO TV ′, i O MmITVr_W N!´NµpqZ UXWr O¶bO V2s−iD b WKM
V/V ′⊥ ¨y©6ªV« dW ` Wr ` W#dR Ot· f!f ¬ ²Sf& KWdW `¢OklO UXd NPO XV ′,0 = V/V ′⊥ fg GL[ ` IyG[lI
σ ∈ SV ′ a GLKcKG#I OO KQVG `JO GV ′, i NO MJQSRTVUXWM O U Z^ WYT NPZO Kc[ ZO MmI Nxp|Z U+W#r O` TVQ Z d ` G[ ObO GV ′, i
dW `NPO UXG ` dR Z MJU OMσ, n,0 −→ BV ′, n,0 .¸ p W#d `Ju M N W `JO UXW ` [ O [ Z MJ[ Z I NPO IyRTVG `Ju U O '(µw(¹(' a Nxp|Ok I O KM Z GK bO»º W4¼KW#[ bbOGi −→ Mσ,n,0
O MmI GV ′, i fi NOklZ MmI Oob GLKQoMJ[ ` Mσ,n,0 bO M Z MJGLUG ` dR Z MJU O Mωi
∼−→ det(ΩGV ′, i
)¨ '('( ) fq½o²
∼−→ det (XV ′, i)⊗ ωAV ′, i.
"!$#&%'#)(* +",.-0/0%21"354"!76"8:9**,.!0!035"!$;.%&!7<=32/0>?%&!$>@3A6"4";B2,./0%&!$6"%DCA,B%&E2%&;F2/G,.-0(*>?8:-H,.IA4"%&!$6"%J",LKM %N 4O+"/0,."#&,B+* ;P%&O4"O%&Q-H,.%&/
n ≥ 3"32O6", M ,B!0,BR";.%S+*2/
p1"%:-T6"%S", M %N 4O+*2/02("32/0,.IA4"%S6"%U-WVA+X% D%&
p YZ %@+32,B6"![6]\^4""%?-0%&;B;.%?<=35/G>@%@%&!G-J4""%?<=2>@,.;B;.%(k0, · · · , ks)
6]\_%&`-0,B%&/0![/G%&;ab-H,L<=![IA4",c#&3`6"%@;B%<d ,B!0#&%N 4e,. M %&/G!0,BR";B%f6"35Q-?;ag<h35/0>?%i%&!G-?4""%f!0%&#:-H,B32jE5;B32R*2;.% Ylk 324"!?6"8:9*",B!0!032"!?;B%f6"8 M %&;.35+"+%mK>@%&Q-n6"%po"354"/G,B%&/WKrq5 #&35R",s6]\_4""%O<h35/G>@%O>@3A6"4";B2,./0%O%:-n>@32`-0/032"!n4""% M %&/0!0,.356"4t+"/G,B"#&,.+X%O6"%u7v`#)("%&/&1 M 2;B2R";.%?!0,
g ≥ 21 ,B"!0,wIA4]\_4"x+"/0,."#&,B+%?6"%?6"8 M %&;B32+"+X%&>?%&`-S6"%@o*324"/0,.%&/rKrq2 #&35R", Yy 324"/#&%&;az1"324"!|4-0,B;B,.!035"!n;B%~#N2/02#:-0&/0%~/08&6"4",L-|6"% Ag,n,0 × Spec(Fp) fbX 1;B6"%&"!H,L-H8~6"%~!035
;B,.%&4g35/G6",B* ,B/G% f~z %:-4""%'6"%&!0#&/G,B+-0,B32g6"%'!H%&!$#&32>@+32! `-0%&!,./0/G8&6"4"#:-H,.R";B%&! J*Y 9*P1"354"!#&35"!G-0/04",.!H32"!U;B#&35>?+*2#:-0,.9*#Nb-H,.35>@,.",B>@2;.%D6"% Ag,n,0 +354"/n ≥ 3
"326", M ,B!0,.R";B%+*2/p Y %:-G-H%?6"%&/G",B&/G%?%&!0-S4"x!H#("8&>nf+"/030%&#:-0,.< !0!H3`#&,B8@i;d\ ;BE2&R"/0%?E2/H 6"4"8&%?6*%&!U<=32/0>?%&!S>@3A6"4";B2,./0%&!
6"%7+X32,B6"!(k, · · · , k) +354"/
k ≥ 0 YAk 354"!4-H,.;B,.!035"!6"%7>n *,.&/0%7%&!0!0%&`-H,.%&;B;.%7;a[*32/0>@2;.,.-0876"% Ag,n,0 +"/G35+ Y ¡* £¢X ¡z¤)Y
¡* n¥¦b§¨A©@§¦PªFF«h¬*=¨A©@ª¨x®]=¨`¯¨z«° 9±%@4"x%&`-0,B%&/n ≥ 3
"326", M ,B!0,.R";B%?+*2/p1P4"x8&;B8&>@%&Q-
k = (k0, · · · , ks)6"%
Zs+1 1P4"Z[1/n]
K²>?3A6"4";.%M
%:-T;d\^32O+X32!0%ωk =
s⊗
i=0
ω⊗ki
i
IA4",]%&!G-T4"i<d ,B!0#&%N24O,. M %&/0!0,BR";.%S!04"/ Ag,n, 0 Y³´0µU¶¸·º¹r·0»"¶½¼w¾²¿X¾º¿ YFÀ Á "%|<h35/G>@%|>@3A6"4";B2,./0%|6"%iCA,B%&E2%&;6"%E2%&"/0%
g1F6"%|+X32,B6"!
k1F6"%|", M %N24+"/0,."#&,B+* ;]%&
n1"6"%U", M %N 4i+*2/02("32/0,.IA4"%U6"%-ºVz+% D %&
p1%:-T M 2;.%&4"/0!Â6*2"! M %&!G-s4"i8&;.8&>@%&Q-
6"4OE5/G354"+%H0(Ag,n, 0 , ωk ⊗M) .
ÃnÄ2ÅÇÆ*ÈÊÉ5˸ÄO¼w¾²¿X¾dÌ YFÀ Z ?<d >@,.;B;.%k = (k0, · · · , ks)
,."6"4",.-74""%S<d2>?,B;.;B% (detk0 , · · · , detk0
) 6"%#N2/02#:-0&/0%&!s6"4iE5/G354"+%[2;.E58&R"/G,BI`4"%
GLg Y ° O+354"/G/H ,.-s/G%&>@+";B2#&%&/Â;B%S#(*32,L±f6"%k+* /s#&%&;.4",]6]\^4""%
!"# $%&('%)+*-,#,-.0/.
s+1 1 .32 15436 .789':8;*-<=7 6 '>,-? 43@1 *#AB. 6 /. GLg C .8D,FEG<=7+< @ 8;*-.37/ 1 '%*-8H/. 6 &I< 1 )J. 6HKML '>,-.3B 1N6L .3O8N< 1 *#.3,-,#. 6P ,-. 6Q15436 B,-8;'%8 6 /.RO.858;.R2' 1 8N*#. 6 ES'>/'%28;. 1 '%*#.378 6 '>7 6 2T.3*-7.>UV*M
. 6 8MB7 Z[1/np] W )+</B,-. C ,-. 6 &('>* 6 O3.'>BYX ω⊗ki
i ⊗M 6 <>78Z8;<>B 6 * 6 <=)J< 1 2[. 6 U]\^'_/<=77 4 ./.k. 6 8Q/<>7O 4 AB* L '%,#.378N. K O3.3,-,#.R/. ∑i ki
U`bac>d^ce:fhgifNcjlknmhoTmqp Usr t^uwv:x9yzu
H0(Ag,n, 0 , ωk ⊗ M)u9|~JThuT%%z9y:u#
9;y:y|Fqyy:;%xN%uΣ
y|FxN%Fqy V<>*-8Σ′
B7.'%B8 1 ./ 4 O3<>)+2T< 6 *-8N*#<>72<>,- 4 / 1 '%,#. GL(X) W '%/)+* 6;6 * @ ,#. 6/.C(X)
U<>8N<=7 6A′
g,n, 0,'O3<>)+2'%O8;*O':8;*-<=7w8;< 1 <%q/'>,-.bAB*,#B*. 6 8M' 6N6 <O3* 4 .>U Z.3BYXw/ 4 O3<=)J2T< 6 *-8N*#<>7 6 '>/)J.8N8N.378B7 1 ':& W7.3)+.378QO3<>)+)bB7 P <=7¡2T.3B8Q/<>7O 6 B22< 6 . 1 AB. Σ′ 1 ':¢7. ΣU£h,].XY* 6 8;.¤'%,#< 1N6 B7¡)J< 1 2[* 6 )+.
φ : A′
g,n, 0 −→ Ag,n, 0AB*¥. 6 8M2 1 <=2 1 . C 6 B 1¦ .3O8N*-&H.8 @ * 1 ':8;*-<=77.3,FU§\^<O'>,-.3)+.378M2T<>B 1 ,'8N<=2T<>,#<>?=*#. 4 8;'>,-. C φ 6 ES*-/.378;*. KB7¨)+< 1 2[* 6 )+.M.378 1 .M2,#<>7?=.3)J.378 6 8N< 1 *#AB. 64 AB* L ' 1 *#'>78 66 <=B 6 ,#.M)J©3)+.Z8;< 1 .>UªD7¨B8;*-,#* 6 '>78,#. 61N436 B,89':8 6 /.+«G¬¬w®°¯± C £9U³²:´ C <=7 L <=*8QAB.
φ∗ OA′
g,n, 0= OAg,n, 0.8QAB.R2<>B 1 8N<=B8 j > 0 C <>7w'
Rj φ∗ OA′
g,n, 0= 0 .
<%8;<>7 6 ω′i
,#.i W¶µ )J.¤ @1N4 /.b·</?=. 6 B 1 A′
g,n, 0
U¸¹<=)J)+.ω′i = φ∗ωi C ,'J&I< 1 )ºB,-.»/.»2 1 < ¦ .3O8;*-<=7.8Q,#' 6 B*8;. 6 2T.3O8 1 '>,-.R/.¤\^. 1 ')+<>78 1 .3780ABES<>7w' @ *-.37
H0(Ag,n, 0 , ωk ⊗M
)= H0
(A′
g,n, 0 , ω′ k ⊗M
).
¼^½F¾½¼^½ ±¿ÀÁÂÃTÄÄDÁÅ!ÁÆÇÉÈÁËÊ]ÃTÌÍ:ÎÁÍÏ;ÐÑÒ=ÃӥΠ½ÕÔ \].EV ′, n,0 W 8N< 156 .3B 1 MV ′, n,0
. 6 8º':¢7.6 B 1 BV ′, n,0
2T<=B 1 V ′ ∈ CU£h,].X* 6 8;.R/<>7ORB7¨&('>* 6 O3.3'>B A .37w'%,#? µ3@1 . 606 B 1 OBV ′, n,0
8N.3,]AB.MV ′, n,0 = Spec(A)\].R8;< 1 . EV ′, n,0
'%?=*8 6 B 1 A .8Q,FEG<=7w2T.3B8Q/ 4 O3<=)J2< 6 . 1 O3.R&('%* 6 O3.'%Bw.37'>,-? µ3@1 . 66 .3,-<=7w,-.¤? 1 <=B2T./. 6 O' 1 '>O8 µ31 . 6 SV ′, n,0
/.EV ′, n,0
UTÖ7¡< @ 8N*#.378A =
⊕
λ∈SV ′, n,0
L(λ)
<=× L(λ). 6 8¹,-. 6 <=B 6 OBV ′, n,0
W )+</B,-.Q/. A 6 B 1 ,-.AB.3, EV ′, n,0
'>?>*-8¹2' 1 λ U¸¹<=)J)+. MV ′, n,0
. 6 8B7EV ′, n,0 W 8N< 156 .3B 1 C L(λ)
. 6 80B7¨&('%* 6 O3.'>B¡*-7 L . 156 * @ ,-. 6 B 1 BV ′, n,0
UØ¥<>B 1 σ ∈ SV ′ C 6 *,FEG<=7¡2T< 6 .σ∨ =
λ ∈ Sym2(V/V ′⊥ ⊗ R) | b(λ) ≥ 0 ∀ b ∈ σ
,
! "$#
%&('*)+',-/.0+&/1321%&
Mσ,n,0 = Spec
⊕
λ∈SV ′,n,0∩ σ∨
L(λ)
.
4 %5%&/5ωket =
s⊗
i=0
(det(XV ′, i))ki et ωkab =
s⊗
i=0
ωki
AV ′, i.
6 '&/587'9-/.0+&/1:2;13%&<5=/1:>'?&@2ABC%&<=2;137135AD7'E)/,%)%51:2;13%&GFHFHJI+HLKMA2873ADN$'73N=/7POQFHLFHKRSTP-/A ωi '=>U%1513&+'VA-WX=/&/A5Y2,;'?2AR ZC%132 f ∈ H0(Ag,n, 0 , ωk ⊗M) R[]\^`_badceaf/_hgPikj igPikj Rml noA`-/.>UA73%)/)ApqA&@2-/AEr/%=/,Y1A,tseu'N%v/1-/A f A& σ ∈ SV ′
A5Y2w7'*5.,Y1Ax %,pyA773A
FJσ(f) =∑
λ∈SV ′,n,0 ∩ σ∨
aλ,σ(f) ∈∏
λ∈SV ′,n,0 ∩ σ∨
H0(BV ′, n,0 , L(λ)⊗ ωket ⊗ ωkab ⊗M
)
%v2A&C=/AA&.>?'73=+'&@2 f 5=/,7AN%pq)/73.2;. x %,pyA7o-/A Mσ,n,073A7%&/Vq-/A5;' σ sk52,;'z2;AR
4 %5;%&/5 σ> =λ ∈ Sym2(V/V ′⊥ ⊗ R) tel que b(λ) > 0 ∀ b ∈ σ
. nW1-/.$'?7/-/Aw7' σ sk5Y2,;'z2;A-/A Mσ,n,0
A5Y2A&/VA&/-/,Y.y)+',`73A5 H0(BV ′, n,0,L(λ)) %| λ ∈ SV ′, n,0 ∩ σ> Rm~&A&-/.-/=/132C=/A)/,A5@=+A2;%=/5w73A5 aλ,σ(f) 5%&@2&@=/75w)%=/, λ ∈ σ∨ − σ> Rff/zadceaf/_hgPidjigPi Rml @+M+@!k +¡U¢£m@ f σ ∈ SV ′
@d+¤ ¥@ f ¦@ V ′ §¨ <©Q¡?Q §Dª ZC%1A&U2 σ, τ ∈ SV ′
R«~=/1:22A¬(N%&/51-/.,YA,=/&/A<5=/132A®-/A<N¯&/A5,A731'?&@2 τA2 σ B+%&) A=2w5=/)/) %5A,wC=/A τ A5Y2=/&/A x 'NA-/A σ R/°t7A±1352AE'73%,Y5w=/&/A1pqpyA,513%&%=>UA,e2;AMτ,n,0 → Mσ,n,0 .² %pqpyA7A5w5N³/.pq'5w5Apq1´sd'v.713A&/5 GV ′, i
5AN%,,YA5;)%&/-/A&@2EµJ¡MNA)/7%&/VApyA&@2$B/%&G>%1:2@=+A
FJσ(f) = FJτ (f) ∈ H0(Mτ,n,0 , ω
k).
~&('?)/)A773A+z¶+dE%=·¢@©d]73A5.73.pqA&U2;5w-/AC/Γn,0
R¥n'*)/,%) %51321%&)/,.N.-/A&@2A)A,pqA2w-/A-/.0+&/1,7A-/.>A7%)/) ApqA&@2w-/AEr+%=/,Y1A,tseu'N%v/1
FJV ′(f) =∑
λ
aλ,V ′(f)
-/A f A&7') %1&U2;A V ′ ∈ CRnoA`N%,%77'13,A5=/13>?'?&@2,Y.5;=/pyAC=/A7@=/A5)/,Y%)/,Y1.2.5.73.pqA&@2;'13,A5¦-/=
-/.>UA73%)/)ApyA&@2D-/A9r/%=/,Y1A,tseu'?N%v/1JR~&¸G=217315AE73A x '1:2D@=/A ∩σ∈SV ′σ∨ = C(V/V ′⊥)∨ A2C=/A
7A`V,%=/)A ΓV ′, n,0'V132w5=/,w7JW¹'?7Vºv/,A
∏
λ∈SV ′,n,0
H0(BV ′, n,0 , L(λ)⊗ ωket ⊗ ωkab ⊗M
).
!"#$%'&(&)*),+-*(.0/21*31"/2154687 9;:=<>@?A:@BCEDD:@FG:@HIJ<:LK;CNMPORQ5:@ORSUTVWXCZY@Q[<:
f:@H
V ′:X\@I]QH<>*D:@H<VNHI
<:Σ ^ :RI_:X\@I`QH?$V$ORQaVNHIJ\bCM\cBed VW@IaQ5CNHf<: ΓV ′, n,0 g 9hd VRDDBiQ5WEV$IaQaCNH λ 7→ aλ,V ′(f)
:X\@Ikjl\@MDZDCNORI<V$HP\
SV ′, n,0 ∩ C(V/V ′⊥)∨ .m!HnVFγ·V ′ (f) = γ · FV ′(f)
DCNMPOhIoCNMIγ ∈ Γn,0 g
p .Aqr+(bsZtu.v/21U31/21"/w687 xyV ′ ∈ C z|~eZUz|eA|zlE$X$"z|z| y eAeeAzG y A,"z
eR|G BV ′, n,0 z|~|$ Spec(Z[1/n]) z"zkz|Zz|z|zA y z|AA|A y z f z|~|"|z|]z
Z[1/n][[SV ′, n,0 ⊗ ωket ⊗M
]],
¡Z¢$z|"zh$ y ez|"z|z| q <>@?A:RBCXDD:RF=:@HI z f 6£;¤a¥¤¦§¤G¨k©Nª¬«8ª®°¯¢±¯v²c³wN´;¯©¤¶µ · $!|z`AeAAE$zAA|ZeezJzJz|~ y A y P y zlz¹¸[ºRz|=z|f y y EAc"»ezlz½¼¡¾¿ÁÀÂ;Äà 6"ÅÆ zc»~ZA y y Zfe y $$EzA ye|eÇEzhz¼ÉÈ0Ê¢ÂË Æ z[z¼ÍÌΠ£ Æa6Ï (&AÐ;&ÑN-*Ò-e&Ó/21*31UÔ§1U3687 ÕCNMIo:_WXCNFDCb\VNHIo:'QOROE>E<$MWRIaQaYB:<: Ag,n,0× Spec(Fp)
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y "z|!z Ag,n,0 × Spec(Fp) EAz|]Aez||z|z|!|ZAEAez| y e~| y "z|z Aordg,n,0
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Aordg,n,0 −→ (Z/nZ)∗ × (d′1 < d′2 < · · · < d′s) | 0 ≤ d′i ≤ di ∀ i ≤ s
y L0Zæez (A, c : (Z/nZ)2g ∼→ A[n], H•
) z Aordg,n,0 Aee y z'"z`ç,$Ez|z' y y y ez'z c
z!"ç,$ y "zJçZ~|zJz|!eAA!_ y y N y ç¬ d′i z| Hi
6ß | y !zJ_~ZA y y Zvè ¤ è ¤,¥¤ Ë z_R zL|AZE$ezL|Az@zLz Aordg,n,0 ~z||Zeez_zLAez Ag,n, 0 × Spec(Fp) é W¬êR6 NE$ZeAz ¥¤ Ë ¤ëì 6
p .Aqr+(bsZtu.v/21U31UÔ§1¬í687 î ïç, y |JZðf|AE~ z»EAEzr|ZZEAEz y ~e| y z½zAg,n,0 × Spec(Fp) ~z||ZE~zhz V ′ U~eENEzhZv|z|~E y V ′ ∈ C G y G$ 6%'&(&)*)¬+-*(.0/21*315Ô§154687 ñQ
FJV ′(f) = 0DCNMOhIoCNMPI
V ′ ∈ C ^ CNHnV f = 0 gÖ >@F=CH\@IaOeV$IaQ5CH g× x ;keR|G A y U|Z|~z|b"ç¬Ae$ y AzÙ"]|ZAA"A y zÄ|A|ez|ez|Z_Ez$P y y ez| y y Zz|'zLç¬ y E|z$P 6;ß òz|'ZLEAEz| z M z|~'zL*óPz
! "$#%'&)(+*,)-
Z ./ ,)(0*213,)4 M 4*652789:<;Z/pZ . ; Z[1/n] = ,>;
Z/lZ .=? l 4*652,)&@& =ACB -4 / -64 A (04-1D,)(&)4E)(0F!(0*4 / 9* np G'H & / 9I-65(J,):0(4- . M 4*65K,)&L9I&)&)4M9,45K:ON = &P9H0(Ag,n, 0 , ωk ⊗M
)= H0
(Ag,n, 0 × Spec(M) , ωk
).QD(
M = Z[1/n] = , Z/lZ .'/= ,)-R5 = ,5 σ ∈ S:SNT( A 9I84E), AU= - /)V (0* A 4
Mσ,n,0 × Spec(M) −→ Ag,n, 0 × Spec(M)E = &)&)7 / 9-W:9 / - =/ = *(05( = &>XYXYSZ[Y\L9],)&)4^(0&354-*4J5_( = &P& = &PF!(0E)4C9MF`4Ja5 = ,54*b:4*bJ =Ac/= *_9&`5_4*(-6-7E),)J5( B :04*dE)4 Ag,n, 0 × Spec(M) Gfeg= ,)-hJ = &)J:,)-64 . (:b*,ic5dE)4j-64 A 9I-1D,)4-h13,)4 /= ,)-]5 = ,5*J V 7 A 9a-7E),)(k5 X .3/ = ,)-l5 = ,5 = ,F`4-m5fE)4&)*4 U 45 /= ,)-l5 = ,5lnS9(0*4M9, F : = JM9:04 A 4&35o:( B -4b*,)- X .:9p-4*65-(0J5_( = &H0(X,F) −→ H0(U,F)4*65^(0&rq4J5(0F4 G QD( M = Z/pZ . ,)&s-_9I(* = &)&)4 A 4&35^9&'9I: = 8,)4U*Nt9 /)/ :(01D,)4 Ggu &s,5_(0:(0*_4U:9 / - =/= *(kv5_( = &xwyYSZ'YzYSZ|45:4dJM9-9J5~-4d-67E),)(05pE)4 Ag,n, 0 × Spec(M) [ X! . 4&x54&'9&`5pJ =Ac/ 5_4dE)4*E)7F`4: =/)/ 4 A 4&35_*KE)4a = ,)-(04-mvm9IJ =B (4&5 = ,)*W:04* V ′ ∈ C G
]'$+< Gg Q = (k5 R ,)&x*6!*65_~ A 4dE)4d-64 / -67*4&359I&35*E)4* = - B (054*E)4d* = ,)*mv4* / 9J4*5 = 59I:4 A 4&35(0* = 5- =/ 4* A 9( A 9I,!nS9IJ5_4,)-*E)(0-4J5*E)4V
* = ,)*:ONT9J5( = &E)4Γn,0 GK Nt9 / -~*:9-4 A 9-61D,)4cwyYOZ'YzY\ . :ONT9 /)/ :0(JM95_( = & f 7→ FJV ′(f)V ′∈R
4*65a(0&rq4J5_(kF`4 G QD(<:ON = &|*_4c-64*5-4(0&35^9I,!n = - A 4* AU= E),):9I(-64*W;FI9:04,)-*KE'9I&)*K,)& AU= E),):04*_9&)* p v5 = -*( = & . (0:-7*,):054E)42:9pE)7 Ac= &)*65-_95_( = &E),J = - = :0:9I(-64wyYOZ'YzYX1D,)4 f 7→ FJV ′(f)
4*5K(&rq64J5_(kF`4 /= ,)-R5 = ,5 V ′ ∈ C Gu &E)7E),)(k5WE),J = - = ::9(0-4wyYSZ[YzYX,)& / -6(&)J( / 4E)4E)7F4: =/)/ 4 A 4&35WE)4a = ,)-6(4- vm9J =B ( G
¡2¢ ¢)£¤£ '¥¤r<¤ Gg ¦§¨©«ª)¬ M ⊂ M ′ ©®¯ Z °¤±]§ ®!²©_³P©«¬ f ∈ H0(Ag,n, 0 , ωk ⊗M ′) ´¦[¨ FJV ′(f) ©³«¬µ¶§$©¸·¶«¨O©«ª'¬¸³ `¹ ª³ M º'§r®!»¬ §®!¬ V ′ ∈ C ¼§rª ¹
f ∈ H0(Ag,n, 0 , ωk ⊗M) .
u &E)(0* / = *478`9I:4 A 4&35WEN,)&)42F`4-6*( = &E', / -6(&)J( / 4E)4a½b¾DJ V 4- G¿C ¢Ày¢)Á ¥¤Âm¥ ¢)à <¤¤Ä Gg ¦[¨ g ≥ 2 ¼l²6Å ¹ º`º²Æ¨¶ ¹ ¬O¨§rª ©»©_³«¬O»¨¶«¬O¨§rª@»_Ç ¹ ²¨³M©a®!ªj¨³$§±]§»¤º)ÈD¨³«±]©
H0(Ag,n, 0 , ωk ⊗M
) ∼−→ H0(Ag,n,0 , ωk ⊗M
).
