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Lecture 4 Local dualityforGaloiscohomologyandEulercharacteristicOctober 12
1 LocaldualityforGaloiscohomologyLet k be a localfield i.e afiniteextensionofQporFpKtDLet E zOe Defoe he Ferbethecoefficientrings s t l charK
so when K Qp l p is wed
Let M be a finitelengthOemodulewith continuous GKcutin
Recall I Ik Gk Gk I
o H Gkn MIK H GK M H LIK MGkk H2GmM11 11 Hq
Hur K M unramified thing K M bioGHEEpart singular hascohomdim1
partI 0 H'urGKM H GK M HingGK M 0
Note when K Qpand l p this isnot agoodsetup
Pontryagindual for N anOemodule N i HomonfN Eloe
theorem Localduality Let M be afinitelengthOEmodulewith cont Gkcutino For i o n 2 Hi Gr M is afin lengthOEmoduleand H GK M o
1 For i o 1,2there is a canonical functorialperfectpairingHi GK M x tf GK M G H Gk Eton EKESo Hi GK M tf GK M as
2 whenftp underthelocalduality theexactsequence E isselfdual
o Hh Gk M H Gen thingGKM o
Ils Ils Ils
o thingGeMED H okMHD HbrGrMID o
gls Eulercharacteristic
X Ge M TECDilengthoefHikemy0 if HpLkQp length M if l p andHop
Bernard when M is afinitedim l E vectorspare thetheoremholds if wereplaceEGE no E lengthy a dime so M i HomeM E
Example Unramifiedcomputation warm upexercise
AssumethattheGKaihin on M fatusthroughGkk and l pThen in H Hw Ge M H GkK MII
Mhere 1 MFrida
HsingGKM H IK tf ago.