É Ç«±h§rª³«¬O» ¹ ¬O¨§rª ´gÊ u &h*_4b-67E),)(05fJ =AUA 4 / -7J7E)4 AcA 4&35f9,hJM9* =? M = Z[1/n] = , Z/lZ . 45=? l &)4WE)(0F!(0*4 / 9I* n GDË 4W*J V 7 A 9 Ag,n,0×Spec(M)
4*65o,)& = ,F`4-m5oE)4&)*_4WE)4 Ag,n, 0×Spec(M) GË 4s*J V 7 A 9 Ag,n, 0 × Spec(M)
4*65P-7E),)(05 GÌu &Í4&ÎE)7E),)(05P1D,)4|:ONT9 /)/ :0(JM9I5( = &ÍE)4|-4*65-(0J5_( = &4*65c(0&rq4J5_(kF`4 GÐÏP= &35- = &)*c1D,N4::044*65c*,)-Ñq4J5_(kF`4 G Q = (05 f ∈ H0(Ag,n,0 × Spec(M) , ωk) GÒeg= ,)-5 = ,5 V ′ ∈ C A 9r( A 9: .M= & / 4,5n = - A 4-Ó,)&E)7F4: =/)/ 4 A 4&35yE)4Ð = ,)-6(4- vm9IJ =B ( FJV ′(f)9&'9I: = 8,)4;cJ4:,)(ÓE)4a:9cE)7%'&)(05( = &PwyYOZ'YkwyYOZ G[Ôb= 5 = &)* Λ
:ON4&)*4 AB :04aE)4* λ ∈ SV ′, n,0
5_4:0*W1D,)4aλ,V ′(f) 6= 0 GÕ :f4*5p(0&3FI9-6(9I&35p* = ,)* ΓV ′, n,0
45p* = &x(&`5_4-6*_4J5( = &Ö9MF4MJSV ′, n,0 \ σ∨
4*65%'&)(4 / = ,)-C5 = ,5 σ ∈SV ′ G<u &E)7E),)(k5CE)4 ×ØpÙ)y. :04 AGyÚCGÛGtÜ o1D,)4 Λ ⊂ C(V/V ′⊥)∨ Ý 4&4Þ 45 .ß= &-64 A 9I-1D,)4]1D,)4
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aMV ′,n,0
Qbc3d053e1_51?):6f>?0f45'*_6B0531
MV ′′,n,0
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=h@?):4f>?0f45'*_6B0531
a Ag,n, 0 Q
i]j(kjlmj KonpqsrEtupvwfr/nvyxwzvw jP| FE):0536~?30uFED7+-08+-F1?&#,/):DA0f ∈ H0(Ag,n, 0 , ωk ⊗M)
=?3d4AF/?4054B@,%950
V ′ ∈ C06f?3d95/30
σ ∈ SV ′
QPX,%34f950676B0@,/D6B):0E=h3F/?4f+2FE36AD7FE34f>?0&,DA05476ADA):96B):FE310
fa &,
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σ06[1_5;3)6?30YuFED7+-0+2F1?&,%)#D7010
.E053D70g − rang(V ′)
Q/]WA`gg Q ] [¡B¢¤£¦¥(¡¨§ª©¦¥(§(«W¬®¢
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C(V/V ′⊥)Q
bc3®DB,%@@g05&#&:0>?0[@gFE?D0 ≤ i ≤ s
=y�[4A9¨G_5+[,,%Äg_5&#):053AV ′, i
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D
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Q'U,%@DAÈ54&#,@DAF/@F/4A):6A)#F/3 kjiyjlmj(k =y&#,ÉÈ59¤G0 BV ′, n,0 → AV ′, n,0
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 Ag,n,0 QX,%34¼?3@D705+-):05D6B05+2@4=FE3Ú4A0RD7054A6ADA05):36,%?Ú9,/4ÛFEÀ
n > 2pg√
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S05476X?3Z4B9¤G_5+[,f3FED7+[,%&0¦Ë9505&:ᓵ=4A)
G →S05476-?3I4B9¤G_5+[,Z4A05+2),%Ä_5&:)#053I06-4A)
ωG/S1_54A)#./30Y�Y1_6B05D7+-):3,/36-1?sÇ,/):4A950,/?I10542FED7+-054
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m6B05&>?0
ω⊗mG/S
4AFE)6[053.E0531D7_@,/D-4A054[4A0596A)#F/34[.E&:FEÄ,%6-4B?DSQè\N04A9¤G_5+-, Ag,n, 0 054760¦ËÌ9505&#&:0536[06[3FED7+[,%&
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6A05&ê>?0ω⊗mi
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ϕ : Ag,n, 0 −→s∏
i=0
P(H0(Ag,n, 0 , ωmi )
).
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g,n, 0(.+81(SPQ$ -.,0&%)/5'j$k.,/5041DO( ϕ (*1
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ψ−→s∏
i=0
P(H0(Ag,n, 0 , ωmi )
),
04l ψ (*').m81GCE04%A#F/')CG(n1/<;40&lE3(*'on6%A(*'mO( π ')041T.m10&1Gpq/5O(*'m(.r2&M*04CGM.,%A/JT8(*CE(*1T.m-*0411(?>q(*';(.i0&l π∗OAg,n, 0
= OA∗g,n, 0
9rs 048%t.,0&8@. 0 ≤ i ≤ s ;u/3(?>@/'A.)(D81vPw$&/5',-*($ 8x/1Tp4(*%)')/563( Li ')8%P(H0(Ag,n, 0, ωmi )
) J8/7(*'A.h.)%)N*'$&CG#3(y(.h.)(*3J8( ϕ∗Li ∼→ ωmi 9@z ef(?>@/5').)(*1-*(yO8I#30&12&(*CE(*1.O(R(*24%A(GUXWY|[]_]=` 9~a c=CE0&1T.,%A(SJT8(
si=0Li (*'A..)%)N*'$&CG#3(S')8% ∏s
i=0 P(H0(Ag,n, 0, ωmi )
) 9s 0&')041'L = ψ∗ s
i=0 Li O=e~$ #%)N*'UfWY|[ ]_]L` 9~q9~a c_;m-&ef(*'A.i81vPQ$&/5')-*($&8$&CG#3(D')8% A∗g,n, 0 9m 3%)M*')83.,(O(EUfWY\[]_]` 9b@9ba cJ8(
A∗g,n, 0 = Proj
(⊕
k∈N
H0(A∗g,n, 0 , Lk )
).
0&CECG( π∗L ∼→ ⊗ωmi (. π∗OAg,n, 0= OA∗
g,n, 0
;041i$ H0(A∗g,n, 0,Lk) = H0(Ag,n, 0,⊗iωkmi ) #70&8%
.,0&8@. k ≥ 0 9 H 1D10&.)( k = (k, · · · , k) ∈ Ns+1 9 e~$ #%)N*'UXWZY\[]_] 9 ` 9 c<;0&1$
A∗g,n, 0 = Proj
(⊕
k∈N
H0(Ag,n, 0 , ωk )
).
z $G#%A04#704')/.,/5041DQ|CE0&1T.)%)($ 304%A'+J8( A∗g,n, 0
(*'A.+/1OM*#7(*1O$&1.LO( Σ 9r4|oQ 9 =_? ¡ g ¢ m ?¡ N ¡Q£jk¤B? ¡<_£j¤K¡¥?¦b¤S§¨ m > 2Ng
√2N ¢¤k¡ K ¨q V©,k£ª¤
?¡m¤«_? ¡ G ?¡ G′ ¬ ?¨ i¤«©j®¯°i±¤²±4³¯?¦´_ @¤ ¬ ¶µT £, g ¤¨q£ Spec(K) °t¨q ¤ ¬ ª¦± £¤± ¡Q_k @¤ λ?¡ λ′ ?¡ ¬ S¤?¡<£¨©¡<¨q£,j¤ ¬ B ·&,±k¨Gª7£ ¸©~ª± ¦j¤²? m ¹º λ j¤¡ª7£ ¸©?~ª@± ¦j¢r¦»A4¤¡¥²± ¨Gª¦´¨T¤¨ ¸¤«)µT¯? _ f : G→ G′ ©,k°yª±k¡<<³¦¼± ¨&E¤?¡<£¨©¡<¨q£,j¤ ¬ ·&,± ¨½? m ¡¥?¦¦²§¨ f t λ′ f = Nλ ¢k¾ f t ¬ ¯j¤?5µ ¸¦A¿ ¤«,µT¯ _ ¬ ¨± ¦ ¬ f ¹À ¯?°g @¤?¡<£)± ¡<_ ¹Á R0&/(*1T. f (. g O(*8q>/5')042&M*1/(*'g',$k.,/5')Pw$&/5'j$ 1T.g3(*'iFTÂq#70&.)FN*')(*'gO8Ã35(*CECG( 9 0&CECG( Hom(G,G′) '*eb/5O(*1T.,/n(ZÄG81D%)M*')($&8DO( End(G)⊗Q /1-*358'LO$ 1' End(G)⊗ Z[1/N ] ;041#7(*[email protected]/5% f (. g -*0&CECG(EO(*' Q Å (*1O04CG04%A#F/')CG(*'O( G .)(*3'J8( Nf (. Ng ')04/5(*1T.O$&1'End(G) 97 eb$&#%AN*'¼UfÆÈǺÉËÊÌ;-*04%A0 9Í«Îq9~a c_;3$ Q Å $&3524N*6%A( End(G)⊗Q (*'A.K')(*CG/ Å ')/5CE#35(O(%,$&12≤ 4g2 9Ï 0 .,0&1' Tr 3$¶.)%,$ -*(+%AM*O8/5.)(+JT8/38/$&')')0-*/M*((. q 3$yPw04%ACE(J8$ O%,$k.,/5J8(',8% End(G)⊗QJ8/(*1p0&/( u ')8% Tr(utλuλ−1) 9 (.).)(yO(*%)1/5N*%)(y(*'A.OMn1/(y#704')/.,/p(y#$&%yUbÆÐÇÉËÊ@Ì;T.,F 9 Í495Í c(.#%)(*1ODO(*'p $&35(*8%)'+(*1T.)/N*%A(*'+')8% End(G) 9s $&%FTÂq#70 .,FN*')(&;0&1$
q(f) = Tr(p) ≤ 4Ng2 , q(g) ≤ 4Ng2
!"# $%
&')(+*,&.-/(#01'2& u ∈ Hom(G,G′) '2&3*45& f − g = mu 687:9<;>=?@9Am2q(u) = q(f − g) ≤ q(f) + q(g) ≤ 8Ng2 .
B ?@CDC & m > 2Ng√
2N E;@FGH/E8? ' GI 02&@J ?K9 & 9L=M3= 5(N')45& q(u) < 1/N2 687OF?K9<;Nu ∈ End(G)
&' q EF & 9=L= &30P ; *#&35 F 0Q& 9 '2( I3F &30Q0R5 F End(G) J =?K9A u = 0 &' f = g 6SUT@VXWY[ZK\]TL^`_bac_da 6fe gh9iE8? 0 ;@9 ' N = 1 J G = G′ &' λ = λ′ =;@9 0U*+&<*#& C>C & EF1M3A3M3= & 9 '3J?K9jC>?K9 ' F &L45&k' ? 5' ; 5' ?@CD?@FREG (+0 C & = & G EFRM 0R& F P ;%9 'U5 9 & E8? * ;@F (#0 ; '2( ?K9jEF ( 9A ( E; *+&L&'U5 9 &01' F 5 A '25 F & EF ( 9A ( E; *+& = & 9 (+PK& ; 5 > 2g
√2 &301' M3lK; *nmk*dop( = & 9 'R(+' M@6 B & A (h&301'q5 9 &sr ?KF1C &sr ; (#t*#& = 5*#& C>C & = &vu& F1F &Dwyxz@|~ 6
vY@cR^`_da,_ 6fe c)2sU.k π %/ k¢¡R£ U[k Ag,n,0 ¤ . [¥2.¦§¦¨/¦ ¤ ¦`U.3©ª .¢¦.R%¦©¬« B ?@CDC & ϕ∗Li ∼→ ωmi
Jh*#&r ; (+0 A & ; 5 ωmi &301'>' F (NP/( ; *#(+0 M *+&L* ?K9l= &30s®t F &30= & π E8? 5 F 0 ≤ i ≤ s 6]¯ o ;@EF1I 0°wp±³²D´µ¶J·' G6c¸¹6»º6y¼ ~ ? 5½wp¾"¿>À,J EF1?KE6cÁ¹6ÃÂ6à~J ?K9 & 9X=M3= 5(N'45& E8? 5 F ' ? 5' E? ( 9 ' lKM3?@CDM ' F (+45& x = & π(Ag,n,0) &'D' ? 5' 0 ≤ i ≤ s Jn* ; ®t F & π−1(x) &301'( 9A *+50R& =;@9 0 Ag,n,0 &'¢* ;A * ; 0R02& = op(+0 ?@CD?@FREG (+0 C & = 5j0 AGM3CU;!; t M *+(#& 9 Gi
&301' A3?@9 01' ;%9 '2&L0R5 Fπ−1(x) 6Ä &:r ; (#0 A & ; 5 = &30"& 9=?KC>?KF1EG (#0 C &30 = & Gi
&301' A3?@9 01' ;%9 '"0R5 F π−1(x) JmvP ; *#&35 F 0 =;@9 0"5 9Z ÅÆC>?= 5*#&Ç*+(#t F & = &)' H/E &)® 9 ( 6 B &3* ; ( CDE *+(#45&Ç45&:* ;ÈA * ; 0R0R& = op(+0 ?KC>?KF1EG (+0 C & = 5¢0 AGM3CD;v; t M *+(#& 9E8? * ;@F (#0 M Gi
&30R' A3?K9 01' ;@9 'R&Ç0R5 F π−1(x) 67:9¢F2;%EE &3*+*#&Ç45o ?K9 0R5 EE8? 0R& n > 2pg√
2p 67:9°;%EE *#(+45&*#&k*+& CDC &kÉ·Ê˶ÊdÌXm G = G0Jnm G′ = Gi
Jnm¨* ;¨E? * ;%F (+0 ; '2( ?K9ÍEF ( 9A ( E; *+& λ = λ0 = & G0JhmÎ* ;
E8? * ;@F (#0 ; '2( ?K9ÏA;@9?@9 (#45& λ′ = λi = & GiJm m = n &'Om N = p E8? 5 F 0 ≤ i ≤ s 6,7:9 & 9Ï=M3= 5(N'45&Ð* ;>A * ; 0R0R& = op(+0 ?KC>?KF1EG (+0 C & = &Ð* ;>AG;Ñ9 & = o»(#0 ?@lKM39 (+&30
G0 −→ G1 −→ · · · −→ Gs
& 9 ' F &:0 AGM3CD; 0 ; t M *+(#& 9 0 E8? * ;@F (+0 M 0&01' A3?K9 01' ;%9 'R&:0R5 F π−1(x) 6 B &3* ;vC>?K9 ' F &:45&:* ; ®t F & lKM3?@CDM.Å' F (#45& π−1(x) E8? 0R0 I3= &Ð5 9 5 9 (#45& E? ( 9 ' l@M3?KC>M ' F (+45& 6Ò YÓÓdWYTk^`_dac_#^ 6fe ,U2.Ô.¦! π Õ Ag,n,0 2.§/¦Ï[U ¤Ö ¢¬/O[¦¨ U[3©ª .¢¦.R%¦©×« ¯ o ;@EF1I 0Ø* ;ÙEF1?KE8? 02(N'2( ?K9 ÉcÊËÊÚÛ&'Ø*+&ÝÜ %¦ Þ Ö 2. = &Ýß ;@F (#0Rà(w»áqâãåäää º6»º6pæ ~J* ;ÈF &301' F ( A 'R( ?@9°= & π m Ag,n,0 &301'§® 9 (+&:0R5 F 0 ?@9 ( CU;%l & 6Ä·;qEFR?@E? 0R(+'R( ?@9 ÉcÊËÊÚU&'*#& A;@FR;@A ' I3F & F1M3= 5(+' = & Ag,n,0 C>?K9 ' F & 9 '¬45&s* ;°F &301' F ( A '2( ?K9Î= & π m Ag,n,0 &301'Ð5 9 & ;@EE *+( A; '2( ?K9t( FR; 'R( ?@99 &3*#*+& 6 B ?@CDC & π∗OAg,n, 0
= OA∗g,n, 0
&' Ag,n, 0 &301' 9?KF1CU; *J¶*+&¹' GM3?KF1I3C & = &30Or ?@9A '2( ?K9 0r ?@FRC &3*#*+&30çwèáqâkã äää º6#éK6y¼ ~ CD?@9 ' F &X45&X*#&X0 AGM3CU; A∗g,n, 0
&301' 9?@FRCD; * 6:êf? 5'R& ;@EE *+( A; '2( ?@9® 9 (#&Èt( F2; '2( ?K99 &3*#*+& = op( CU;@l & 9?KF1CU; *+&È&30R'Ç5 9 (+0 ?@CD?@FREG (+0 C & 6 B op&301' =?@9A *+& A; 0 = 5 CD?@FREG (+0 C &π : Ag,n,0 → π(Ag,n,0) ? t'R& 9 5 E;%FQF &30R' F ( A '2( ?K96
!"#%$'&()+*-,.0/213546.576,8$09:23576.;$535)+<$')+*-,=(7
π:>&@?=: &A$'3;76.0.;$'35: $576.0(7
Ag,n, 0 BC *-D2DE7π76.;$0(*-,,41:>3F9HGI4J>: 9&:K$')+*-,L(7NM*-3;DE76.OD2*(&9:>)+3576.O(7=P)+76Q-769HRS97N<6*-3;*-9+9: )3;7UTWVYXVZ[V\])D219)+^&7N^&7U1_*-&3$'*>&A$
V ′ ∈ CR9:23;76.;$'3;)<$5)*>,=(7
π/29:
V ′ ` .;$'35: $57a.57bMY:><$5*-3;).57a76,πV ′ : AV ′, n,0 −→ A∗
g,n, 0 .c ,ED2*-,8$5357<6*-D2DE7d13;46<646(76DED276,8$e^&7πV ′
76.;$f&,2DE*>351g)+.'D27h,)i)35: $5)*>,,769.5&3j.5*-,2)DE:>Q>7>R(*-,$k976.lhi3;76.kQ>4*>DE4$535)+^&76.m1*>.5.5n6(76,8$k&,o&,)^&7F1_*-)+,8$kQ>46*-D24$'3;)^8&7 B_p J :>,8$m(7q13;*-&AJ-763m9:,*-3;Dr: 9)s$'4a(7πV ′(AV ′, n,0)
R)+9,*>&.MY:>&A$d4$t: i9)+3d^&769+^&76.d3546.5&9s$t:K$'.d135469+)D2),: )3;76. BuFvw>xWwyKz|z5w~%Y BS F>bH5 ' 5 rH'ttS 08tlr0 m'08 C/Γn,0 8tar8tqKKtA π FAH5 H P*-)s$
x&,¡1_*-)+,8$aQ>46*-D24$'3;)^&7O(7
A∗g,n, 0 B£¢ G¤:>13;n6.E¥I¦§O¨©WR[135*>1 B[ªqB¬«B¬« R9:N3576.;$535)+<$')+*-,¡(7
Gi
/π−1(x)
1_*-&30 ≤ i ≤ s
76.;$b&,¡.'<tg46Dr:N.'76D2) ` : i_469)+7,¡(7O35:>,QN$'*>35)+^&7<6*-,.;$':>,8$ Bp )+,.')YR976.S35:>,Q>.SDF&9+$5)19+)<:K$')sM.S7$:>i_469+)76,.®(4h,).S(: ,.S9:l1: 3;$5)7lXV\[VZ_¯£(7°9:0<g:K±,7®TWV\[V² BIp ¯
G0 → G1 → · · · → Gs.5*-,$j<6*>,.5$':>,$'.%.5&3π−1(x) B c 3%(£G¤:>13;n6.%976.%3;46.5&9+$': $5.%(79:m1:>3$')+7bXV\[VZ£R <676.%3': ,Q-.%1: 3': DEn$53576,8$7?: <$'76D276,8$09YG¤76,.576DFi97
C/Γn,0 B³ 7´.'<tg46Dr:
A∗g,n, 0
76.5$N(*>,<oDF&,)m(£GI&,7o.;$'35: $5)+h<:K$')+*-,µ),(7?A467o1: 3C/Γn,0 B¶ * $'*>,.
ZV ′9:·.;$53':K$'7(7A∗g,n, 0
1: 3': DE4$5354671: 3V ′ ∈ C B P*-)s$ x &,¸1*>),$qQ>46*-D24$'3;)^&7r(7
ZV ′ B¶ *>$5*-,.76,<6*-3;7x9YG¤&,)+^&7a1_*-)+,8$0Q>46*-D24$'3;)^&7a(7a9:Ohi3;7a(7
πV ′ : AV ′, n,0 −→ A∗g,n, 0
76,x B¶ * $'*>,.
OA∗g,n, 0 , x97b<*>DE19+4$'4a(&=9+*<:>9+).54a.;$535)+<$
OA∗g,n, 0, x
(7A∗g,n, 0
7,x7$
BV ′, n,0, x97N<6*>DE19+4$'4M*-3;DE769°(7BV ′, n,0 × Spec(OAV ′,n,0, x
)97N9*>,Q¹(7N9:´hi3;7U(7
BV ′, n,0 → AV ′, n,076,x B
uFvw>xWwyKz|z5w~%Yº BS »tS;-a@¹¼K|½rO¾' KH¿OA∗
g,n, 0 , x∼−→
∏
λ∈SV ′,n,0 ∩ C(V/V ′⊥)∨
H0(BV ′, n,0, x , L(λ)
)
Γn,0
¿@% °@o¼KrK½OZV ′ , x
∼−→ OAV ′, n,0 , x
!"#$! %&
')(+*-,./+02143&0256,.798 :);=<?>@4ABDCFEHGJI K6K6KMLN$O?NQPSR6TVUW;X<&YYZ<?[]\^?_`>UaZb OA∗g,n, 0, x
ZBcbed$B4^?_`^?@4>fZg U2;=<?YYZ<?[DhZBjik^lY\b4d$^?YBm@4no?[U$daA@4ZBmB4[@jU$Zp\^?_`>U$nb4n)ik^?@4_qZUrhZ Ag,n, 0 UaZpU$^?YohZ
π−1(x)Ntsu<
>@4^?>v^lB4dwbxda^lY)yz|z9_`^?Yb4@4ZV~[Zπ−1(x)
ZB4bdaY\Ua[Bh<&YBU<V ′ B4b4@x<bxZVhZ Ag,n, 0×Spec(OA∗
g,n, 0, x)N
s;=Z+d$BcbxZY\Zh[>@cZ_`daZ@daBx^?_`^?@4>fdaB4_`Z@cnBx[UwbxZhZUW;=daB4^l_q^l@c>fd$B4_`Z
MV ′, n,0 / ΓV ′, n,0∼−→ A
V ′
g,n,0
Zbh[Jik<?dwb~[ZUW;X<&YYZ<?[hZBik^?Y\bxda^lYB@4no?[U$daA@4ZBB4[@ MV ′, n,0
ZBcb∏
λ∈SV ′,n,0 ∩ C(V/V ′⊥)∨
H0(BV ′, n,0 , L(λ)
).
s<qB4Z\^?YhZH<?B4B4Z@cb4d$^?YBxZhnh[dwbhZU$<q>@cZ_`daA@4ZZbh[\^l@4^?U$U$<?da@4ZyzWz tz|N
WJW N ,10,0V ′ ∈ C 31x ¡/+0W1+56¢£0256,.¥¤
π ¦ 3 V ′ § /+02143&0 ¨¤ Ag,n, 0 /S © 3¢£0,1£5/S ª £.«.D5/S,*¬,1®¯5/+*¬
πV ′ : AV ′, n,0∼−→ ZV ′
7')(+*-,./+02143&0256,.7#8 °^?dab
V ′ ∈ CNu±jY<-²[³~[ZªU<e@4ZBcb4@4da\bxda^lY³hZ
πg U$<
V ′ Bcb4@x<bxZªhZ Ag,n, 0B4Z)iW<&\bx^?@4daB¡<&dabmZY«[Y«_q^l@c>fd$B4_`ZπV ′ : AV ′, n,0 → A∗
g,n, 0
Nt´r<?@jhnµYdab4d$^?YTtUaZpB4^l[B B4\£fn_`<ZV ′hZ A∗
g,n, 0
ZB4bUW;=da_¬<&olZªhZπV ′
Nu±jY"<_q^lYbx@cnq~[ZªU$Zª_`^?@4>fdaB4_`ZπV ′ : AV ′, n,0 → ZV ′
ZBcbµYd¶ d$@4<&b4d$^?YYZU2Nr:);X<&>@4AB)U<¨>@4^?>v^lB4dwbxd$^?Y¸·tzWz ¹zWTUaZeB4\£fn_`< AV ′, n,0
ZBcbpY^l@c_¬<&U2N±jYºhnh[dabph[B4Z\^lYh¸d$B4^l_q^l@c>fd$B4_qZ»hZ»U<>@c^l>v^lB4dwbxda^lY¼yz|tz ¸~[Z»U$ZBx\¡fn_¬<
ZV ′
ZBcb-Y^?@4_`<?UWN¾½jd$YB4dWTπV ′d$Yh[dwb[YdaB4^l_q^l@c>fd$B4_`ZhZ AV ′, n,0
B4[@ZV ′
N°^?dab
n′ [YeZYb4d$Z@¿hdw²daB4d ¶ UaZj><&@n
_`<?daB¿Y^lYe><&@pNsÀZjBx\¡fn_¬< Ag,n′,0
ZBcb¿_p[Ydvh;=[YZm<&\bxda^lYh[Áo?@4^?[>Z
GSp2g(Z/n′Z)
TZb)UaZ¬~[^&bxdaZYb9><&@9UW;X<&\bxda^lY"h["Bx^?[B ol@c^l[>vZΓn,0/Γn′,0
B;=dahZYb4daµZ\<?Y^?Yd$~[Z_`ZYb g Ag,n,0 NVÂV^l_q_`ZJUW;=n²lZYb¡<&d$U
ΣZBcbqn~[da²&<&@4d<&YbqBx^?[BqU2;=<?\b4d$^?YÃhZ
GL(X)TUaZ
B4\£fn_¬< A∗g,n′,0
ZBcb¬_p[Ydh;=[YZ¨<&\bxda^lY¸hZGSp2g(Z/n
′Z)>@4^lUa^lYo?Z<?Yb-\ZUaU$Z»B4[@ Ag,n′,0
NVsÀZ~[^&bxdaZYbhZ A∗
g,n′,0
><&@Γn,0/Γn′,0
ZBcb\<&Y^lYda~[Z_`ZYbdaBx^?_`^?@4>fZ g A∗g,n, 0
N°dn ≥ 3
Td$UVZ+d$BcbxZ-[YºZYbxdaZ@n′ > 2pg
√2p
~[dVZBcbphda²d$B4d ¶ UaZ-><&@n
_¬<&d$B)Y^lYº><?@pNrľY
>v^lBx<&YbA∗g,n, 0 = A∗
g,n′,0 / Γn,0 ,^lY^ ¶ bxdaZYb[YZH\^?_`><&\bxdwµ\<&b4d$^?Y_`daYd$_`<?UaZhZ Ag,n,0 ~[dZBcbdaYhn>ZYh<&YbxZh[\¡f^ldQJhZn′ N
Å ?ƺSÇlÈrk6É N :jZHU<q_`Ê_qZ9_`<?YdaA@4Z?T^?Y»^ ¶ b4d$ZYb[YZ9\^?_`><&\bxdwµ\<&b4d$^?Y¨_qd$Yda_¬<&U$ZHhZU2;FZB4><?\ZhZJ_`^h[UaZB`ol@c^lB4B4d$Z@`<?B4B4^\dan g Ag, 0 NV°<hZB4\@4da>bxda^lY¸ZBcb`_`^?d$YB`nUano<&Yb4Z~[Z\ZUaU$ZhZ A∗
g,n, 0
\<&@daUik<?[bbxZYda@\^l_q>bxZhZB<&[bx^?_`^?@4>fd$B4_qZBh[\£f<?_q> Ag, 0 N±jY³Y^&bxZ A∗
g,n
U$<\^l_q><?\b4daµ\<bxda^lY³_`daYd$_`<?UaZªhZ Ag,n \^lYBcbx@c[dab4Zªh<?YBqC=Ë̪ÍÎÀTtÏpNÑÐNÑPR2Nu±jY@x<&>>vZU$UaZ¬~[Z Ag,n, 0 hnB4d$o?YZ¬U<\^?_`><?\b4daµ\<bxda^lY"b4^l@c^?Ò6h<&U$Z¬<?B4B4^\d$nZ g
ΣN:m<?YBHU$Z`b4fn^l@cA_`Z
B4[da²&<?YbTY^?[B9@cnBx[_q^lYBHU$ZB9>@c^l>@cd$nb4nB9hZ`U<\^?YB4b4@4[\b4d$^?YÁ~[Z¬Y^?[B²ZY^?YB9h;FZÓtZ\b4[Z@NusÀZ\<?BhZB\^?[@ ¶ ZBZU$Uad$>b4d$~[ZBnbx<?Yb ¶ d$ZY\^?YY[T^?YB4[>>^?B4Z
g ≥ 2N
!#"$&%(')*+-,/.1032546287:9<; =>
n ≥ 3 ? >@<ACBD>EGFHAJIKLEMONPGQSRTMURVKXWVK>ZYIA A∗g,n, 0 []\ WH^AOMGF_> `bacAdFZe [ AgfhK>EGI \ Spec(Z[1/n]) ikj @lMUWVKFZ>mAKF Ag,n,0 MUWVQnQoAnWVIp-A \ FacAKqErA ? AOEGF&KXW \ QoRV@lAGF&>KacP [ AGKaDR KFacA Σ is AtQoW \u[ N>EGQoA1a6vwWVIxG@y>zacA Ag,n,0 p-A \ E Ag,n EvwPGFHAGK]aAK|I:KQoW \u[ N>EGQoA1acA A∗
g,n, 0
p-A \ E A∗g,n i~ WVI \ F8WVI:F
0 ≤ i ≤ s ? @AJfx \ PacATWacAωi
EI \ Ag,n,0 EvwPGFHAGK]aAKIK`R >ErMOAR I|>Kp-A \ EG>ZxG@AEGI \ A∗g,n, 0 i s A []\ Wa I>FFHAGK:ErW \ >mAG@ ⊗si=0 ωi
AUEGFR Q [ @AzEI \ A∗g,n, 0 ? AGF>@lACBD>EFHAJI:K>ErWVQoW \u[ N>EGQoA
MUR KWVK>ZYGIAA∗g,n, 0
∼−→ Proj
(⊕
k∈N
H0
(Ag,n,0 ,
s⊗
i=0
ωki
)).
s AnErMUNPGQSR A∗g,n, 0
AOEFQI:K>a6v I:KAnEF \ RVFZ> fMURVFZ>mWVK@WrMURV@AGQAKF(`#A \ QoPOA [ R \ R QPF \ PUA [ R \ C/Γn,0 @REGF \ R FHAdREOEWrMG>mPOAdV ′ ∈ C
AOEGFMUR KWVK>ZYIAGQAKFh>ErWVQW \8[ NAd AV ′, n,0 i]j @ACBD>EGF8AI:K1QoW \u[ N>EGQoAMUR KWVK>ZYGIA
π : Ag,n, 0 −→ A∗g,n, 0 .s v >QoRcAd>Kp-A \ ErA [ R \ π acAz@REGF \ R FZ> fMUR FZ>mWKaA A∗
g,n, 0
MOWVKXMG>ZacAR p-AUMd@REGF \ R FZ> fMUR FZ>ZWVKacA Ag,n, 0YI:>AOEGF [ R \ R QPGF \ POA [ R \ C/Γn,0 i PGQWKqEGF \ R FZ>mWK i hDDCt# c¡UC A∗g,n, 0 ¡ Proj
(⊕k∈N H0
(Ag,n,0 , ⊗si=0 ω
ki
)) ¢G£¤¥¦¡d§d¥ZDDC
A∗g,n, 0
∼−→ Proj
(⊕
k∈N
H0
(Ag,n, 0 ,
s⊗
i=0
ωki
))
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g,n, 0 -¤ 0 ≤ i ≤ s9º -¦¡ j ¥ZS#C- |§ Ag,n,0 §´-
A∗g,n, 0
9X» -S A∗g,n, 0 #C¡ -S´-¥¡ A∗
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g,n, 0 # b¡D¤q¡l]D¦ c¡l¾D¢#-S¢¡¦²¤§kÁ¤ V ′ £uC¡U´ ¡ 9» #¥´n¢#¤¥Â¡Ud§ÃZÄkÄXÅ 9ÆdÇ ¡È§¤1¨#DCD¥¦¥´ Cd¬(ÅXÉ 9
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S
Àh`ijn<ayzKX[VYg[ZÁzKX[h¦opV<XoZc_[^`irn¼h`g»<acV<X[2h`ZÁZch`ijbeHZyopn<[qmhxÂrl,[ach`ZÁ_`VYg[g[h¬h`ZÁZpX[aSh`ZyoZch`ijbº
n<?^`qmbsh`g~ VYbeoGX[gÃZc_[^`irn²h`gÃYayVYX[?h`Z§Zch`ijbºHZcopn<[qmhÄZcX[a
S<Å©n<[ay¿`Z¯ÆÈÇ]ÉÊËNÌ4ͯÎÐÏKÒѺÏ8ÓÔªbsqh¬bmZyoph
X[g©ZcV<X,ZHZc_[^`ijn¥h`g©YayVYX[?h`ZG0 VYXdYh`ayoS\,n<g[Z G och`qKzKX[h¨?VYX[a4ocVYXo s ∈ S ªqsn¥_`VYix2V<Zpn<gkoch¨g[h`Xocach\[h
Gs
ZcV<bmoG0s
,Õh]Zc_O[^`irnG0 h`ZyoWZch`ijbºn<?^`qsbmh`gtZcX[a S _JnaZch`Z¯l,[ach`ZZcVYgfoW_`V<g[g[h¬h`Z`
cSc[" ¡H¢2¡¢2¡+Ö,F£ ¤¥g®Zc_[^`ijnjZph`ixbeHZyopn<[qmh0irnaczKX[^]ZpX[aSh`Zyo¥qNnj\[VYg[g[^`h0\~ÈX[g®Zc_[^`ijn
h`g©<acV<X[2h`ZGZch`ijbºHZyoOn[qsh¨ZcX[a
Sª:\~X[g©Zc_[^`ijn
H^6opn<qmhl,g[bKZcX[a
Sª:h6oS\~X[g[h¨bmijixh`acZcbsV<g
H → Goph`qmqsh¾zkX[h¾qsh¾zKX[Vopbmh`gkoG/H
ZcV<bmoZch`ijben?^`qsbmh`gtZpX[aS
× oOngko\[VYg[g[^`Z¯X[gtZc_[^`ijnÃh`gt<acVYX[?h`Z¯Zph`ixbeHZyopn<[qmhGh6oX[gZpV<X[ZHYayVYX[?hl,g[b~^6opn<qmh
H\[hGª
qshzkX[V<ocbsh`gkoG/H
h`ZyoØZch`ixben2^`qmbsh`gjZcb,h6o|Zph`X[qmh`ijh`gfoHs
ach`g[_`V<gkopayh_,n<zkX,h_`V<ij?V<ZOngkochW_`V<g[g[h¬h\[h
Gs
2V<X[a¥opV<Xos ∈ S VYbeo (G,H)
X[gDZc_[^`irn¼Zch`ixbeHZyoOn[qsh¦irnaczKX[^<2ÙqFh¬bmZyoph¦X[g[h©bmZcVY<^`g,bmhG0 → G/H
h`gkopayh²Zc_[^`ijn<Z¨Zch`ijbºn<?^`qmbsh`g[Z` VYgvg[V8«fn<Xvh`Zyo¨qsh¥YayVYX[?hH ∩G0 zKX[b?h`ZyoÄzkX,n<ZcbºÔl,g,b
h6oÚ[qsno§ZpX[aS V<bmo
(G′, H ′)X[gxnXopachÄZc_[^`ijnÁZch`ixbeHZyoOn[qshÄijn<ayzKX[^ÄZpX[a
SY¤¥g.ÛrÜ:ÝÞ[ßKàNáÛrâ¯h`gkopayh
(G,H)h6o
(G′, H ′)h`Zyo],na¾\[^6l,g[bmocbsV<g·X[g·ijV<ac[[bmZpixhx\[h
G\,ng[Z
G′ zKX[bÚh`gfdfV<bsh H \,n<g[Z H ′
!#"$%"$&$'( )+*,+!+-.0/21435687,9;:<=?>46@3A=B/CD=B1ED/C= FHG,94F686@3AI76@6JDK1LFM1N=PO
G0 //
G/H
G′0 // G′/H ′
C3J687,9;:<=?>4683>QC3J>4I<R6@F>Q>4368=PSTFHUR5B=?3/>Q>4D9S V .0/3XW Y[ZN\!]^W`_J3/!1N9;3 (G,H)
3a1(G′, H ′)
3>;1:F9bCRac/=P1N=?7/dD/d687,9;:<=?>46@30>ND9+e43Ia14=BOfC3Q/7[ghFDdiDFH>4=PS`c/=3a1j:5?FH1 Vk R/R9NFH5?=B>N7/>jlJ:9;R>43/!1j5B31N<R794m683@nopno qrFHDs2>4I<R6@F>E>436@=tSu>;1LFHU5?3>E6XFH94iDR> VvJw)xyfxzMT|4x~TT? V ZMW`_^ S ^Y[L]X^ZMXH U ^%Z _b!_^KY[_!_ S _
(G,H)_(G′, H ′)
!_ Y[L]X)Y_^2\HNZ [ _LYQY_WpYTha _NYXMN ]LYQY S 8N_LYpW`pWpZM^%¡ U W^ W ^¢W Y[ZMXZM W YX_
HomS ((G,H) , (G′, H ′))∼−→ HomU ((GU , HU) , (G′
U , H′U))
£J]¤Z)^KYp4HpW`Z)^ ¥ ¦¨§ FH::5?=BIFH14=?7/©C3j9;3>;1N9;=?Ia14=?7/©3>41ª=?/)e43Ia14=B«,3¬IF9 U 3>;1ªC3/>N3¬3a1S3>41ª>4R:F9;R V7,/!14947/>®iD § 35?5B3¯3>;1®>4D9e43Ia14=B«,3 V° 7,=P1 f D/@6@794:<=B>46@3¯C3 (GU , HU)
CF/>(G′
U , H′U) Vh± 5=?/CD=B1D/r687,94:<=B>46@3
aC3
HU
CFH/>H ′U ² D/r687,9;:<=?>4683 b C3 G0
U
CFH/>G′0
U
3a1¯D/r687,94:<=B>46@3cC3
(G/H)UCFH/>
(G′/H ′)U VK¦ 3>j>4I<R6@F> H 3a1H ′ >N7/!1¬c/=?>¬Ra1LFH5?3>j>4D9 S ² C7,/I a > § Ra1N3/Cd3/³D/6@794:<=B>46@3@C3
HCF/>
H ′ V´¦ 3>>4I<R6@F> G0 ² G′0 ² G/H 3a1G′/H ′ >47,/h1J>4368=PSTFHUR5B=?3/>C7,/I5?3>¯6@794:<=B>46@3>
b3a1c> § Ra143/C3/!10>4D9 S :FH9¯5?3A1N<R794m683¤noµno q V± 5f3s=?>;1N3:F9¯I7/>4RiD3/!10D/6@794:<=B>46@3C3
G0 ×H CF/>G′0 ×H ′ iD=´3/!«h7=?3 H ∩ G0 CF/> H ′ ∩ G′0 V¶J§ FH:94m>J·¸º¹@»¼ ²:947: V[±u½©V¿¾V?ÀaÁ ² I76@683®5B3®687,9;:<=?>46@3ÂC3 H∩G0 CFH/> H 3>;1´D/3Â=?686@394>4=B7,/7,DK«,39;143 ² 5?3ÂiD714=?3/!1C3
G0 ×H :FH9¯5?3G947,D:3i!DF>4=PS`c/=f3a1Q:5?FH1H ∩ G0 3>4109;3:9;R>43/!1LFHU5?3310RGhFH5´l G V± 5f3/#3>;1C3A6@Ã683A:7,D9
G′ VÄ /7,UK14=?3/!1¯F=B/>4=D/687,9;:<=?>46@3AC3 G CF/>G′ iD=Ra1N3/C f V
° 7=B1 (U → S ← S0)D/¢7,U!e43a1QC3J5 FXIFH14RG,794=B3
OuvCompCRac/=B3JCF/>Q5?F¤:FH9;14=?3dno ÅoBÆ VÇ /:F91N=BID5?=B39 ² S 3>;1¯D/>4I<R6XF/7,9;6XFH5fFMÈX/3AI7,68:5?3a1E:7,D9 5 F147,:75?7G,=B3CRac/=B3A:F9 5 § =?CRFH5CD>47,D>Su>4I<R6@FOÉ39;6@R
S0 V!Ê 71N7/> DegSt5 FAIFM1NRG7,9;=?3EC3>Â>4I<R6@F>Â>436@=tSu>41NFU5B3>Â6XFH94iDR>
(G,H)>4D9S1N35B>¤iD3
GU
>47=B1dFUR5B=?3/ ² iD3 GS0
>47=B1¤G,5B7,UFH5?3683/!1¤3s1N3/>4=B7,/ËC § D/Ë>4I<R6@FÌ>4368=PSFUR5B=?3/#C3J:FH9;14=?3147,9;=?iD3=?>47H1N9;=B«=?F5B3J>4D9S0
:F90D/#G947,DK:3c/=´Ra1LFH5?3>4D9S0 ² 3a1QiD3 H ∩ G0>47,=P101494=P«= FH5FDSuC3>4>4D>0C3
S0 V¶ 36@Ã683 ² /7147,/> DegStample
5?FXIFM1NRG7,94=B3C7,/!105?3>07,U!e43a1N>0>47,/h15?3>EOÉF6@=B5?5B3>
(G,H,L) ² 7,Í (G,H)3>;1¯D/r7,U!e43a10C3
DegSt3a1 L D/OÉF=?>4I3FHDr=?/!«,394>4=BU5?3IDU=B>;1N3>4D9
G·¸º¹@»¼ ² CRaO V¬±LVÏÎVÑÐVtÒ[Á 9435?FH14=B«,36@3/!12FH6@:5B3Ó>4D9 U V ¦ 3>³687,9;:<=?>46@3>dC3Ó5 FÔIFM1NRG7,9;=?3
DegStample
943>4:3Ia1N3/h1¯5B3>EOÉF=B>NI3FHDs29435?FH14=B«,36@3/h10FH6@:5?3>E>4D9U ² C7/IA>N7/!1EC3>¯=B>47,GR/=?3> V° 7=B1 (G,H)
D/Ô7,U!e43a1©C3DegSt Vª¦ 3¤>4I<R6XF GU/HU
3>41©FUR5?=B3/Ô>ND9U ² C7,/IdFC683a1©D/OµFH=?>4I3FHDF68:5?3¬9;=?G=?C=PcR LU :FH9j·ÖÕ¤×!تÙKÚ ² ½Q±VÏÎVÑÐ 31Û ±VBÀ,VÑÐÁpV,Ü 3bOµFH=?>4I3FHD©> § Ra143/C©3/©D/JOÉF=B>4I3[FHD=?/h«h39;>N=BU5?3r9;=?G=?C=PcR L >ND9
G/H Vb¶©§ F:9;m>¢·Ñ¹©Ý)ÞK»Å ² :947: VjÎVÑÐÁ ² L 3>;1¤6JD/=¯C § D/3#>;1494DIa1ND9;3IDU=?>;143IFH/7/=?i!D3 V¦ 3A6@794:<=B>N683 f : G→ G/H3>;1Ec/=3a1¯5B3AOµFH=?>4I3FD
f ∗L >ND9 G 3>;1¯6JD/=C3E5?F>41494DIa14D943 IDU=?>;1N3 7,UK143/D3E:FH9Â=B6XFHG,3 =?/!«,394>43 Vh¶J§ FH:94m>¯·Ößà³áãâpâ ÐVäKVåHÁ ² 5B3 OµFH=?>4I3FHD f ∗LU
!"#%$&'()* ++,.-0/210,.3&45/7698:,.;<,.=>/?4@;BA3%,C-D1 U E F =G,.=IHJ.HD69/2K>D,C3MLNDO3P6H, L HJRQ=69/2D=STN*=UR/7,RD1
DegStample −→ DegSt
KVD6W,.-0/C,.--,.=>/76%,.3P3%,.;B,.=:/C-7D1X,.UR/6%S EZY ,[STN*=UR/7,.D1\=L,.-/CA4@-CA3%,.6P=,.;<,.=>/\QH].3P,[AD6P-KVD,[3P,.-\;BN*1_^A`6P-;<,.-aH,b3P4<U45/7J.c*N106P, DegStample-N*=>/2H,.-a6%-7N*cJ.=6%,.- E
dfe*gihjlkmnepoqsrqtrquo Ewv x DAA N-N*=-yK>D, S -N69/yD=z/174@6%/H,|AN*6P=:/ST,.10;BJ S0 E<x N6%,=:/ GD=|-7U`J.;f4i-,.;<69^s-0/4@O3P,~-D1 S KVD6,.-0/[email protected]%6P,.=|-D1 U ,R/ LU D=S4@6P-U.,4*D|U.DO6%-/,~-7D1 GU E L4@A1].-B /` EE%E%. 6%3w,"6%-0/7,D=,NO-0/1DUR/76%N=fJR/7,.=H10, LU ,.=~D=S4@6P-U.,4*D~U.DO6%-0/7,-D1 G E\Y ,R/0/7,!NO-0/1DUR/6PN*=U.N*110,.-AN*=HD=,4@AA3P6%U4@/6PN*=KVD4*H14@/6PKVD,!-D13%,ic*1N*DA,!H,.-U.N;<A N-74@=>/7,.-2U.N*==,",.-2H, GS0
E ND-?48:N*=-2;BN*=>/1JbK>D,bU.,R//,4*AA3%6PU45/76%N=G,.-0/2=VD3%3P,b-6 G ,.-0/;D=6 HLD=I;f4@1KVD4@c,CA4@1 H ,R/2-6 LU H,.-U.,.=HG GU/HU EM& T¡¢£>¤ ¥¢¤Z¦n T¡R§l£V¡B¡¤Z¨ª©T£V¡ 1 «¬p §® °¯t¡*²± x N6%,.=>/ S D=G-U`J.;f4³,R/ M = [Y
i→ G],R/ M ′ = [Y ′ j→ G′] H,.D³ 1 ^s;BN@/769ST-a-D1 S E´µ¶\·n¸t¹_¸º·»oqsrq¼½qsr Ewv ¾ =, Gm
^sO6P,³/,.=-6PN*=BH, M ,R/ M ′ ,.-0/¿D=,ÀST4*;B6%3P3%, (E , t, t′) N*Á E ,.-0/D=, Gm^sO6P,³/,.=-6PN*=BH, G ,R/ G′ N*ÁB3L6%-N;<N10A`6P-;B, t ,.-0/¿D=,À/1698³6&4@3P6%-74@/6PN*=BH,À3&4\O6%,RV/7,.=-6%N=
(i× Id)∗E ,./aNÁI3L6%-N;<N10A`6P-;B, t′ ,.-0/aD=,\/1698³6P4*3%6P-74@/6PN*=IH, (Id× j)∗E /,.3P-aKVD,(Id× j)∗ t = (i× Id)∗ t′ .
x 6 Biext1(M,M ′ ; Gm) HJ.-6Pc*=,\3P,Âc10NDA ,\H,.-ÀU.3P4*--,.-ÀHL6%-N;<N10A`6P-;<,\H, Gm^sO6%,³/7,.=-6%N=-
H, M ,./ M ′ N*=Ã4 Biext1(M,M ′ ; Gm) ' Hom(M ′,M t) A4@1[£V¨5ÄÅ EÇÆVE Å ME 3n,"6%-/,HN=UD=,ÈRÉuÊ_ËÌÍÊÎ"ÏÉuÐ5ÎpÑ:Ê\ÒaÐ5É°ÎÓÔ@ÕÖ PM H, M ,R/ M t KVD6 U.N*110,.-AN*=HG<3L6%H,.=>/769/7JCH, M t E N*/N=- m p ,R/ q 3&4!;D3%/6PA3%6PU4@/6PN*=|,R/ª3P,.-ªH,.D³×A1N®X,.UR/6PN*=-ªH, G × G -D1 G E#Ø N*-N=-
Λ(L) = m∗L ⊗ p∗L−1 ⊗ q∗L−1 AN*D1B/ND"/BS4*6%-U.,4*D×6%=>8:,.10-76%O3P, L -D1 G EF =×14*AA ,.3P3%,GKVD,HL4@A1].-f¨®£"Ù A1N*A EnÆVEÚ* ½3P4IU45/7J.c*N106P,BH,.-CS4*6%-U.,4*D³ÃU.DO6%-0/7,.-b-D1 G ,.-0/bJ.KVD6%8@4@3P,.=>/,fI3P4U4@/J.cN*16%,CH,.-ÀS4*6%-U.,4*D³6%=>8,.1-6PO3%,.-a16%c6%H6%QJ.-a-D1 G E´µ¶\·n¸t¹_¸º·»oqsrq¼½qT¼ Ewv ¾ =ÃST4*6%-7U.,[email protected]/,B-D1 M ,.-0/bD=,<S4@;B6%3P3%, (L, s, t) ½N*Á L ,.-/bD=S4@6P-U.,4@DÃU.DO6%-0/7,[-D1 G N*Á s ,.-0/CD=,/1698³6&4@3P6%-74@/6PN*=HDST4*6%-7UR,[email protected]/, i∗L ,R/CNÁ t ,.-0/CD=,/7106%8V6&4@3P6%-74@/76%N=IH,C3P4<O6%,³/,.=-6%N= (Id× i)∗Λ(L) /7,.3%3P,CK>D, (i× Id)∗t = Λ(s) EÛ =A4@10/6PU.D3%6P,.1R-6 L ,.-0/ÂD=~S4@6P-U.,[email protected]%-/,-D1 M N*=H6P-A N*-7,HLD=,4*UR/6PN*=H, Y -D1 LU.N;<A4@/6PO3%,b<3L4@UR/76%N=IH, Y A4@1À/174@=-73P4@/6PN*=I-D1 G E
dfe*gihjlkmnepoqsrq¼½qMÜ Ewv x DAA N-N*=- i : Y → G 6%=®X,.UR/6%8, E L4*A10].-G3P,/`J.N10].;B,ÃH,pH,.-_^U.,.=>/,ª¨®£"ÙÅ EPÝ@ 3&4fHN*==J.,HLD=GS4@6P-U.,4*DU.DO6P-0/,-D1 M J.K>D6%8@4*D"/ÂfU.,.3%3P,HLD=ST4*6%-U.,[email protected]/,C-D1a3P,\ST4*6%-U.,4*D H0(M) Ex N*6%/ M D= 1 ^s;<N*/6%S¿,R/ L D=S4*6%-U.,4*D~U.DO6P-0/,-D1 M EÞ ,S4@6P-U.,4*D Λ(L) ,.-0/Â=45/7D10,.3P3%,.;B,.=:/;D=6VHLD=,²-0/1DUR/D1,²H,ßO6P,³/,.=-6PN*=H, M ,R/ M E® 3V,³6P-0/7,ßHN=UßD=;BN*1A`6%-;B, λL : M →M t
!#"$%"$&$'( )+*,+!+-.0/1324/
Λ(L) = (Id × λL)∗ PM 5687:9; .<2!4/ L />=?.A@CBEDF GH=04I M =>J ; 13/=?.<KLAM1N/H=O4I G 5P K 7 =Q / Q K=R1N/SLAT,I?MU ; =OLV/ λL /=?.W4 7 /SMT,1NKI ; =0KX. ; T 79 / M 5YZ\[Z ]Z_^a`bdc3ef3gihj`bdk(el)mjnobdc3`qprfsk(nteuXcaZwv x TLVLA/ 7y T 7 =zMKCI 9|7; IV1NK 7 T. ; T 7~9 /9 T,41/aLAT. ; =O4I4 7 = Q U LzK S 5C687 I0KCMM/11N/24/1N/ 7 T)K4 Ker(f) 9 J4 7 / ; =OT, 7; / f : M1 →M2/ 7 .OIO/ 1 LVTC. ; ==04I S /=?.LH4 7;<9 J4 7 / | 1.0IOKC. ; T 7 .OIOT ; = Q I0K 7 = 5W6879; .24/ f /=?.o@CF G= ;Ker(f) = GrW2 (Ker(f)) 5P K 7 = Q / Q K=R1/H,I?T,4M/ Ker(f)
/=?. |7;3 .0K1/H=O4I S 5 K_I Q ; MI?T24//=?. K4=O=O/ Q KCI1N/8IOT4M/ |7; /.M1NKC. Ker(f)/=?. .0K1/8MT,4Ia.0T4.0/ ; =OT, 7; / f 9 / 9 /,I MI?/L ; /IK4 Q KIOK Q . I ; =O. ; 2!4/=WI = ;N9 4/11N/= 9 /=MT ;N7 .0= 9 / S 5
Oio ¡O¢¤£¥t¦¥¨§¥t¦ 5s© ªE79 T,41/«LVTC. ; =O4I S /=?.W4 7 / KCL ; 1N1/ (M1,M2, f)RT,¬
M1/.M2=OT 7 . 9 /4 1 LVTC. ; ==O4I S /. f 4 7 / ; =OT 7; / .KC1N/ 9 / M1 9 K 7 = M2 5
687 M/4. ®T ; I4 79 T41N/SLAT. ; M1f→M2
Q TLVLA/S4 79; KC,IOKLALV/ Q TLVLH4.KX. ;Y1
f
// G
Y2
??
T,¬¯1\JT 7 K(MT= M1 = [Y1 → G]/.M2 = [Y2 → G] 5w687~9; .z24/ G /=O.z1 K&MKI¡. ; /%°±GB³²j´@µF¶²¨G··G 9 / M1 → M2
R/.E2!4/¸1¹J ;7)º / Q . ; T 7: Q T 7 T±!KC4 |7;9 / Y1 9 K 7 = Y2/=?.W=KAMKI¡. ; /o@CF G 5
ªE7 Bz»X¼tD½² °BzG 9 / (M1 →M2) 9 K 7 = (M ′1 →M ′
2)/=?.W4 79; KI0KCLVLV/ Q TLVLH4.0KC. ;
M1//
M ′1
M2
// M ′2
ªE7¾9 T,41N/LVTC. ; @CBEDF G/=O.A1NK 9 T 77 / 9 J4 7¾ K ; = Q />K4 Q 4 ; =?.0/KLAM1N/=O4I¿4 7À9 T41N/LAT. ;o5Á T ; . (U → S)4 7 T, º /. 9 /¸1NK Q KX. T,I ; / Ouv 9|7; / 9 K 7 =W1NKVMKI¡. ; /VÂ 5ÄÃ5687 IOKMM/11N/ÆÅ 9|7; . ; T 7 [Z ]Z¹[Ç 24J4 7 1 LVT. ;9 /_È4L T,I 9 =O4I (U → S)
/=?.«4 7 1 LAT. ; =04I U 9 T 7 .S1 KMKCI?. ; / .KC1N/S/.1 K¿MKI¡. ; /S=O/L ; K 1 ; / 77 /S=0/SMI?T,1T 7 / 7 .=04I S 5Oio ¡O¢¤£¥t¦¥¨§¥ÊÉ 5s© ªE79 T41N/SLAT. ;i9 /HÈ4L T,I 9 =04I (U → S)
/=?.4 7 .0I ; M1/.(G , Y1 → Y2 , M1, U →M2, U) ,
T,¬G/=O.A4 7 = Q U LVK%=O/L ; KC 1 ; / 7À9 /I0K 7 %.0TI ; 24/ Q T 7 =?.0K 7 .A=04I S RËT,¬ Y1
/.Y2
=OT 7 . 9 /=I =O/>KC4Ì=O4I S L³4 7; = 9 J4 7 / ;7)º / Q . ; T 7¾ Q T 7 T±!KC4 |7; Ri/.¿T,¬ M1, U → M2, U/=?.¿4 7¾9 T41/LVTC. ; =O4I U 9 /SMKCI?. ; /S=O/L ; KC 1 ; / 77 / GU
/. 9 /«MKCI?. ; / .0K1/ Y1, U → Y2, U 5 K 9 T 77 / 9 J4 7¾9 T,41/LAT. ;W9 /ÍÈ4L T,I 9 /=?. 24 ; ®CKC1N/ 7 .O/ Q /11N/ 9 JÄ4 7 / ; =OT, 7; / .K1// 7 .OIO/ 1 LVTC. ; = 9 /ÆÈ4L T,I 95i687 KMM/1N1/ 9 T,41/VLAT. ;9 /_ÈÍ4L TI 9 @CBEDF GV1 K 9 T 77 / 9 JÄ4 7
9 T,41/wLAT. ;9 /È4L TI 9 M1 →M2/. 9 J4 7H K ; = Q />K4 Q 4 ; =?.O/KLAM1N/ L =O4I M2, U 5 x / K ; = Q />K4
!"#%$&'()* +-,
.-/013254687:9<;=%6>2-?@A4B6-CD2/EF=37G;7*EH4B1%6>26-981JIK7:0@-=%136-CC6LD6 M1 → M2;7:E>MONQP<RSTUUWVOX"VYV[Z]\^T6-4
D@A_C1%4`/C6a; b=37*E1%2B7:413b*C λ2D6 M2
Vc b*136-Cd4 (U → S ← S0)
/Ceb*0gf56A4hD6i=378.7:4@-jb*E1%6 OuvComp VklCmDb/0=36a98b*41%noD6qpe/9>nrb*ED7*98;=36 (M1 → M2 , L) 2B/E (U → S) 6-254astvuwyx^w zE56-=&7|4B1Jd6-986-C~4q S0
21=36 1 I9<b:4B1JnD6Gpm/9>Inrb*ED;b*=&7:E1%2@ (M2, λ2)
6-254F;b*21%41%n7*/26-C2FD6L=37D@A_C1%413b*CJVlb:4Bb*C2iZvI Mumample,+=37
.7:4@-jb*E1%6D6-2Db/0=%6-298b*4B1Jnr2D6ipm/98nybE5Dm7*98;=36-26A4;b*21%41%ny22/E (U → S ← S0)V6F9<-986T
Cb*4bC2ZvI Mum+=37.7:4@-jb*E1%68D6-2Db/0=%6-2a9<b:4B1Jnr2D6pm/98nybE5D2/E (U → S) 7:D9<6A454W7:C~4a/Cn7:132.-67:/m./01%254B6i7:9<;=%6ag/1 =%6-2`E6-CDm; b21J4B1Jnr2-V1%C21^T=36nrb*C.A4B6-/ED?Ob*/0=31
ZvI Mumample,+ −→ ZvI Mum+
6-254o6-226-C~4136-=%=36-986Cd4o2/Ef6-.A4B1JnV|UK=~C?O6-254o;7:2;=36-1%C6-9<6-C~4_D-=36.7*E=%6-2o98bE5;132986-2oD6=37l.7|4B@-j*bE5136ZvI Mumample,+
2Bb*C~4mD6-21%2Bb*j@-C1%6-2-V`7:C2m=%6E6-2546D6.-6A454B6;7*EH4B1%6*TCb*/2.-b*C24E/1%2bC2D6-/nrb*C.A4B6-/E52
ZI Mumample,+ −→ DegStample
ZvI Mum+ −→ DegSt
6A4Cb/2Q9<b*C~4BE5bC2Qg/?O1%=32QE@7:=31%2B6-Cd4D6-2@-g/1J*7:=36-C.-6-2QD6.7|4B@-jb*E1%6-2-Vhb/2Q2B/1Jdb*C2D6;E-2iM >¡¢£T.VUU5UK\ 6A4MN¤¥¦§"¨\^V
S^J£Nª©£«£¬y®°¯]®gy±²v³´®g¥µ®g¶²¸·© ¥µ¹oy®~²º »CmD@A_C1J4hD?O7*0 bE5Dm/C6i7:;;=31%.7:413b*CmD6a=?O6-CI26-9q0=36D6-2 b0~f6A4B2 D?O/C6¼2Bb*/2HI.7:4@-jb*E1%6D6#ZI Mumample,+
D7*C2£=?O6-C26-9q0=36¼D6-2£b*0gf6A42£D6 DegSt Vc b1J4 (M1 → M2 , L) /C½Db*/0=369<b:4B1Jn`D6mpm/98nybE5D7:9<;=%6G6A4L; b21J4B1Jn`2/E (U → S ← S0)V
lb:4Bb*C2 G 2B7;7:E54136m26-9<1¾IK7*0 @-=31%6-CC6*Tg/1`6-254¸D6eE7*Cj4Bb*E1%g/6e.-b*C254W7:C~4¸2/E S VUK=`6-254¸DbC.j=%b07:=36-986-C~4`64B6-C21%bC0 −→ T −→ G
π−→ A −→ 0D?O/C¿2.@-9<77*0 @-=31%6-C A ;7:Eq/C!4Bb*E6 T 2/E S V¼»C¿Cb*46 X =36jE5b/; 6D6-2L.7*E7*.A4-E6-2qD6 T V»C½D1%4>~/6 M1 → M2
6-254eÀdÁKs Â&t|ÃdÁ2B1 T 6-254>/C¿4bE6GD@-;=%b]Ä@2/E S T2B1 Y26-254>/C¿n7*1%2.-67*/
@A4W7:=36i.-b*C254W7:C~4l2/E S 6A4l2-?13= 6"132546/Cmn7:132.-67:/e.-/01%254B67:9<;=%6 M 2B/E A 6A4l/Ce132b*9<b*E;1%29<6L ∼→ π∗M VÅ 6-2Db*/0=36-2#9<b:4B1Jnr2#D6apm/98nrb*ED7:9<;=%6-2Tg;b*21%41%ny26A4#D@-;=3bvÄd@-2nrb*E986-C~4=%6-2b*0gf56A4B2D?O/C6a2b*/2HI.7:4@-jbE5136a;=%6-13C6qZI Mum
deplample,+
D6qZI Mumample,+V
c /;; b*2Bb*C2l~/6 (M1 → M2 , L) 2b1J4lD@-;=3bvÄ@*V»C7:;; 6-=3=%6ÁxKt|wyÂ&ÆD6 X 4Bb*/"4l2b/2HI6-C26-9q0=36E ⊂ X 46-=og/6 0 ∈ E T46-=g/6 −E = E 6A4F46-=o~/6 E 6-Cj*6-CDE6 X .-b989<6Lj*Eb*/;6>7:0 @-=31%6-CVlb:4Bb*C2 φ : Y2 → X =37;7*EH4B1%6q@A4W7:=36qD6q=37; b*=&7:E132B7|4B1%bCÇ7:22Bb~.-13@-6L L 6A4 B2 : Y2 ⊗X → DivS=^?O7*;;=%13.7|4B1%bC<01%=31%C@7*1%E6h7:22Bb~.-13@-6h M2
;7*E¼=&7;Eb*; b21%413b*Ce&SV»C<D1J4g/6 (M1 →M2 , L)d@-E51%_6a=378.-b*CD1%413b*C (∗) 2-?O1%= 6"1%254B6a/C6a@A4b1%=36 E D6 X 4B6-=%=36ag/6a=%6aD1%g1326-/E
B2 (y, φ(y)) + B2 (y, x)
2b1J4`6AÈ6-.A4B1Jno; b*/E4Bb*/"4 y ∈ Y26A44Bb*/"4 x ∈ E V »CCb*46ZvI Mum
depl,∗ample,+
=&7L2Bb*/2HI.7:4@-jb*E1%6a;=%61%C6D6qZvI Mum
deplample,+
nrbE59<@-6aD6-2`b*0gf6A42`g/1@-E1J_6-C~4`=&78.-b*CD1%413b*C (∗) V
!#"$%"$&$'( )+*,+!+-./0012314265/7
(M1 → M2 , L)2819;:6/4<1,=!>87?:A@7CBED
Mumdepl,∗ample,+ FHGJILK 0MON2QPSR TVUWX
0M810 FYOY8YZF\[F\[^] X_;7 1Da`b1c:39edf@7Vg/`Vdh1,MO@
(M2 , L) K @`i7?:j/4lknmEo!pq rtsZrqvucwyxzrk|r~wZm^kjq r~wyX5!/9 1,4289;2O:3774_ K @1,4479@ I /42 ` K π : P → A9v4:3NM87_v1 K _v7`i74!:j@7:07495/9 14!:39;74!: G 1`b`i7t1/7M:@74287X
9;9@ I /4Hd K 9;2 7 K /9v47M8289;=_v7 L 28/M P 5!/9?:874@_;7d K 9v2 7 K / /=9;2O:37 L 28/M G X9v9;9@ I /47 K :39;1,4@7 G 28/MP5!/9?:874@_ I\K :39;1,40 K M :8M K 428_ K :39;1,428/M G X9e@ I /47 K :39;1,4i@7 Y2
28/MP5!/9?:874@i_ ILK :89v14i28/M U 0 K Mf:8M K 428_ K :39;1,4Hzxu Y → GU _ I\K :39v14@7
y ∈ Y 72O:41:87SyX
@ I /47 K :39;1,4@7 Y2
28/M L 1`b0 K :39;=_v7#_ K 0M8 @74!:87|7?:J5/9?:874@_ I\K :39;1,4@7 Y 28/MLU _ ILK :89v14@7 y 72O:41c:37 Sy X97?:@ I /47 K :89v14@7 G 23/M N = L⊗π∗M−1 1`b0 K :39;=_v7t2314 K :39;1,4|28/M P X5/9MO9;74:_v72 14@9;:89v14228/9e K 4!:872 9;_7~9;28:87/4 1/7MO:
U@7P@7¡:07¡4928/M
S7?:¢9v4 K M89 K 4!:£0 K M G :37_5/7 P = ∪y∈Y2
Sy(U)X
B, L 72O: K `b0_;7 28/M P @ K 42¤_;7 28742¤1,¥V_v72¤287 :39;1,42¤@7 Ln @?49v282874!:¤/47 = K 237¦@7 _ K :3101,_;1,9v7@7 § K MO9v28¨9@7 P 01,/Mn ∈ N
X[ :81,/: v ∈ Σ(G/S) K @`i7?:/4 74!:3MO7 28/M P 289©7?:ª287/_v7`i74!:289£01/Mj:81,/:
x ∈ X X9v_©7~9;28:87y ∈ Y2
:87_5!/7v(Iy,x · (Ox)cv) ≥ 0 F
GªK 42¤_ K 14@9;:89v14 [ X Σ(G/S)@289v47_ K 28/MOd K 7 @7§ K MO9v28¨9«D¬9v7` K 44i@7 G 28/M
SX I 72O:D^Da@9vMO7_ I 74287`J=_;7@72¢ K _;/ K :89v142f28/M¢_;7 1M802¢@72¢dM K :39v14J@7 G 5/9281,4:f01,289;:89;,72f28/M¢_ I\K 447 K /J@7 S F® 1/:
v ∈ Σ(G/S) K @`b7?: /4 74!:8M87 cv 28/M A K M A 72O: 0MO1,0MO728/MS F¯¢K M@49e:39;1,4X Iy,x 72O:_ I 9v@ K _dhM K :89v144 K 9;M87@/ 1M802@72dhM K :89v142@7 S 5/9 1M8MO728014@ K /°@9e9;237/M B2(y, x)
X7?: Ox72O:J_;7|281/2OD±d K 9;2 7 K /@7 π∗ OG dh1M8`i|@72tdh14 :39;1,42 5/9¡287n:8M K 428d1,MO`b74!: 0 K Mt_;7 K M K :8NM87 x281,/2_ I\K :39;1,4@7 T F²n³´µ¶·,¸¢³(¹ºa»º ¼fºa» F£½ ¾ 1`b`i7 Y1
72O:b9;4 _v/2i@ K 42 Y2
Xf_;7H2 ` K P 72O:i`J/49¦@ I /47 K D:39;1,4@7Y1 F¢¯ 1/MO: K 4:EX P 4 I 72O:J0 K 2J/4`b1!@N_;7MO7_ K :39e7`i74!: 1,`i0_v7?:J@/ 1
Da`b1c:39ed@7Hg/`VDdh1M8@M1 ¿ :81,/2_v72 K 9;1,`i72©281,4!:,M89e2©2 K /d_ K 14@9;:89v14 [FcÀ 4t07/:©7~0_v9;5/7M_ K M K 9;281,4@ ILÁ :8M87@72@1,/=_;72`b1c:39edh2@7g/`Vdh1,MO@|@7j_ K ` K 49;NM8723/9e K 4!:87 ¿ M2
d1,/MO49;:_v7`b1@N_;7jM87_ K :89;,7`b74: 1,`i0_v7?: P 7?:M1
_;7M8287 K /@70M89;1@720 K M _v75!/7_@9e9v287M P2 F K 0M8101289;:89v1428/9; K 4:3772O:@`i1,4!:8M87@ K 42PLR TVUWX YOYZFLÃc] 7?:@ K 42PLÄÆÅ£ÇÉÈÊË¡X [^]±FÌ ¶)ÍÎÍÏ^ÐÑÐ8ÍÒÓ¹º»º ¼fºhÔ F£½ ÕÖ x~q«×Ø~x±m^Ùo!rG
oucØP
rZØw×Úx Øm^knm^saÛx Ø~knr6Ø~×sU Ü Õ rØZÛÝ~knu
PS0
rZØ~wÞZm^rOßrr~wØrZØZm^kmØEucwrZØxs~s3Ý3o^×~wyxyà~q rZØ Øm^ws3mZs3rZØr~wÞorjwyáZrâÞxØ~×sS0 Ü
Õ rªãcs3m^×)r Y2
uãcx+wq«xyà?s3r~knr~wÞØ×sPS0
Ü
!"#%$&'()* +-,.0/*13246578/:9;=<;&>?;A@BDC EGFHJI
eG : S → G K&LNMOQP IHRF*STS OQU IWV O B6XZY\[^]G_F`IW[a MOZbO _ F M HJIWHJcdH%Ie bO M2HRfg_ K H%h UO h UOKROM F UMji=MP [eQf L Sy eG (S0) bO PS0
S O V O S P F*S^IV O _ L*M GS0
_ F U V8IWF U IZe K eQf O S^I?SFSS UK y bO Y2
BklS O S b e bU HJIh UOmK%OQMM F UMni=MWP [eQf L`M Sy(GS0) O I Sz(GS0
) M F*S^I b H Mpo F*HRSqI M#bL S M PS0_ F U V bOQUdr e K eQf O S^I Msb H M IWH%S P I M y O I z bO Y2
BtuAvu&tuwyx zD|~-D*|6pxzDx|6pdz|u# EGH
Y1 = Y2 K%OQM Vje MWUK I L I MgbOPQO _ L V L* V L _[ OMOIWVjF U c O S^I b e o!bL S M\ D" O I bL S M\ w P [Bjj BNEGH Y1 6= Y2 H K%MMWOb eQfgFS^IV O SqIOrL`P I O f O S^I P F*ff OPQOQUdrbOy D" F U\ w P [Bqj H KM Y L* H%I bO V OQP F_H O V K%OQM _V OQU c OQMO S¡V O f_ KRL*¢L S^I Y I L SqIW£`Is_ L V Y1 I L S^IW£`Is_ L V Y2
B¤F UM SF UM¥P FS^I O S^I O VFS M¥b FS P¦bO Ve§¨eQV O S PQOQM_Ve P H MOQM B© L S M¥PQO II O _ L VnIWH O SF UM¥b FSS O VjFS M¥U S MO S MK YFª oO I¥HRS^I U H%IH%§(⋃
y∈Y2
Sy(G)
)/ Y1 .
klSFª"IWH O S b V LmU S MP [eQf LmO S VjF U _ OQM G MWO fgH i=M I L ª K%ONMU V S O I L ª e K H O S MU V U BX ON VF U _ O G MO V Lf L Vh U e_ L V KRO VF U _ OP F*S M I L S^I H = Y2/Y1
BDEGFH O S^I G1 O I G2 KROQM h U F`IWH O S^I MyK&L«¡U f§¨F*V bbO M1 O I bO M2
B¥klS L`U V L G0 = G1 O I G/H = G2
B¬ZS_ L VnIWH PQUK H O V G MO V L H M F*fF*V_[ O G1 MU V U B®EGFHJI R K Y L SS OL`UbO S O I I K YH b e L`K#bUM F UMni=MWP [eQf L § O Vjfe S0
B®klS¯_ F MO Sn =
Spec(R/In+1)_ F U V n ≥ 0
B°±4³²*´µ²¶-·¹¸n·²º»9;¹< ;=9;=<½¼ \ D"?|³¾®u®¿uÀv^Á BDC Â¥Ã-ÄGÅ#ÆÇÃ-ÄdÆ
n ≥ 0 ÈÉpÊËjÌÉ&Í Æ Ë ÄGÎ Í6ÏÑÐÒÓ0Ô PnÕ ÅWÃÇÖ ËÑÏ Æ É ×mÍ ÄdÅ Sn Ë ÆÄdÎ Ó Ã-Å Õ ÐGÉRÍÓËØÒ Æ Ô`Ê&ËmÍ ÄdÅpÖ ËÑÏ Æ É ×pn : P ×S Sn −→ PnÙ Ä ÉmÉAÚ^Ë ÎÆ É ÛË Pn Ô Ä Ù ÄÃ³Æ ÉÜË ÎÆ ÚqË P ×S Sn Õ Ô Å ÊÝ Ô^Ï Æ É Ã-Î ÚqË Y1 Þ¥ß Ë Õ Ê Ä ÍÑÈ Pn ËÑÍ Æ Ó ÄGÎ ÉyÚµÝ ÄGÎ
×QÔ`ÉRÍ6ÏÑËWÔ Ä Ô`Ó Õ Ê&Ë Ln Æ ËÊ Ù Ä Ë p∗n Ln ∼→ L Þ.0/*13246578/:9;=<;=9;¨à BDC X L±b HJáeQV O S PQO _ L V#V L __ F*VjI \ D IW[Bdâ"B%ã6ä O I å " _VF*_BjÑB\æ"Bèç OQM I¥h U Y\F*S Lb HJcdH M eé_ L V Y1 O I¥SF*S_ L V Y2
B ê OQKRL _ O Vjf O IjIWV Lb Y\F*ª"I O SH%V U S MP [eQf LMO fH i=M I L ª KROTL`U¡K H OQUb Y U S MP [eQf LMO fH i¹L ª e K H O SBX LK HRfgH%I O H%S bUP IWHJc ObOQMDMP [eQf L`M Pn b eëSH%I U S MP [ef L §¨F*Vf OQK P h U H OQM I8f U SH b Y U Sy§ L H MPQOL*U
L fg_ KRO L B©±Y L _Va MìTíîðïÜïÜï®ñ BæB ñ P M Y L`KR eQªVH MWOØO S U S MP [eQf L P _VF oOQP IH%§ MWU V S O Imf U SHb Y U S§ L H MPQO6L`U¡L fg_ KRO L BklS_ F MO
Gn = pn
(⋃
y∈Y2
Sy(G)×S Sn)⊂ Pn
_ F U V n ≥ 0BX OMP [eQf L Gn OQM I PL SF*SHRh UO f O S^IyH M F*fF*V_[ O0 Y2/Y1 × (G ×S Sn) b Y L _Va M
K&L V O f L Vh UO tuÀvu%òµu&¿ BNEGF*H%IBn KROP F*f_ K eQf O S^I L H%V ObO Gn bL S M Pn f U SH bOMWL3M IV UP I U V OVe bU HJI O B X L0K H%fHJI O HRS bUP IH%c OybOQMlMP [eQf L*M Bn b eëSHJI U S M F UMni=MWP [eQf L §pFVjf OQK § O Vjfe B bO P h U H M Y L`KR eQªVjH MOóO S U S M F UMni=MP [eQf L § O Vjfe B bO P B*klSy_ F MO G = P−B P Y OQM I U S MP [eQf L h UL`M H i_VF ojOQP IWHJ§ MU V S B6X®YHRS PQK%UM HRF*S bO Y2/Y1 bL S M Gn
HRS bU HJI U S O HRS PQK%UM HRF*S bUTMWP [eQf LNP FS M I L S^I Y2/Y1
!"$#%&# %'%()!*,+-,",./012
G 34 05!671298:59;6<8:=>7-1?@>AB67-CED@>F<8:FHGI7-5JCKA@/AG
@>AB@L71MK/AS0
A298N=>2:7CE75JDOAP0Q;D597/;=R8/AY2/Y1
A<8S/;T67-CED@>F<8:FUGV7-59CEA@"/AG 3W 1TD5J7-;YX-AZ@L0[D5J7D72J=>8J=L71\2J;=>XQ0Q1"8:A]67-CKCEA_^a`cbedgfhYi]?
j 3lk A<8 j 3nmQo A<8T^ap!qsrtu?D5J7D 3v9vJvw3 j 3>xykzo3|\*~u~yJ~c 3S ePzw G w :zK\w] U¡¢£ ¤ S ¥Z PzwE P wI¤¤:¦Q£§< \"¨wU ©¡ª£¤ S ¥!ZJ« I¡£¬y¡'" G y¡Y P wZ£¡ z¬*®¬y¤L®B£ ¤ U ¯° 7=>8
(M ′1 → M ′
2 , L′);1±7-²"³JA<8E/A k´ Mum
depl,∗ample,+
/AµD058:=>AF<8:0@>AY ′
1 → Y ′2
A<8E/AµD0Q598:=>A2JACE= ´ 0²F@L=>A11A G′ 3¶ 78J7-12 G′
@>A·2:6wOFC0T7²Y8:A1";¸A1¸0DD@>=L¹";01"8º@L0T6712J8J59;6<8:=>7-1®D5JF6F/A1"8JAH»(M ′
1 →M ′2 , L′) ¼ =L@0¨CK½CEA[67-CKD@>F<8:FNGV75JCKA@@LA·@>7-1MT/A
S0
¹;A·@>A·D597/;=R8º/AY ′
2/Y′1
A<8Z/AG′ 3° 7-=R8
α;1CK7-59DO=L2JCEAP/A
(M1 →M2)/0Q12
(M ′1 →M ′
2) 3v @=L1/;=R8[;1CK7-59DO=L2JCKAα : Y2/Y1 × G −→ Y ′
2/Y′1 × G′ .
|\*~u~yJ~cL¾À¿ ^n`ÁbSdgfhYi o¢Â*ÃUĪÅeÄ ÆÇ*3S ÈwZÉ- Ês£¡Ë£ ¡Ì£s®¬y¤" ® β : G →G′ Ì<£ ]®Í®s:¬y_LÊUά*¤SÌ£ α P ¬y¡ ϸ" S0 ¯Ð ACE059¹;7-12H¹;AÑ@ÓÒn711AÑGV0=R8P0Q;6;1AsOÔ D78JOÕ2:AÑ2J;5_@LAÑ67-CKD7-58:ACEA18B/A
αX =L2 ´ » ´ X =L2H/A2GÓ0Q=L2J6A0Q;YÖ×6;²=L298JA2[0QCED@>A2 L A<8 L′
7;/A2[CK7/Õ@>A2N59A@ 0y8:=RXACKA1"8[67CED@>A<8:2P
A<8P ′ 3
ØÙÚÜÛzÝ-ÞªÙ) 3S v @·1ÒnA298¸D02¸1F6A2J2:0Q=L59Aß/AßCK7-1"8J5JA5¸»Ë17;X-A0;à@ 0)á1=R8:;/Aß@>760Q@LA010Q@L7M-;A¨»µ^n`ÁbSdgfhYiZ? j 3 j 3 ² o3â 1A<ãA<8?71ä0
Y ∗1 = Y ∗
2 ∩ Y1
0XA6P@>A2N178:0Q8J=L712[/A ¬¯U,3å ~*~ææÓÛ*Ù'Êç 3S G :N£ ¡&zwß¡&ÏQ¤:¬y£zw\£ ¤ S Ì<£ Nw!I¡"¡y¡_y£Üw¬*RÉ" L !" P ¯è ®¬*¡Y¤JQ¬*¡¯Té W 1ê/FCK7-1"8J5JA$@Òa=L1/FDA1/016A$/A
GA1 L A<8¸A1
PA1ê0DD@>=L¹;0Q1"8¸@ 0
D5J7D7-2J=R8:=>7-1gë ÄìÄ ë Ä>Å 0;ËCK7-59DO=L2JCKA=L/A1"8J=>8JF 3¢W 1Ë17Q8:AM1 ×M1 = [Y1 × Y1 → G × G] 3uv @AÖY=>298:A];1TCK7-59DO=L2JCEA]/A
M1×M1
/0Q12M1
/71"8e@ 0ND058:=>A]2:ACK= ´ 0Q²F@L=>A11A]A298e@ 0NC\;@>8J=LD@>=L60y8:=>7-1/AG 3uí AKCE½CEA?=L@¢AÖY=L298JAK;1'CE75JDO=>2:CKAK/AKCÑ;@R8:=>D@L=>60Q8J=L71'A1"8:59A
(M1 ×M1 → M2 ×M2)A<8(M1 → M2) 3W 1A1/F/;=>8H;1ßCE7-59DO=L2JCKA
m/A
G × G /0Q12G 3v @SAÖY=>2J8JAÑFM-0@>ACEA1"8H;1
CE75JDO=>2JCEAi/A
G/0Q12
G7-²Y8JA1;ß»D0Q598J=L5[/AT@L0@>7-=u/Òn=>1"X-A592JAT/A
G 3î A2·67CED@>F<8:F2[GV75JCKA@L2/A
mA<8B/A
i/F<á1=>2J2:A18B@L0¸@>7-=S/AÑM-597-;DAÑ/;67-CKD@LF<8JF\GV75JCKA@S/A
G × Y2/Y1 3ï =>12J=? m A<8i=L1/;=>2JA1"8[;1AP@>7-=/APM-597-;DAP67-CKC\;Y8w0y8:=RGª2J;5
G 3î Òa=L16@>;2J=L71/;ä2J6wOF<C0K671298w0Q1"8
Y2/Y1
/012G
A298[;1CK7-5JDO=>2JCEAP/APM5J7;DA2 3å ~*~ææÓÛ*Ù'ð 3S GU w$£ ¡ñzwEc-§¡ò£¤ U s£¡ß« £ ¡êÎQ z::Q£ó£§< ÊQ· L ¯è ®¬*¡Y¤JQ¬*¡¯_é W 1±0$/F³:»ß/FCE71"8J5JF¸¹;A
GU
A298s;1ô2J6OFCE0ßA1gM5J7;DA2s@>=L2J2JA×2J;5U 3õ 7CECKA
PU = GU
?u@LAE2:6wOFC0GU
A298TD5J7D5JAE2J;5U 3ev @UA298T=L595JF/;6<8J=L²@>A60Q5¨7-;YXA58T/012¨@LA
2J6OFC0ß=L595JF/;6<8J=L²@LAP 3©î A×GÓ0Q=L2J6A0Q; L A2J8K0CKD@LA×2J;5
GU 3¢v @A298KCÑ;1=º/Òn;1A×5J=>M-=>/=>á60y8:=L71
!"#%$&'()* +,-."/02143065798;:<3-*/=>021</;?06@202A>A%06?3CBD7*=>E@20793@23.=>EF/0 L E38 G G HJI 7958K2EMLNPORQTS"U V/XW GYYFYZG>[\9] VA>0^B_7*=%E@207*3 L 79?`a0b/c310dEF/X8F3@b/X38F0e@23.=%EF/X0d@791-1=%:430 Gf -*=>021g/ G1
0b/ G2A%02E:<3-9/X=%021</EihJA>7Mjk3`6BD-*8?l?02E 1 m `M-*/=%BnE?0ejo3`6BD-8F?79`a5A%02E M1
0b/ M2 Gp 7*8e@2-1EF/X8F3@b/X=%-1V Gi02E/qA>0r:43-9/X=%021</q?3!?-3.A%0r`a-*/=%B?0sjk3`6BD-*8? Mi → Mi
5-*38 i = 1-3 2 Vt-uvA%0o`M-8F5W=>E`a0?0 Mi?7*1E Mi
02E/aA I =%?021</X=w/Xx Gzy 0o@2-8F-A%A&79=>8F0oE3=%97*1g/r8Fx2E3A%/0o?0oA&758-*5 -E=w/X=%-1|~_~|~w021@2-*1EX=%?x28X791</sA%0C`M-85W=%E`a0 (M1 → M1) → (M1 → M2)
53=>EsA%0`a-*85W=%E`a0 (M1 →M2)→ (M2 →M2) G;_ GR Z F&c4lr ¢¡£4&r¥¤g#¦9X §¡Z¨¤< G1 ¤g© " G ª4lr ¢¡£4&r¤<¦9X ¡X¤< G ¤g© " G2 «¥¬ o§Z£®a©v¯¦ Z¡ZZ "©9 Y2/Y1 Za¤g©9°±&² ³©9´¤<G→ G2 «p -*EX-*1E G∗ = ∪y∈Y2
Sy(G) V:43=µ02EF/¶=>1@2A%3E¨?791E P G p -38i/X-*3"/ y ∈ Y2V-*1k1-*/0 σy : S → PA I 31=>:<30JEX02@b/=>-*1±:43=ª9793"/ y ∈ G(U) E38 U · 02A%A>0J02EF/^x2¸7*A%0¹7*3²/8X791EA&7 /XxJ?0JA&7rE02@b/X=%-1±1023"/80
?0 G 57*8iA I 7*@b/=>-*1k?0 y E38 P G p -38/X-*3"/¶021</=>028 n ≥ 1 V-*1o1-*/0 Z(n)y
A>0d58F-4?3=%/iº.8x
Z(n)y
//
G∗
n
S σy
// G∗
ytI 7955A>=%@79/=>-*1 Sz =%1?3=%/»31¼=>E-*`a-*85W=%EX`M0¨?0 Z(n)y
E38 Z(n)y+nz
5 -38z/X-*3"/ z ∈ Y2 G"½ 1¼021¼?x2?3=w/:430dA I 31=%-1o?=%ED¾F-=%1</X0∐
y∈Y1/nY1
Z(n)y
5 -8¿/Z791</ªE38µ31qEFÀ°EF/K2`a0z?0Á8F0258x2E021</X7*1</Eª?0 Y1/nY102EF/ª`J31=>0z? I 310zA>-*=g?0Á¸*8-*350z@7*1-*1=>:<30 G
H 0Á5A%3EV§A>7c@2A&79EE0Á? I =>E-`M-8F5W=>E`M0Á?3e¸8F-35 0Á-*."/X021<3q02EF/ª=>1?x25 021?7*1g/X0z?3q@W-*=wÂd? I 31qEFÀ°EF/K2`a0?0d80258Fx2E021</Z791</E Gà *ĵŠƿÇP ¢¢ÈÊÉ LËNÌORQTSÍ"Î ]ÐÏÑ ~Òª~_<UÓ GR ¬ d¦ Z §¡ G[n] ¤<X¶¡ DM¤< n Ô XÕ ¤< G X»& ¼ ¢¡£¹Ör× ° ¤9&DØ2 n ∐y∈Y1/nY1
Z(n)y «
f -*=>021g/ y ∈ Y10b/ s ∈ S G½ 1k76A>02Ecx2:43=w*79A>021@202EcE3=%97*1g/X02E
Z(n)y ×S Speck(s) 6= ∅ ⇐⇒ ∃ z ∈ Y2 | y − nz ∈ G(OS,s)
⇐⇒ y ∈ Y1 ∩(G(OS,s) + nY2
).
!"$#%&# %'%()!*,+-,",./1023254
G1 687 49;:< G2 = 49;>@?A4CBD0E>GFA4H">JI9;9;0KFALMID3N0-DO?A4C230E>GFQPN4SRD25P80 7 N M1 →M1 687 49;:<M2 →M2 =T 0-H0OU>GFA4H">V> 7 0FW9V9;DFQ>G49V4YXIEK>G49VK0-D 7 >G49
0 → Gs[n] → Gs[n] → (G(OS,s) + nY2) ∩ Y1
nY1→ 0
0 → G1, s[n] → G1, s[n] → (G(OS,s) + nY1) ∩ Y1
nY1→ 0
0 → G2, s[n] → G2, s[n] → (G(OS,s) + nY2) ∩ Y2
nY2→ 0
Z\[]*[^_^a`b_]*c'de_fegde_fh<ji kmlnpo*q_rst uYnvlxwyEz| qGlYM~"lG
~yEwUuG2
luzu qlYz|t uowwo-yEluYz
Y2/Y1 w\ryEqz|tgYtglYqG G luz wusnvylYwEqGorluJulYnt|uYz_yWlJuY qSnvyEq;jryEq
Y2/Y1 kmlnvoqrst unplwyEz|ql~"lG1
~ywUuG
luYz1 wltnnplqutgowo lqYzl5lYz@u t|~"lYwzat l¡pWyonSrouyEwzlwl z|qGl
G0~"lG ¢ qGyt ulw£t uo*npoqrstWuYnplyE ¤~"lGuuY"u~"l
U ¥ 4K0D:?W4
(G, Y2/Y1)NL¦HFA>VN0-HKDH0O§;4>VN4
DegStample
L>G4HNIH>G1, U
<¨ c©`]ª-«¬c)defedeffU<ji 0FA>
M1 → M2
DH4pFA9;0-®LHFW4vH0-HHLK49;9GIEF 7 4254H">L>GI?W4v4H"> 7 4 1 ¯230E>GFQP89SN45RxD2P80 7 N°:0-9;FA>;FAP9±9;D 7 (U → S ← S0)<²SH°H0E>G4
Mi = [Yi → Gi] T 4>±?a³´0H£H0E>G4Gi?W4µB"D0>;FW4H">5I9;9;0KFAL¶
Mi
:0D 7 i = 10-D
2<·²SH¸:4DU>4¹4K>;D4 7 DH4µK0H9;> 7 DK>;FW0HºIHIE?W0®-D4
4>»K0-H9;FWNL 7 4 7 P2/Y1 T 0¼ P2
49¤>»DH230"N½?W4 7 4? I>GFQ¾4234H>µK023:?A4>µN4M2
<!²SH0-OU>;FW4H">»DH9;K¿L2vI
G9;425F ¯ 9¤>IEO?W4C9;D 7 S 4>JFA9;0-250 7 :¿4S¶ G1 /Ker(G1 → G2)
9;D 7 U < /14>¤>G4C®-LHL 7 I?AFW9GI>GFA0-H49¤>CFA?W?AD9;0-F 7 4KI 7 G 49;>V?W4B"D0>;FW4H">VNDN0-DO?A4230E>GFQP¬N4RxD2P80 7 N[Y1 → G2] −→ [Y2 → G2] .
ÀÁ|ÂÁ ÃÁpÄÆÅÈÇ£É ÊËÈÌÍUÎÏÇÅjÐ-Ç&ÑÈÇ&Ð-ÎUÒÉÓÔÕÌÇÁJÖ 0FA>(U → S ← S0)
DH$0O§¤4>SN4OuvComp
<× 0D9¾-4H0-H9N4ØNL¦HFA>mDH\P80HK>G4D 7 N4JÙ ¯ Mum
depl,∗ample,+
NIEH9DegStample
<F(G,H,L)
49¤>m?a³´FA2vIE®-4N4
(M1 →M2, L) T H0D9 I¾0-H9J230H"> 7 LSBD4 (G,H)49¤>1FWHNL:4HNIH">1N4 L < Ú±IH91K4>¤>G4M:I 7 >GFA4 TH0-D9!¾-0-D?A0-H9VL>;4HN 7 4?W4\P0-HK>;4D 7 : 7 LKLN4H>C4HN4D XÆLBDFQ¾EI?A4HK49CN4KI>GL®0 7 FW49
Mum :Ù ¯ Mum+ −→ DegSt
Mumample :Ù ¯ Mumample,+ −→ DegStample0FA>
(M1 →M2 , L)DHÆ0-O"§;4>VN4Ù ¯ Mum
deplample,+
<Ú³ÛI: 7 ½9Ü´Ý!Þßà T ?W42Æ<á¤á;á<Ûâ <Ùã T FW?4YXUFA9;>;4\DH4H">;FW4 7 n ≥ 1>;4?ÈBD4
(M1 → M2 , Ln ⊗ [−1]∗ Ln) ¾L 7 FA¦4? IpK0HNFA>;FW0H(∗) <²SH°0-OU>;FW4H>±N0-HK
DHÆP80HK>G4D 7 Ù ¯ Mumdeplample,+ −→ DegSt .Ú³ÛIE: 7 ½9\Ü´äæå3ç À T K0 7 0U<Uá<è"<Ù"<Ûâã T >G0DU>!0O§;4> N4Ù ¯ Mumample,+
9;4±NL:?W0FW4\I: 7 ½9 K¿IH®4234H>!N4OI9;4p:I 7 DH230 7 :¿FA9;234
S ′ → SL>GI?A4 T 9;D 7 §;4K>GFQPV4>¦HFa<Ø/J0-25234p?W4p250 7 :¿FW9;254»49¤>¦HF T ?A49;K¿L2vI
S ′49¤>VK023:?A4> 7 4? I>GFQ¾4254H">C¶5?a³´FANLI?N4?|³éFW23I®4FWH"¾-4 7 9;4N4 S0
<²SH0OU>GFA4H">VN0-HKDHP80HK>G4D 7 Ù ¯ Mumample,+ −→ DegSt
! #"$%& ')(
*,+-./021/3542/76849+;:/<=?>A@BC+D/E./ (M1 →M2 , L) /[email protected]*H/3.,+3542/7./ L <IJ+-K.68L,3@[email protected]:O/7BCN;42@MTJ./VUWRBXTYN-Z.[+;.BC/84KR3\T]+;@0Z1/)+RW1RS@O0F42/E+BX*:/<^_@O302@@:/a`@O0Z4Z/7R3\TYN31842/R-
Mum : bdc Mum+ −→ DegSt .
e 3WBCN34Z-Z/71NBCBX/E.,+;30EfhgikjlmQ,nFnZn9<hoX*<Hpq8r$sR>A@:@3.R@M4GR3\TYN31842/R-
Mumample : b)c Mumample,+ −→ DegStample .
tvuxwzyz|~H < $9)%a2 Mum Mumample9Z;&da59V28%;&a29V5Z;
5%a¡9)¢£ a¤¥za¡Z;¡¦z¢X§ e 3¨1NP30F4Z-ZR@M4[R3¨TYN31842/R-WsR,+0Z@ c @O3©P/-Z0Z/D-2ª;1/¬«:+1NP30F4Z-ZR1842@ONP3¨./UWRBXTYN-Z.¬0Z/BC@ c +SH6:O@/33/®@3©P/-Z0Z/<¯?3¬/°/84Q«W4ZNPR4V0Z1±6BC+[0Z/BC@ c 0F49+;S:/®B®+;-ZsR6 (G,H) /0F4+0Z0ZN1@O6/²R3/²@O02NDP63@O/ G0 → G/H /342-F/²0Z1±6BC+0#0Z/BC@ c +SH6:@O/30<G³@ (G,H) /0F4\R3´NSµF/84./ DegSt QJ:/00219±6B®+;0 G0 /84 G/H 0ZN34X./0NPSµZ/8420k./ DegU,S,S0
<n¶:·:/R-1N-Z-F/02*HN3.¸./R`1 c BXN4Z@OT0V./#U[RBkTYN-Z. M1
/84 M2.>¹+;*-Zº0»:/C42±6N-ZºBX/¼,½&¾½O¿$½&¾<x=À>A@0ZNDP63@O/ β : G0 → G/[email protected]@M4CR3/W@0ZNDP63@O/ α : M1 → M2< e 3Á02+;@O4CsR/ β /84C:&+!*,+-Â42@O/W0Z/BX@ c +;S6:O@/33/W./ α NP34
BCÃBX/1NPBX*:684Z@N3²TYN-ZBX/::O/k:O/:N3D#./ S0<¯?3!*,+;-F4Z@1R:O@/-8Q/::O/0Ä1N;Ŧ31@./34Ä+R c ./0Z0ZR0Ä./ S0
<IJ+-À.68L,3@O4Z@N3Q5:&+ÄL,S-Z/K./ H ∩G0 0ZR- S0
/0F4À4Z-Z@M©@&+;:/<=$+7*,+-Â42@O/G0Z/BC@ c +SH6:O@/33/G./ α /0F4v.NP31R3[@0ZNPBXNP-F*±@0ZBX/»0ZR- S0
<ÆÇ/:&+CBXNP34Z-Z/EsR/E:¡>A@0ZNDP63@O/ α /0F4È6849+;:/E/84È.68L,3@O4GR3É.NRS:/EBCN;42@MT./#UWRBXTNP-F. M1 → M2
42/:ÀsR/ Mum(M1 → M2) = (G,H) < e 3¬-Z+@O0ZNP33/®./®BXÃBC/®*HNR-1NP30F4Z-ZR@O-Z/7R3WsR,+;0Z@ c @O3©5/-F0Z/7./ Mumample
<
Ê ½Ì˽CÍEÎzÏHÐYÏHÑxÒÓÔÓÑÕCÓÑÖÑx×ØHÓÙÚÜÛxÓÝ[Ð]×YÑxÒÓÑÞ +;30X1/84F42/#*,+-Â42@O/QNP3¸0ZR**HNP0Z/ D = 1 ß NP3¸*,+;-Z:O/\./Éa¡a]9[5#,Z;à¥áEÌ a<
^_@O30Z@¡Q:/»1±,+;BC* Ag, 0 *,+-Z+BXº842-F/»:/0È0Z1±6BC+0_+SH6:O@/30È*-Z@O31@*,+;:/BX/34È*N:&+;-Z@O0Z60_BVR3@O0È.>hR30ZNPR0 c D-ZNR*/G.,/G-2+;3D p <³N@O4 Ag, 0 :&+E1NBC*,+;1842@ML,1)+;4Z@N3®42N-ZN;Ŧ.,+:O/È./ Ag, 0 +0Z0ZN1@O6/_«E:&+E.61NB c*HNP0Z@M42@ONP3 Σ ./ C(X) <âÈN4ZNP30 G :O/0Z1±6B®+¥0Z/BC@ c +SH6:@O/3²1)+3N3@sR/02R- Ag, 0 <$³+¥-Z/0F4Z-Z@O1842@ONP3« Ag, 0 /0F4G:O/70219±6B®+X+SH6:@O/3WR3@O©P/-Z0Z/:¡<,âÈN;42N30 H :O/7DP-FNPR*H/7./7-2+;3D p R,3@M©5/-F0Z/:m0ZR- Ag, 0 <tvuxwzyz|~Yã$¶ < ä9ÂåP&a®¸,]8Xd9æ,¤®Wda¤k̦a¶5ça& G a Ag, 0 5%,_9%¤_è%);#a¡2É9a G é a»2%ê¡[;9 G ;F92a[ Ag, 0 é a7aK8#Zë 5dæ,a2a9[ H5; G d%ÌHìÀ,¡kamèH;ía Ag, 0 ¢£ a¤¥za¡Z;¡¦z¢À§ =?>hR3@O1@O4Z6E-F6)0ZR:M42/7./E:&+X3N-ZBC+:O@O4Z6E./ Ag, 0 /84G./E:&+X*-FNP*HNP0Z@M42@ONP3 Ê ½]¼,½¡¼,½M¿$<UWNP34Z-ZN30À:]>h/a`@O0F42/31/<5îK/*-Z/3N30À:+Ä1NP30F42-FR1842@ONP3X./ Ag, 0 /84À1NP30F4Z-ZR@O02N30 G 6842+*H/K*,+;-?6849+;*H/P<îK/*-Z/3N307:/073N42+;[email protected]±,+*@M42-F/ b <mINPR-í42NR4 σ ∈ Σ .@M°6-Z/347./ 0 Qm:]>h/30Z/BVS:/ Wσ/0F41N30Z4Z@O4ZR6í./Ä./R`ï68:6BC/35420 wsab
/84 wet ß :]>h@O02NDP63@O/í/342-F/í:/0 1 c BXN4Z@OTY0R3@M©5/-Z0Z/:O0/0F46849+:O/
!"$#%&# %'%()!*,+-,",./102 Mwet
σ, 0 3 4 / 35 687:9;<=>6 3??3 /102 Mwsab
σ, 0 @ ACB 02 <ED 6 4 3 2 4 B 0 4 3 9 5 ;6>F 0HG 4 < 3? 4 2 3 I 3 0 J)/ 3? / I 6LK,7KM< 2 3? 4 / I 0N/ 6>F= 3'O < 4!PQ B ?R? B 4 3 Malg
σ, 0
= 9TS9 2 4 3 = B S9U= 3 9=VF-<; 2 6>W 0 3 S B ? / 4 2X0 6 4 3 I 9 ? / = 3ZY 9 2 9[7F 2 9 Y \ 3 ]^V_`^ ] Q 3 4 Galgσ
= 3 / S \ < 5 9 / 35 687:9;<=>6 3? /102 Malg
σ, 0
B ; 4 3? 0 Y 9 2 9 YY 2 B J 6 5 9 4 6 B ?aY 2 < / 3 2 DU9 ? 4= 3 / B 0/ 7bF 2 B 0 Y3 I 3 p 7 4 B 21/ 6 B ?@ c ? 9 I B ? SMalg
σ, 0 = Malg, wsab
σ, 0
∐Malg, wet
σ, 0 .
c ?'Y B / 3 Galgσ = Galg
σ/102 Malg, wsab
σ, 0 d S 3 S6 3 / 4 =><F6 4 6 5e3fY 0 6 / W 0 3 =hgi9 I\ < 2 3? S 3 I 3 H I 9 ? / Galgσ
3 / 4kj ? 6 3l3 4 Y =>9 4 3 /102 Malg, wsab
σ, 0 @ m ? 0 4 6>=V6 / 9 ? 4 = 3 / ? B 4 9 4 6 B ? / I 0 Y 9 2 9UF 2 9 Y \ 3on ^V_`^ p Q B ?D B 6 4 W 0 3 =qgr9 I\ < 2 3? S 3 I 3 H I 9 ? / Galgσ → Malg, wet
σ, 0 3 / 4 0 ? / B 0/ 7bF 2 B 0 Y3 B 0 D 3 2 4 I 3 Y1/Y0 @s gi9 Y 21t/vuxwzyf| Q Y 2 B Y@~ @>E Q =>9 / B 5f5e3 9 5 9=VF9 5 < 3 I 3 Galgσ 3 4 I 3 Y1/Y0
9 0 7 I 3 /1/10/ I 3=qgr9 I\ < 2 3? S 3 I 3 H 3 / 4 2 3Y 2 < / 3? 4 9U;= 3Y 9 20 ? / S \ < 5 9 3? F 2 B 0 Y3 // 35 687 / 4 9U;= 3 W 0 3 =hg B ?? B 4 3Galgσ −→ Malg, wet
σ, 0 .
3 /2 3 / 4 2 6VS 4 6 B ? / I 3 Galgσ 3 4 I 3 Galg
σ Malg, wet
σ, 0
S B G ? S6 I 3? 4Q 3 4 = 9ZS B 5eY B / 9 ? 4 3?3 0 4 2 3 I 3 Galgσ
3 / 4 Galgσ @ gi9 I\ < 2 3? S 3 I 3 H I 9 ? / Galg
σ 3 / 4 <F9U= 3 Y1/Y0Q I B ? S j ? 6 3f3 4 Y =>9 4 3 /102 Malg, wet
σ, 0 @ B 5f5e3 9 0 Y 9 2 9UF 2 9 Y \ 3~ ^ n ^ p Q B ?Y B / 3U0 =
∐
σ∈Σ
Malg
σ, 0 et U0 =∐
σ∈Σ
Malgσ, 0 .
c ? B ; 4 6 3? 4 0 ? / S \ < 5 9 / 35 687 / 4 9;= 3 G /102 U0 @ 9S B 5fY B / 9 ? 4 3?3 0 4 2 33 / 4 = 3 / S \ < 5 9G
I <7j ? 6 I 9 ? / =>9 Y 9 2 4 6 3eH@V Q = 3 /2 3 / 4 2 6VS 4 6 B ? / I 3 G 3 4 I 3 G U0
S B G ? S6 I 3? 4Q 3 4 =qgr9 I\ < 2 3? S 3 I 3 HI 9 ? / G 3 / 4j ? 6 3!3 4 Y = 9 4 3 /102 U0 @- = 3 J 6 / 4 3 0 ?3 I B ?? < 3 I 3 I 3 / S 3? 4 3 I 3 G 2 3 =>9 4 6VD 3 9 0 F 2 B 0 Y B G I 3< 4 9U= 3 R0 ⇒ U0 @ B 5f5e3 R0 3 / 4 B ; 4 3? 0 Y 9 2 ? B 2 5 9=>6 / 9 4 6 B ? I 3 U0 × U0
I 9 ? / R0Q = 9 Y 2 B Y B 7/ 6 4 6 B ? | ^ n ^ n ^L_ K B 02 ? 6 4 0 ?3 I B ?? < 3 I 3 I 3 / S 3? 4 3 I 3 G 2 3 =>9 4 6VD 3 9 0 F 2 B 0 Y B G I 3 < 4 9U= 3 R0 ⇒ U0 @ 6 ? / 6 Q G I 3 / S 3? I 3? 0 ? / S \ < 5 9 / 35 687 / 4 9U;= 3 /102 Ag, 0 = [U0/R0] @
-)M`h @C ¡ B 0/ ? gi9D B ? / Y 9 / 3 0 ; 3 / B 6 ? I 3 = 9S B ? / 4 210 S 4 6 B ? I 3£¢ 0 5 K B 2 I / 35 6L7/ 4 9U;= 3@ m =V= 3 / 3 2 D 6 2 9U6 4 S B ? / 4 2X0 6 2 3 I 3 / S B 5fY 9US 4 6 j S9 4 6 B ? / I 3 G = 9 5 9 ? 6 t2 3 I 3 ur¤!¥¦§ Q S \ @*¨©bq@
| ^ ^fª¬«*®M¯±°²³)² ¯´f²¯µ¯±¶¸·²"¹º¼»C«[¶M¯C½¶¸»C¹® B 6 3? 4 n ≥ 3 0 ?3? 4 6 3 2 3 4 Σ 0 ?3 I <S B 5fY B / 6 4 6 B ? I 3 C(X) @¡ B 4 B ? / Ag,n = 9¾S B 5eY 9S 4 6 j S9 4 6 B ?4 B 2 B G I 9= 3 I 3 Ag,n 9 /¿/ B S6>< 3À Σ 3 4 G = 3 / S \ < 5 9 / 35 687:9;<=V6 3? S9 ? B ? 6>W 0 3 /102 Ag,n @ 3 / S \ < 5 9
G< 4 3? INI B ? SÁ= 3 / S \ < 5 9&9U;<=>6 3? 0 ? 6VD 3 21/ 3 = /102 Ag,n @ s gr9 Y 2Xt/ur¤À¥e¦§ Q Y 9 2 9 @b¨Â@rÃU Q = 3 / S9 2 4 3 /
! "#$&%'!() *,+-./0-213546.87:9;1-2-135<=1 Ag,n > 0!7?0!9;@BAC7C1DA5-135<=E0F7?0!9;9G13H/.9;9JI"A?0!A?E&4K35<=1#-0L4K.7C9G1
Y //
G
1nY
??
0E2D=3CEMI=1N-13O3CAC7CI=/BACI=7C13P<=1ND=E&QR10!I > 7:ED=/E > 0!-3CI=7 G S!T D > 0!7:ACE/I=-2E17UV/13P/0!7:AC13P-./0!-13O46.87:9;1-2-13> 07C09G@BA?7:1DAW<=13X<=.8I=Y=-213X9;.!A?E2463W<=1[ZXI=9G46.87:< S]\ D^<_9G.8DRA?7:1`-0 > 7:. > .83CE&A?E2.8D^3CI=E2Q!0!DA?1a<=19b0!D=E@7:1#/.89 > -@BAC19;1DAH0D0!-.F8I=1c0!IdA?e=_.7C@9G1Jfgihgkj SlJmonp'n=qVrtsurCn=vxwyzyt| SO ~5:ikBb,b,6t8Bi G Ag,n !kW G Bc V W!! G V ? Ag,n J#5Bi; ! n ? G[n] 6¡¢=N !¢ Ag,n £
fg2¤'g¥¦§¨©oª8«?ªR¬ª®`¯O°Kª²±´³§Oµ;°6ªµ;¶8³µ;§´¦§«·W°K¸¹§'ºª§» I >=> .3?.D=3 g ≥ 2 SR¼ .8I=3¢9;.DAC7C.D3¢MI=15-21H<=7C0 > 10!IbI=D=E&QR17:3C1- H•
3CI=7 Ag, 0 D=1H3½¾_BA?1D=<]¿?0!9b0!E31D²I=D<=7?0 > 10I<1F87:.8I > 13ÁÀD=E23]1BA > -0!AC3]3CI=7 Ag, 0 -.87:3CMI=1 Ag, 0 13:A]I=D=1cQ0!7CE2_BA?_<=1 » E21F81-´<=1At > 1 > 0!7?0!e=.87:EMI=1 D 6= 1 1BA Ag, 0 I=D=1/.89 > 0/BACE2À/0VA?E2.8DLA?.7C.!Ã<0-215<=1 Ag, 0 SRT D > 0!7:ACE/I=-2E17BU8E2-D½Ä1ÅE3:AC1 > 03Æ<=1$3C/e=_9b0J3C19GE&ÇÈ3:A?0Y=-21$3CI=7 Ag, 0 MI=E_BAC1D=<"1$-1$3C/e=_9b00Y_-E21DI=D=E2Q817:3?1-1BAÆ<0D=3-1MI=1-'-k½Ä0<=e=_7C1D=/1c<=1H•
3C.8E&AHÀD=E11BA > -0!AC13CI=7 Ag, 0 ST DX4¹0E&AUI=DWAC1- > 7C.-.D=F819;1DRAÁD½Ä1Å"E23:A?1> 0359GÉ9;1c0 > 7C@35/e0!D=F819G1DAH<=1#Y0!3C1cÊ Spec(C) S» .E1DRA
` 6= pI=DWD=.89JY=7:1 > 7:19;E217H1BA 2 ≤ d ≤ g
I=D1DRA?E217 S » .E2A B ∈ Matg(Z)I=D=1c9b0VA?7CE2/1
3:Â9G_BA?7CE2MI=1<=_BÀD=E21 > .83CE2ACE2Q81ÆAC1--21MI=1 B ·εg 3C.E2AË/.8D=F7CI;Ê ε1
9;.<=I=-.pU.8Ì
(εi)<=_3?E2F8D=1Æ-i0$Y03C1
/0D=.D=EMI=1b<=1Zg S » .8E&A b ∈ Matg(Q`)
I=D=1;9b0VA?7:E/1;3:Â9G_BA?7:EMI=1;<=1GQ!0-2I0!ACE.D`Çt0<=EMI=1;_FR0-21
ÊB SRÍ 0 9b0VA?7:E/1 b ED=<=I=E&A¢I=D 1
ÇÈ9;.!A?E&4<=1ÁZXI=9G4K.7C< > .3CE2ACE24 > 7:ED=/E > 0-219;1DA > .8-07:E3C_ [Zg → Ggm]3CI=75-¹½Ä.Y¿C1BA
(Spec(Q`) −→ Spec(Z`) ←− Spec(F`))<=1OuvComp SÆT -2-1²ED=<=I=E2A<=.D=/ÎI=DÏ3C/e=_9;0 G 3C19;EÐÇt0Y_-2E1DÏ3CI=7
Spec(Z`)MI=E$13:Ad0Y_-2E1D
> 7CE2D=/E > 0!-19G1DA > .8-07:E3C_]3?I=7 Spec(Q`)1BAÆE23?.9;.7 > e=1$Ê Gg
m
3CI=7Spec(F`) S » .8E&A Q`
I=D=1$/-2ÑACI7:10-2F8_Y=7:EMI=1<=1
Q` SP¼ .AC.8D=3 Γ-21F87:.8I > 1<=1WÒ 0!-.E3<=1 Q`/Q` SÓÍ 1F87:.8I > 1 GQ`
[p]<=13 > .8EDRA?3<=1
pÇA?.7C3CE2.8D<=1
G(Q`)13:AI=D=1
FpÇÈ7:1 > 7C_3C1DA?0!A?E2.8D3:Â9 > -1/BACEMI=1$<=1 Γ
<=1$<=E9G1D=3CE.D2g S=¼ .8I=3
> 7C_3C1DAC.8D=3Á<=_3?.7C9;0E23]-13]9;0!AC7CE2/13]<=1A?0E2--1 2g × 2g > 0!7ÁY=-./3]<=1A?0E2--21 g × g SÔ -´1Å"E3:AC1JI=D=1Y03C1(ei)i
<=1GQ`
[p]<0!D=35-i0!MI=1--21#-i0L9b0VA?7:E/1#<=1#-i04K.7C9G1#3CÂ9 > -21/BA?E2MI=1#13CA
(0 Id
−Id 0
)
!#"$%" $&$'( )+*,+!+-
.0/21354687359:.7;7<.=7?>@3A0/CB;D41. Γ EF .4173HGID FCJ .(χ · Id ∗
0 Id
),
D,K χ 1L6CB<M,4.N7<.NA3 F 3A0/CO F .PA0QA7;D5/RD J B<9:.TSVUW5XYU p Z\[ >@3A0/CB;D4&1:&6CD,:6^]_M F D: E .N1>`B;4. F /RB<.N1. Γ.6a/H1D44L. E 3 F 7;3 J 35/ F B<A.b1:c7;D,M,3 F B</Cd J .b1. J D,4D1 F D J B;.feg1>@3 EF O6ih`j kPlm\n EF D EZgoaoCopZrqZ@s5t n.7;7<.vu53:w/ (0 B
0 0
),
D,K B 1L6CB<M,4.=7;3 F L1:A0/RB<D,41. B J D!1:7;D p Z[ .=6CD:6^]_.6 E 35A.
H =
(d−1⊕
i=1
Fp · ei)⊕ Fp · e2g
1. GQ`[p] .6C/x/RD5/p357;. J .4!/vB<6CD/ F D E .H1.H1B J .46CB;D4 d Zo 7y.6a/z6C/R37<. E 3 F Γ J 3B<6v7<.H6RD:6^]_M F D,: E .1>@B<4. F /CB;.|1. Γ 4>35M,B~/ E 36/ F B~uB357;. J .4!/P6C: F H Zf[ .|M F D: E . H 4BL0/R37<.i6C: F Spec(Q`)
4.|6C.EF D7;D4M,.N1D,4A E 36.4&:4(M F D: E . 4BL0/p357;.N6C: F Spec(Z`) Zg D JJ . ` 6= p n\B<74.6C. EF D7;D4M,.E 3684D4 E 7;:68.4:4M F D,: E . 4B.0/ E 73/26C: F Spec(Z`) Z D6RD46 D = d n4D5/RD46 Ag, 0 7;3u3 F B<L0/RL1.NB<.M,.7g356R6CDAB<L..0/ Ag, 0 :4.AD JNE 3A0/CB A3/RB<D,4/RD F D5137<.=1. Ag, 0 Z[ .=6CAdL J 3N3L7;B<.4 GQ`
.0/27<.6CD:6^]_M F D,: E . H B<41:B<6C.4!/2:4 J D FCE dB<6 J .=1.Spec(Q`)
1346 Ag, 0 ng9!:B6>@L0/C.41.4c:4 J D FCE dB<6 J .b1. Spec(Z`)13546 Ag, 0 E 3 F 7;3 EF D E D5]6CB</CB;D4+ Zf 4.41L1:B</9:.V7<.b6CD:6a]_M F D: E .b:4B~u. F 6R.76C: F Ag, 0 4.V6>@L0/C.41 E 36.4:4M F D,: E . 4B.0/ E 7;35/6C: F Ag, 0 Zg 4c.0.0/n7?>`B J 35M,.VB;4!u,. F 6C.V1>@:4 EF D,7<D,4M. J .4!/L0u,.4!/R:.7 E 3 F 7;.
J D FCE dB<6 J . Spec(Z`)→ Ag, 0 6C. F 35B</8:4 EF D7;D4M,. J .4/21. H ZDB</ D = d1, · · · , ds :4.46C. J 7;.D F 1D,44L4D4#uB<1.9!:.7<AD,49!:.1B;6a/CB;4A0/x1. 1 Z 43ds > 1 .0/7;. F 35B;6CD,44. J .4!/ EF LAL1.4/ J D4!/ F .9!:.7<.6CD:6a]_M F D: E .2:4B<u,. F 6R.7 Hs
4.6C. EF D7;D4M,. 3 J 35B;6 Ag, 0 7<D F 6C9:. Ag, 0 .6a/:4.VAD JNE 3A0/CB A3/RB<D,4(/RD F D5137<.b1. Ag, 0 Zo 7.4.6C/1D,4AV1.J J .=1:1 F 3 E .¡35: H• Z
! "#%$&' () (*(
+,.-0/214365 Ag, 0 18797;:<->=@?BA D = g 5214C;1436DFE9CG,IHJ,>7LK418C;=@?FE9?>7M14NO?>HJ=@,>PQPQ,>7M5QC;=@PQ->=J5218HJ,>36,>PREM5O:SHJ1UTC;=@7;?>,>7.3WVQPQ=J,>7YX[Z\VQP]7;:SVQ7^T`_4C;:4V25O,aHb18_SC;14PQ_4=J,>P]XQ,pTcE9:4C;7;=J:4P[d2eB:8E9:4PQ7 Aord−mult
g, 0
H@,aHJ=@,>V!:4C;XQ=@P214=@C;,fT3WVQH@E;=J5QH@=J-g1UE9=ihWXQ, Ag, 0 × Spec(Fp) j -4Zk,>7GE!5218C]XQ?Fl2PQ=iE9=@:SPHJ,m7;:4VQ7^T`79-n/Q?>3o1p:SVqKS,>CGE!XQ, Ag, 0 ×Spec(Fp)
7;VQCrHJ,>sRVQ,>HtHJ,u7;-0/Q?>3v118Nw?>H@=J,>PxVQPQ=iKy,>C;7;,>Hz,>7;Er:SCGXQ=JP218=JCG,%,fEa:SxH@,u79:4VQ7^T`_SCG:SVQ5O,uHb18_SC;14P|T_S=@,>PVQPQ=iKy,>CG7;,>HI,>7GEo?>_y18HI14VPQ:~y14VX2V3v:4C;5Q/Q=@7;3v,XQ,2C;:4NO,>PQ=JVQ7>dMeB:4E;:SPQ7vX2,!3v>36, Aord
g
HJ,HJ=@,>V:4C;XQ=@P214=@C;,oXQ, Ag × Spec(Fp)
dz+LZ\:4VQNQHJ=MXQVPQ=@KS,g14V=JPQXQVQ=iEWVQP=@7;:S36:SCG5Q/Q=J7;3v,oXQ, Aord−multg, 07;VQC Aord
g j HZ185Q5QHJ=@-g18E;=J:4PCG?>->=J5QCG:<s<VQ,!18797;:<->=@,!AmVQPQ,K814CG=J?FE;?!14NO?>HJ=@,>PQPQ,!:4C;XQ=@P214=@C;,]H@,]P2:~y14VXQV3v:4C;5Q/Q=@7;3v,.XQ,.2C;:4NO,>PQ=JVQ7>dyeB:SVQ7?FE;,>PQXQ:SPQ7->,.CG?>7;VQH@E918EM14V|%->:S365214-FE;=@l2-g1UE9=@:SPQ7LE9:4C;:8cX214H@,>7>y,FE,>PXQ?>XQVQ=J7;:4PQ7WVQPQ,PQ:4VqKy,>H@HJ,oX2?>36:SPQ7GE;C91UE9=@:SPXQ,HZk,fq=@7;E;,>PQ->,XQV7;:4VQ7^T`_SCG:SVQ5O,-g18PQ:SPQ=@s<VQ,5O:SVQCXQ,>7h183v=@HJH@,>7.XQ,K818C;=@?FE9?>7I14NO?>H@=J,>PQPQ,>7W^k2,FEk]<n27;,>H@:SP!VQPQ,=JXQ?>,XQ,a=JPQ->,>PREI =@HJH@:SPQ=d
¡¢£2¢o¤¦¥¨§|©ªO«U¬¥®¯¥¨«°§±¬§<²v³SªO´¶µz¯2³4·¥¨¸t³S¯q·°¥¨ªO®z²¹ :4=@E
ΣVQPQ,ºXQ?>->:43v5O:S7;=iE9=@:SP5O:4H@~y?>XQC;14H@,14XQ36=J7;7;=JNQH@,ºXQ,
C(X)d»PPQ:8E9, Ag, 0 ,FE Ag HJ,>7I->:43%T5214-FE;=@l2-g1UE9=@:SPQ7¼E;:SCG:4cX218HJ,>7YXQ, Ag, 0 ,FE Ag s<VQ=HJVQ=7;:SPyEI18797;:<->=@?>,>7>d2½WZ185QC;D>7YH@,rE9/Q?>:4C;D>36,W¾ ¢£2¢À¿¢ ¾=JHO,fq=J7GE;,rVQP36:SCG5Q/Q=J7;36,rXQ, Ag, 0 X214PQ7 Ag s<VQ=w?FE;,>PQXHZ\:4VQNQHJ=wXQVPQ=@KS,g14V[d ¹ :4=@E S HJ,795O,>-FE;C;,X[Z\VQP->:SCG5Q7.XQ,-g14C;14-FE;?>C;=@7;E;=JsRVQ,
pd
ÁxÂ;ÃaÄtÅÇÆ^Å;ÈQÄÊÉOË`ÌwËÇÌqdzÍ ÎIP]79-n/Q?>3o1%7;,>36=iTÇ18Nw?>H@=J,>P]7;VQCS,>7GE.:4C;XQ=@P214=@C;,a7>Z\=@H[,>7GE.,f|E;,>PQ7;=J:4P]X[Z\VQPQ,
K814CG=J?FE;?º14NO?>HJ=@,>PQPQ,:SCGXQ=JP218=JCG,5214CYVQP]E9:4C;,4deI:4E;:SPQ7
GHJ,I7;-n/Q?>3o17;,>3v=ÏTÇ14NO?>HJ=@,>Po-g18PQ:SPQ=@s<VQ,I7;VQC Ag ,FE H H@,I7;:SVQ7^T`_4C;:4VQ5w,.s<V2147;=ÏTcl2PQ=2,FEM5QHb1UE
-g14PQ:4PQ=JsRVQ,79VQC Ag, 0 d2eB:8E9:4PQ7Aord−mult
g, 0HJ,I7;:SVQ7^T`7;-0/Q?>3o1º:4VqKy,>C^EÐXQ, Ag, 0×Spec(Fp)7;VQC¦H@,>s<VQ,>H
G,>7;E¦:4C;XQ=@P214=@C;,B,FE
H,>7GE¦?>_y18HO14VPQ:~y14V
XQV!3v:4C;5Q/Q=@7;3v,XQ,ºQC;:SNO,>PQ=@VQ7FrobG
XQ,Gd2eB:8E9:4PQ7IXQ,36>3v,
Aord
gHJ,a79:4VQ7^T`79-n/Q?>3o1%:SVqKS,>CGE.XQ, Ag × Spec(Fp)79VQCYHJ,>sRVQ,>H
G,>7GEI:4C;XQ=@P18=JCG,4d
!"#%$" '& ($" $)"
*,+.-0/132546+078:98<;=?> @A.BDCσ ∈ SΣ
=FEHGIJCKBMLDBMNKA5ONPLDQNPOA.CSRTCSBDA5ONPUQNPVRTWSRTX5WKR.VYQN[Z=]\^Q%C_Z=]\=`aATCSA.ON
fL<bcBDNSA.X5GOBDQPIOBDd5QWKNKQLDLMQPQOCKWKQ
1 e:f ATCSBhgiNHIOBDd5QWKNKQLMNHNKIW MV ′σ , σ, 0
=jFQPNKA.IN e NKk'YG f R
Aord−mult
g, 0
WKQOkA.OCKWKQlL Rσ e NmCSWKRTCKQnUQ Ag, 0 NKBoQ%CpNKQILDQ f QOoCpNKBoLMQlOA3qRTI_UQ f QNmCpGX5R.LorHVR.WsCSBDQlNSQ f B e RTtGLMBDQOOQ5ukQwvIBNKBMX.OBDxQW1(Ker(f)) = Ker(f)
=ywO!VQIJCzWKQ%gA5W f ILDQW|kQ%CKCKQwkA5OUBDCKBMA.OQO!UBDNSR.OCzvIQ V0DQ%C
V1D
A.OC f~f QPB f RTX5QPURTON V/V ′σ⊥ =
jFR_kRCSGX.A5WmBMQxtWKGQr_LMR[giWmA5OCKBMWmQ(Aord−mult
g, 0 → Aord−mult
g, 0 )d5GWKBhxQwIOQaVWKA.VWKBMG%CKG f AUILMR.BMWmQR.ORTLMA.X5IQrkQLDLMQUQ
(Ag, 0 → Ag, 0)u UA.OOGQVR.WLMRVWmA5VA5NKBhCSBDA5O0 <[email protected]
(U → S)IO
A5tKQ%CHUQOuv
CKQLvIQSNKA.BDCHIONKk'YG f RNKIW Spec(Fp)
=
1.FsK7898=?> F' ¡3¢ £¥¤o (Aord−multg, 0 → Aord−mult
g, 0
)(U → S)
¢J¦TK¦T§^¨©<'wª '«_ .¬%® ©«
G′ ¤ Aordg (U)
©¯°ª±«³²%¡G′ «´¶µ©·¤·#¡·¸«3¹'ºµ§,¦^«3°§»m¦5¬°µª]»·¼ S¤»·¦T»S_«¡
S ½ ©¾©¯°ª±«²%¡Pª 3¹S¦ª §^°·©F¢ ¡PªM¦,©¯ S¢ ª S»µ©:¦Tª ¿¤S ½ »ª?mÀ5» «©
Á V ′ ∈ C©ª?²%¡
V0D/V
′⊥ = V1D/V
′⊥ ½Á ¡·«¡ ® '¹©<» ·
V/V ′⊥ → X(G′) ½Á σ ∈ SΣ
¤ ·©?ªK´ ¦o¤ºµ°'·¹'¹S ·©¥»°·©?ªD¦n §,¬%»ª]»·µS¦T»'pD B(G′,Ker(FrobG′))
¢ ¡H©¯ ¡©¤T»Ã3»M«3¡P»°SµS¤T¡¹°©¥»¥¬%ª P'µS¤¡»©
D¤S Ä
Å6RTW|kA.ONSGvIQOCuLMQ f A5WmVYBMN f QwUQ Aordg
UR.ON Aord−multg, 0
BMOUIBhCzVRTW|LDQwOA3qRTIUI f A.WKVYBDN f QUQ[ÆWmA5tQOBMINHNbÇG%CSQOUÈQOÈIO f A.WKVYBDN f Q
Aord
g → Aord−mult
g, 0 .
Å6R.WUQONSBhCSGUQ Aord−multg, 0
UR.ON Aord−mult
g, 0
Q%C[UQ Aordg
URTON Aord
g
ukQ f A5WmVYBDN f QQNmCPL¥bÇBMOd5QWKNKQUI f A.WKVYBDN f QPU bÇA5ItLDBUIÈOBhdoQR.I =ÉwBDONSB<uLDQNHNKk°YG f RTN Aord−mult
g, 0
Q%C Aord
g
NKA5OCHBMNKA f A.WKVYQN=
Ê !ËÌpÍÎ.ÏÐÑÒÍÓÒÐpÎ5ÒÕÔ?Ì×Ö3ÌpØÙÒ`A.CKA5ON Afor
g
LDQÈkA f VLMG%CKGgiA.W f QL¾UQ Ag LMQLMA.OX#UQ Ag × Spec(Fp)uÙOA.CKA5ON Arig
g
LMRÚxtWKQX5GOGWmBMvIQ³WKBDX5BDUQ³UI#NKk°YG f R!giA.W f QL p e RTUBMvIQ Afor
g
uFQ%C[OA.CKA5ONTLMQCKItQ³UQ Aord
g
UR.ON Arig
g
=ÛlOJxO uOA.CKA5ON
GLMQPNKk°YG f RNKQ f B e R.tGLDBMQOÈkRTOA5OBMvIQPNKIW Arig
g
=
ÜnÝpÞ 1ß.-¼+078i;F89=?> à'ª mÀ.» «©¯¡·ÃT » «»·¦3¾«©¥°»¹©U¤T¤o¦·J« Arig
g
°©6¡·^«3 ¡o«¥TS ¡¢pál·» ½¢ªM¦©_©wªM¦3TK¦T·»·H
¤oGU [p]
©ªz²%¡HT
«3 »©ª ,'°ª ¨Ã.§,·©P¤T¡0· âo¦T¡ã¤T¡ä§^ :¢º»M«§^!¤å' .¬°·»¡o« Ä
!#"%$'&(*),+-(*./"%$021 3547698;:=<?>@A47BDCE6GF4AHI8J8-:=CEKL4NMPOIQR@NS;BDCE:=47BDC9HUTV4W'KL6=CE47BX74RFTV>BYFKZ8-<?:,8;@A@N4X7S?@A@ >'C[8CEK]\
Ag, 0 × Spec(Fp)
// Afor
g, 0
u
Aord
g
88rrrrrrrrrrrr// Ag × Spec(Fp) // Afor
gS?^ Afor
g, 0
F_76,KL<?B4NHL4NX7S?@AJHI_`C,_A\aS;:,@A47HbF4 Ag, 0 HL4NHIS;B<cF4 Ag, 0 × Spec(Fp)Od3e4N@AS?:=JfKI6,@N4
u476=CgJ:,S;J:,4;Odh Ti8-J:,j76lkVmonqpsrt'uev5HL47@wO5x'OIQy-z4`C#k|~Pz/v5KLHb:,_8-HIKL6E4N>BKI6,S?@AS?:=JfKI6,@A4l6,>:>B S;KI6,KLB8;<;4F4 Aord
g
O;bBJ8;:CEKLX7>HIKL47:7vu476=C_`C[8-HI46,>:>B S;KI6,KLB8;<;4F4 Aord
g
OBRBS-CE4 Arig
g, 0
H*T476,J8;X74:,KL<?KLF4 8-6E6,SX7KL_ Afor
g, 0
4`CT0
HL4C,>4gF4 Aord
g
F8;B6 Arig
g, 0
OB8;JJHLKID>8;BDC k'vCEfO5Q?OV'O%z*vS;BS?'C,KI47BCdH*T4W'KI6=C,47BX74FT>B S?KL6,KIB8-<?4b6,C,:,KLX`C U0 :=476EJO U F4 T0 :=476,JOF4 T F8-B6 Arig
g, 0 :,476,JO Arig
g CE47HL6>4uKLBF>KI6,49>BwKI6,S;@NS;:,JfKL6,@N4
u : U0∼−→ U .3Ti8-JJHIKLX8-C,KIS;Bw:,_7X7KLJ:,SD>49F_7 BK]C¡HL496,S?>6¢£<;:,S;>J49HI8;<;:E8-B<?KL47Bw:,47Xf47:=Xf_96,>:
UO
¤ KIB6,KUvHL476 8-:,KL_`CE_76A8-_7HLKI47BB476RJ8-:E8-@N_`C,:,_7476RJ8-: U ∩ Arigg
8-F@N4`C=CE47BDCR>Bs6,S?>6¢£<;:,S;>J4¥X8¢BS?BKL>4;OB¦S;'CEKL47BDCg>B¦:,_76,>H]C[8C @AS?KLB6gJ:,_7X7KL6g>4lX747>PWF4§k¨mo©z4`C#kV¨¥|rz*vdX8-:KIHbB4FS?BB4AJ8;6F4NX7S;BFKLC,KIS;B6,>'ªl6E8-BDCE4NJS;>:D>TV>B4 8;:=KI_`C,_l8-_7HLKI47BB4l8-F@N4`C=CE4A>B6,S?>6¢£<?:=S?>J4X8;BS;BKID>4;O5«qS;>'CE4`\¬S?KL6veKIH6,>'ªNCJS;>:HI8wJHL>J8;:CgF476g8-JJHIKLX8CEKLS?B6g8->PWY\aS;:,@A476@NSF>HI8;KL:,476F4 KI47<;47H
p¢®8;FKL>4767O
! #" !%$& #" % "'&()
*,+.-0/2143657+.808934-0:0;<143>=?-0@BA0/DCE@>-F5HGI@B-0/DJKA03,1434GI5H+>;<-034/DLM@>GNCO34/2CO@K:0A0PQ+.;RGI34/2:03TSK;R34U>34PWVYX4Z.[]\QXp8&@BA0G Ag,n +>:0CO365N5N3^-_5`:034/`GI34PRa6bB34CE34-_5H/34-c1^+.GH+.165Hd4GI;R/I5H;<JKA03e-KA0P<PR3Wf143g80GI@Bh0P<a4CE3e-jik+7:0@>-01g+>A014A0-PR;<34-l+mbB341OPR+nC'+.A?b.+>;</H3oGNd4:0A0165N;R@>-jprqs@>/tGId4/HA0Pu5v+w5H/xUBd4-0d4GN+>P<;R/N34-F5T+.A]yz1^+>/t:034/,LM@>GNCO34/t14A0/N80;R:+.PR34/
:03|U>34-0GN32365
3PR35H~0d4@>GNa4CE30.=K5N~jp9>pFpR69@Bh?5H34-FA8+>Gt+.5N|8&@BA0GPR34/LM@>GNCO34/:03|UB34-0GI3
1p
qs@>A0/7A?5N;RP<;R/N@>-0/7:034/2GId4/HA0Pu5v+w5H/2:03cSK~03480~034GN:]D+.GNGI@B-ljo9]r3657:03tA0PR34Y365|S]+.-0w+>GN+>-lzr=+>;<-0/N;JKA037:034/e5H~0d4@>GNa4CO34/g:ji+>-0-KA0PR+.5N;R@>-':037PQ+t14@>~0@BCO@BPR@>UB;<314@>~0d4GN34-_5H37:00/En ^¡&K¢365s¤£s¥§¦¢¨]p
9©ª©E«x¬9¬`®B¯±°³² µ´¯K¢¶µ·K¸º¹SK@>;R34-_5
g ≥ 2365k ∈ Z
:034A]y»34-F5N;R34GN/4=n ≥ 3
A0-¼34-F5H;<34GY-0@B-¼:;ub];</H;<h0PR3§8+>Gp365MA0-
Z[1/n]½CO@K:0A0P<3Bp³*ik+.80GNa4/¤¾2¿o¦0r=`5N~jp`ÀÁpkÂ]pmÃ=`;RPg3Äy?;</I5H3'A-03'14@BCO8+>165N;<Å1^+w5H;<@B-5N@BGI@>ÆÃ:+.PR3 Ag,n:03 Ag,n 365³A0-/N1v~d4CE+/N34CE;+>h&d4PR;<34- G /NA0G Ag,n JKA0;Fd65N34-0:P<3/N1Ç~0d4CE+s+>h&d4P<;R34-cA0-0;<bB34GN/N34PF/NA0G Ag,n [email protected]@>-0/ωPR3'LÈ+.;R/N143^+.AÉ;<-FbB34GN/N;Rh0P<3Y/HA0G Ag,n :034/ g ÃLM@>GNCO34/Ê:0;uË9d4GN34-F5N;R34P<PR34/;R-_b>+.GN;R+>-F5N34/:03 G pWÌr380GN;<-014;R8&3,:03tsÍF1Ç~034Gx¾2¿o¦0r=?80GN@>8jp]ÁÊpR>pkÎwr+.ÏEGNCO3xJKA03,PQ+ÊGI34/N5NGI;R165N;R@B-n:03 Ag,n Ag,n ;<-0:0A0;<52A0-;R/N@>CE@>GN80~0;</NCE3
H0(Ag,n , ωk ⊗M
) ∼−→ H0(Ag,n , ωk ⊗M
).
Ðe-8+>GÑ5H;<14A0PR;<34G4=P<3TU>GN@>A08&3H0(Ag,n , ωk ⊗M)
34/I5;R-0:0d48&34-0:+>-_5:0AÒ1Ç~0@>;uyÓ:03 Ag,n Ô @B-Ò+>808&34P<PR3/N34/od4P<d4CE34-_5H/ÊPR34/LM@BGICE34/ÊCO@K:0A0PQ+.;RGI34/o:03UB34-0GI3g=³:03n-0;ub_3^+.A
n=W:0389@>;R:0/
k365O±14@K36ÏE14;R34-_5H/
:+>-0/Mp
q@>5N@B-0/DP<3t14@>CE80P<d4CE34-F5H+>;<GN3x:03 Ag,n :+.-0/ Ag,n 3657CÊA0-0;</N/N@B-0/ѽPR3x:03t/H+o/I5HGIA0165HA0GI3t/N1Ç~0d4CE+w5H;<JKA03tGId4:0A0;<5N3>p9Õ|-:0;<57JKAji¤A0-03xLM@BGICE3tCO@K:0A0PQ+.;RGI3t34/I5oÖÄ[_×Ø&ÙZ_Úw\QÛT/N;r34P<PR3t/4i+>-0-KA0P<3t/NA0G
Dp¢[email protected]@>-0/
Cusp(n, k,M) = H0(Ag,n , ωk(−D)⊗M
)
PR3oUBGI@BA08&3O:034/xLM@>GNCO34/t1^A0/N80;<:+>P<34/4p*ik+.80GNa4/E¤¾2¿o¦0=j80GI@B8jpÁÊpR>pkÜwÃ=j;RPÝ-03O:0d48&34-0:l8+./T:0AÞ1Ç~0@B;y:03 Ag,n p&SK; M 34/N57A0-Ò+>-0-03^+.Aj=@>-Ò+
Cusp(n, k,M) = H0(Ag,n × Spec(M) , ωk(−D)
).
"!#%$&!'"()!*&+,-"(./%( 0/1243658798:;=<=>@?BA?CADFE G@H8IJIKMLMKONL
g = 2KOH
3 PRQ k > g + 1 S G@T g = 2 U LRH8IJIKVLMKON"LXWH P M P L QHNZ[ 1
2n] YCZ K[,H\ P]U^PRQ LRT g = 3
LH8IJIKVLMKON"L.WH P M P L Q H1N Z[ 130n
] YCZ K[,H\ PVS@_`PaZ KObBIcT-L ZdP Ne Q Hb P \[ PXf ce,N1g PRZdP N Q [ Pih e8L P TjN[,H1T Q HNkT-LMK Z KObBIc1T-L ZdPM ⊗ Cusp(n, k,Z[1/n])
∼−→ Cusp(n, k,M) .l/monpqrptsus-v,wxqymozrw|,v~Xp~qymosmR"wxzyrm~nmmoqrmsu|Jmm~rzm~=nVvqrvnrqrwuzywu1m
0mojptqymz
ps-v,wxqymz#nzrwxv,sxmz'mnVv,qv,nqrwuzywu1mpD" szm.wu'rptqrmim~%ptzv,~
M =
Fppq#rpt"a~piqrm)qymVwxmq
p1w~m)wu|wuzrm)vz
nD
7987CC98<>@?CA?DFE GKOT P N Q n ≥ 3 U g = 2KOH
3 UiPRQ k > g + 1 Q bKOT-L P N Q T P bL S G@T g = 2 ULRH8ItIKMLMK8N"Lp > 2 PRQ LRT g = 3 U LH8IJIKVLMKON"L p > 5 S_`PZ KObCIc1T-L Z P [ P)f ce,N1g PRZdP N Q [ Ph e8L P TjN[,H1T QHNkT-LMK Z KObBIc1T-L Z P
Fp ⊗ Cusp(n, k,Z[1/n])∼−→ Cusp(n, k,Fp) .¡aptz)p~qypt~z)¢Jv,sxmm~£1wxs^~mR"wxzyrmv,z£mpqrmp1s-v,wxqymm¢m~qrm
2pt
3m£m
ptwuzazyrqrwunmm~a~¢tv,rwu D
¤ ¥¦¨§ª©« ¬86®6¯¯¬°±F±6©°"¬8«j²±³.v,~z4nmym*v,qyrwxm´t~ptzµqv,msxp~z4qywx|tmm~J4sumVzµqyzrsuv,rzµ"wxsuwxzrzµv,~zµsxv)pt~zyrqvOwxp~
%pqrmX D¶ K Z Ie fQ T · f e Q T¸KON"L Q KObKO¹[JeO\ P LRDFE ¡aprpt~z
Bsum
Z º p1sxm»mz4jptqymz¼wusxwu~Vv,wxqymz¼zy½rqrwu1mzm~rwxqymz4zrqZg m C sum*n¾t~m*m
B⊗Rmz^pqrmzµwusxwu~Vvwuqrmzµzrmw º ¿~wxmzµptzrww|Jmz4Àqv,wxnVv,sqvOwupt~~ms D l/mz¼npv,nw¿@nVvOwupt~z4rptqypÁ¸v,sxmz¼m Ag,n zrpt~Âvzrzrp1nwumzÂÀÃmz¼nptptzrwurwxp~z¼p ºsu½tqv,sxmzµv,wuzrzrwxsxmz
GLg(Z) º 1wu|,vqyw-v,~rmz^m C ÄÆÅ#ÇÈÉ ´t DtBÊXDÌË1DÌË8ÍD1Î pt~zrwuqypt~z^~mÂmsusxmnptptzrwurwxp~Σm'~prpt~z Ag,n sxvinptvnrwu¿nVvOwupt~dpqrp,Á¸vsum.wsxwmzy#v,zrzrp1nwxm D l^mzrvnmvsu¢tqywxm Ag,n mzyXqrpqrmdzrq
Spec(Z[1/n])´FmzyXsuwxzrzrmdzrw¼mXzrmsumm~
ΣmzyXsxwuzrzrm ÄÅ#ÇÈÉ ´ DÏCÊiDÐËDÒÑOÍ ´^mmzr~Ózn] vÔqyp8Õrmnrwuzrw
ΣmVzyps-v,qrwuzrm ÄÅ#ÇÈÉ ´^ D¼ËDÆÖ mr DÏÊXDÌ×1DÒØO͸D
¡»p,p~zD
sum.npsum~vwuqrmm Ag,n v,~z Ag,n ´m'X~wuzrzrpt~z º sxmm.zvizyrqrn"qrmm.zrnvqrw Dª s¿~wu~Ùw|wxzrmqXm Î vqÚwumqXm Ag,n 1wÂmzyÀknqyptwuzmm~rz~ptqy v,=zrwΣmzy
sxwuzrzrm D ¡»prpt~z fi ~kzr½1zymXÆv,rwxp~z.sxpnVvsumz.mD
v~z Ag,n D/Î pmXsumz.|tm~msusxmzzrwx~¢s-v,qrwz*m Ag,n zrp~#ptqywxmzV´sum)vwuznmVv,Ω1
Ag,n(logD) = Ω1
Ag,n
(dfifi
)
mzy£sup1nVvsumm~)sxwuqrm D ³XÆvqyz ÄÅ#ÇÈÉ ´ D/BÊXDÆÛ"DÌÜ8Í ´/sÒv,sxwunVv,rwxp~¨mÝ»pvwuqv ºÚÞ m~nmqwx~w~wxzrppqrwuzrmSym2(ω)
∼−→ Ω1Ag,n
(logD) .
¡»p,p~z K = det(Ω1
Ag,n
) sxm#v,wxzrnmVv,vsuwxzv~J4m Ag,n D l^ÆwuzrptptqywxzrmamaÝap1vwuqv ºÚÞ m~nmqwx~wa~wuzrptptqywxzrmωg+1 ∼→ K(D)
D
"!#%$'&'(*),+(.-/102/3(*4-65879:(*;<(*4(*=>@? ACBEDFBEGBIHJLKBEGMFN<DFOPQNSRTOVUWRXKBEY[ZB\M]PQN]P^B\GMSZBE_Q`EXRD<JI_^TMSJ\RVaY6PQM]M]P^b_QTMcRT
C dfecg:h*ijVk >mlonpOGT[RnpTGqN<DFTarT_^_QTMcTMFNsJIZZT_QXTW58$'&'(*),+(.-/0t/3(*4u5Ivw&'xy47S+zI;zI/0|BEO+;<7)/3~;<7L58$'&'(*),+(.-/0t/3(*4u5q79:(*;<(*4(*=TNc_tnJION]D]T-.7'&'(*45q75q$'&'(y)|+(.-/02/3(*4u5879:(y;'(y4(*=>EOM<JJKBEGMFN]D]OPNOGTLN]D]B\P^M]PQY6TRXKBEY[ZBEM]PQN]P^B\GwRT
CRPQN]T5Iv&<x*47&<7402;]zI dmik >oTM%N]D]B\P^MRXaKBEY[ZBEM]PN<P^B\GMsKBIH3GKP^RTGqNsM]P
g = 2BEO
3 dmi mZ>@\E k >|T6Z_^OMmT_Q_^TMsM]B\GqNs_^PQM<M]TMsM]P g = 2BEO3 dmi @N<>¡ >¢>Q£ k >C¤P g = 4
_¥JWM]TKBEGRTLRXKBEY[ZB\M]PQN]P^B\GwRTLACBEDFBEGBIHSTMFN_QP^M]M]T dpmi Z>E£ k TN¦D<J*§GTc_^J[RXKB\Y6ZBEM]PN<PQBEG¨RO¨K©EGTcZJIDFª«J\PN dpi Z>:\E k >¬ )|+:/02v5875Iv¯®°;<$5q7o±²("5.³q7>? ´,BIN<B\GM Aparf
g, n
_^J¦KB\Y6ZJ\KN]PQµK"J*N<PQBEGcN]BEDFB\H3RJI_^TCRT Ag,n JIM<M]BqKP^XT¶ _¥J[RXKB\Y6ZB\M<PN<PQBEG¨RO¨K©\GTcZJ\D·ªtJ\PN"TN¸GBIN<B\GMDparf M<B\G¨bBED]R¨DFXROPQN>:lmTcM]KXY6J
Aparf
g, nTMFN²_^PQM<M]TM]Pg = 2
BEO3TN
Dparf TMFN%OGRPQ¹VPQM]TOD ¶ KDFBEPQM<TY[TGqN]M²GBEDFYJIOVU>¡¤BEPNaOGTGqN]P^TD
D]T_^JIN]PQª>nJ\ZDFM di N]>º>Q£ k _QT»ª«J\PQM<KT"JIO ωa(−Dparf)TMFN¸JIY6Z_QTcM]OD Aparf
g, n
M]PTN¦M<TO_QTY6TG8NM]Pa > 12/n
>¼¦T"Y6J\DF½OB\GM½OTKTDFXM<O_N'J*NRnJIY6Z_QPQN]ORTJwRnJ\bBEDFR¾XN]XXN<J\b_QPSZJ\D¿,O_QTÀTN¤VJIGÀIJID<JIG dpmik RJ\GM[_^TM[K"J\M[ZJ\D·N<PQKO_^PQTD]M g = 2
TN3K"B\GM]P^RXDFXM6RJIGM6_^JM]OPQN]T¨RTKT
N<TUN<T\>TZTGRJIGqN"KB\GqN<D]J\PQD]TY[TGqN ¶ ¤TZTDFRVaÁÂJ\DFD]B\GP^_QM¦GnXGB\GKTGqN¦_^TODÂN]X"B\D]Y[Tc½OT»M]ODSpec(C)
>ÃÄ$'(*;<~)L75q7Å(58zI/;]zI>? ¤BEPN
kOGKBEDFZMcZJIDFªtJIPQNcRTK"JID<JIKN<XD]PQMFN<PQ½OT
p>mSB\GGBEGM·arGB\OMcOG
M]KXYJX
ZDFBEZDFTcTN¦_QP^M]M]TcZODFTY6TG8N¦RTcRP^Y[TGM]P^B\GdM]OD
Spec(k)OGRPQ¹P^M]TOD ¶ KD]B\P^M]TY6TG8N<M
GBEDFYJIOVUD
RTXTN,OGªtJIP^M]KT"JIOWP^Gq¹ETD]M]PQb_^T L M]OD
X>´,B\N]BEGM
W2(k)_2nJ\GGT"JIOWRTM¸¹8TKN]TOD]M
RT»ÆÇPQNFN¸RTc_QBEG\OTOD2RT
kTN¦M]OZZBEM]BEGM¦½OT
XDTN L M]TcD]T_Q¹8TGqN ¶
Spec(W2(k))>
¤OZZB\M<B\GMXEJ\_QTY6TGqN½OnpP^_TUVP^MFN<TSOGTGqN]P^TDν0
N]T_½OT Lν(−D)M]BEPN¦J\Y[Z_^TSZB\ODN<B\ON
ν >
ν0
>:%nJIZD]M dfÈ|eÉ:Ê ZD]B\Z>m£\£E>Ë k B\GJÌrÍÎ Ê Î i >pÏcÐ Hj(X , Ωi
X/k(logD) ⊗ L−1)
= 0ZBEODi+ j < inf(d, p)
>Ñ|G¨RXROPQN¦ZJ\DROJI_^PN<XcRT%¤TDFD]Tc½OTcM]Pp ≥ d
B\GJÌrÍÎ Ê Î i >Á¦Ð Hj(X , KX/k ⊗ L) = 0ZBEODÂN<B\ON
j > 0B\Ò KX/k = Ωd
X/k
RXM]P^\GTc_^T»ªtJIP^M]KT"JIO¨ROJ\_QP^M<JIGqN¸RTX>
ÓÔ\Õ×ÖØ.ÙEÚ¡ÔwÛ:Ü«ÝÜÞ>? lonJ\M]M]TDFN]P^B\G Ì3ÍÎ Ê Î i >pÏcлTMFN%OGT¹ETD]M]PQBEGªtJ\PQb_^TYJIP^Ms¹\JI_¥JIb_^TM]ODkRO
N<XB\D]Y[TRTàßcJ"áÂJ\Y6JIN<J*a3A|P^TqáTZBEOD6_^TMRPQ¹VPQM]TOD]MGTª«M dpÈ,eWÉ:Ê ¯KBEDFB>Ë>Q£*âq>K k ¯½OP¸GnpTMFNRXY6B\GqN]D]X[ZB\ODc_tnPQGMFN'JIGqNc½OT[M]ODC>mlmTN]XBEDFY6T[RT6ß»J.áÂJIYJ*N'J*a3A|P^TqáTM]OD
kD]XM]O_N<TD]J\PN
RT|_tnJIM]M]TDFN]P^B\G ÌrÍÎ Ê Î i >pÏcÐoTNRnOGT,X¹8TGqN]OT_^_QT|DFXM]BE_QON<P^B\GRTMãM]P^G\O_¥JID]PQN]XMRTMãM<K'XYJIMRT,N`VZTµGPmM]ODk>:´¸BEOM¦D]TG8¹8B.`8B\GM ¶ dfÈ|eÉ:Ê D]TY>m£\£E>>pb k ZBEODZ_^OM¦RTcRXN'JIP^_QM>
ÍÎ1äÎåçæVè géêë"hyìëyígéïî@ðë.ñ æ ghyò èólJ,ª«B\D]Y6JIN]P^B\GRT_¥J|KB\BEY6B\_^B\EPQTÂKB\XD]TGqN]TÂKB\Y6Y%ON]TJ\OVU%_^PQY6PN<TM¡P^GROKN]PQ¹ETMôM]OD¡OGM]KXY6J
½OJIM]ParKBEXDFTGqN>"õ_\M<O§[NRBEGKoRToRXY6B\GqN<DFTDm_QToN]XBEDFY6T ÍÎtöÎ2ö ZBEODM
RTôN`VZToµGPEM]ODZ[1/n]
"!#%$&!'"()!*&+,-"(./%( 0/12346572893: M = Fp ;<9=> p > 2 5?4 g = 2 =A@ p > 5 5B4 g = 3 CD 4FE5?4HG4FIJ5?3"K @ML= 2:N893 <9= :7O13 = I =PQ8:B2R4F5BP =)L=)> R ; ES = P = E @*L=)T; 5 = 46E L 34 @ 3E4F5?8PQ8:B2R4F5BP =
Fp ⊗ H0(Aparf
g,n , ωk(−Dparf))
∼−→ H0(Aparf
g,n × Spec(Fp) , ωk(−Dparf)
)
2893: k > 1 + g CUWV ; 2:NX5ZY\[Z]%^`_%a Ccb1Ced =A@ a C agfhG4FI/5B3"K @*L=)Li PQ8E @ : = :*O3 =H1(Aparf
g, n × Spec(Fp) , ωk(−Dparf)
)= 0 .
j = 5 > R i P ; Aparf
g, n × Spec(Fp) = 5 @ 2:B82: =)=A@ IF465B5 = 5B3: Spec(Fp)5B4 g = 2 893 3 Ck I = 5 @ 23: = P = E @
L=%L 4FP = E5B468E g(g + 1)/2 =A@ 5 = : = I6X <9=l Spec(Z/p2Z) CmU = PonP = G Dparf = 5 @ 3E L 4 < 465 = 3: l> :B8465 = P = E @ 5ME"89:BP ; 3pqO134J5 = : = IFX <r=Wl Spec(Z/p2Z) Cs 8 @ 8E5 L I =utv; 465 >=; 3w46E <r= :N5B4 T I = ωk−1−g
5B3:Aparf
g, n × Spec(Fp) ,
O1345 = : = I6X <9=)i S ; I = P = E @7l Spec(Z/p2Z) CxUyV ; 2:BX5ZYz|~G @ R C"C6 fG Lν(Dparf) = 5 @; PQ2I = 5B3:Aparf
g, n
2893: @ 83 @ ν > 12/n(k − 1− g) CxUyV ; 2:BX5y-6 Ce* G89E ;H1(Aparf
g, n × Spec(Fp) , K ⊗ L)
= 0
>; : p ≥ g(g + 1)/2 C E > 89E > IF3 @ I ;oLi Po89E5 @ : ;,@ 468E L 3 @ R i 8:BXP = Hh = E: = P ; :BO13 ; E @ O3 =K ⊗ L = ωk(−Dparf) .
9xh C 84 @ X 3E 5 > R i P ; 2:N892: =¡=A@ IF465B5 = 23: = P = E @qL=¡L 46P = E5B4F89E d 5B3:Spec(C) G5B894 @ UyV ; 2:BX5ZY\¢7£q¤G > 89:N8 Cx¥Ce fG89E ;
Hj(X , ωiX/k(−D)⊗ L−1
)= 0
2893: i+j < d Cr EW283:B: ; 4 @¦Li PQ8E @ : = :3E @ R i 8:BXP =#; E ; IF89S93 = 2893:I = 5¦5 > R i P ; 55B3: Fp5?41I V 89E
> 89EE ; 465B5 ; 4 @ I ; : i 5B89IF3 @ 4F89E L= 5¦5B46ES3I ; :B4 @?i 5 = E >; : ;>A@Bi :B4F5 @ 4FO13 = p Cr E = E LiL 346: ; 4 @ I = : = I6X <9= P = E @= E >; : ;>A@Bi :B4F5 @ 4FO13 = E13IFI =)L=§t 8:BP = 5Po8 L 3I ; 4F: = 5 L= 4 = S = I/E8EE i>= 5B5 ; 46: = P = E @*> 35B24 L; I = 5 C¨ 8E @ :N89E5 l 2: i 5 = E @ O3 V 4FIE V = p"4F5 @?= 2 ; 5 L=Wt 89:NP = PQ8 L 3I ; 4F: =WL= 4 = S = I L= S = E: = ≤ 3 =A@§L=2894 L 5 < 0 5B3: Fp C© E =Aª=A@ G46I5B3"K @*L= 285 = :
X = Aparf
g, n ,
i = 0 G j = 0 =A@ L = ω−k L; E5y«-6 C\D) 2893: Li PQ8E @ : = :#I = : i 5?3I @¬;@ 5B34 <; E @ C®yg¯°±¯²³´µ³B¯¶·xH¸ C ¹*º»¼.½¾ºg»¿ 2 ≤ g ≤ 3 ÀÁ½ k < 0 ÂúÄÆÅ H0(Ag, n× Spec(Fp) , ωk) = 0 Ç
!#"%$'&)(+*-,'.0/!12.43*+$6512.
798):<;>=@?BAC:<; DFEAHGJI6KL?BAMI8@NB8GO8PBNRQ?RPTS:J8UJ?B=)?B8VDFQ?RW:JXH?Y:<8:JZ GOUJI?R:JI[DHSNBGOA]\=@? _^`^ Q:J8XZ>?%P:JXaGOb
g = 4 cd GO=@=)?%Ze:fQNBPBGJ=)EgGOXH;hDC; GO8iQI4PBjJ8?kE:<A]lm:J;hD98no?BX]D9EZ>IXaZ>; XHXC?JpgGJ848?%Eg?BI6D9E:JX:JEEZ ;>qVI?BAkZ>?fDCSNBGJAH\B=@?rQno:J88IZ>:<DH;>GJ8sQ?rtaGQ:<;>AH:4u Aparf
4, n cwv 8xAH?K<:<8PS?JpyZe:iXH?BPBGJ8Q?rQNBPBGJ=[zEgGOXH;hDC; GO8xQ?|GJAHGJ8GJ~9?BX]DZ ;>XHXH?r?DkAC:f8?|Ze:QNBPBGJ=)EgGJXC;hDC; GO8xQIxPBjO8?|E:JAl`:<; DQno:JEA]\BX# pE cO6c G<DCGJ8X
AVor
4, nZe:@PBGO=@E':<PDC;hP:<DH;>GJ8DCGJAHGJ~Q:<Z>?%:<XCXHGPB; NB?%u)Ze:)XH?BPBGJ8Q?%QNBPBGJ=)EgGOXH;hDC; GO8Q?GJAHGJ8GJ~ c' Z!?0;>X]DC?I8=)GJAHES; XH=)?EA]GOEA]??DW; AC:<DH;>GJ88?BZ
π : AVor
4, n −→ Aparf
4, n;>8QI;hD9E:JAaZ>?BX9AH:<)8?B=)?B8VDHX9?B8VDHAH?kQNBPBGJ=)EgGJXC;hDC; GO8X c G<DCGJ8XE
Z ?kQ; K;>XH?BIAa?6PB?BE6DC; GO88?BZ+Q?π
pqI;?BX]DQ?RXHIEEgGOAD
DVor cLv 8@PBGO=kW;>8:<8VDZ ?RDHSNBGOA]\B=)? Oco pJZe:A]?B=f:<AHqI? Jc ?DZ ?YPBGJAHGJZ>Z>:J; AH? Jc>JQ?)o pGJ8E?BI6DY=)GJ8VDCA]?BAYqVI?ωa(−bDV or − cE)
?BX]D:J=@EZ>?XHIAAVor
4, nXH;?DYXH?BIZ>?B=)?B8LDXH;a > 12b
n
?Dc > 4b > 8
9c c :<AY?6?B=@EZ>?JpXH;
a > 36/n:JZ GOA]X
ωa(−3DVor − 13E)?BX]Da:<=)EZ ?XCIA AVor
4, n c 798Eg?BI6Da:<EEZ>; qI?BARZ>?DCSNBGJAH\B=@?Q?taGQ:<;>AH:)u[Ze:KJ:<AH; NDCNEA]GOEA]??DZ ;>XHXH?AVor
4, n × Spec(Fp)EgGOIAAH?BZ ?KL?BA?B8P:JAC:<PDCNBA];>X]DH;>qI?[8IZ>Z ?@Q?BXlGOA]=)?BXQ?f0; ?BUO?BZMQ?[UO?B8A]?4
pQ?@8;hKL?:<I> 36
pQ?EgGO; QX
> 5puRPBGV?fPB;>?B8LDCXQ:J8X
FppqI;OXBn:<88IZ>?B8LD!XHI)XC:J=@=)?B8VD!XHIA
DVor c noGJAHQA]?Qno:J88VIZe:DC; GO8AH?BqVI;>XY?BX]Da:<I=)GJ;>8X
2?DYGO8Eg?BI6DYZ>?QNDH?BAH=); 8?BAY?B8P:<Z>PBIZ>:J8VDY?6EZ ;>PB;hDC?B=@?B8VD
E c
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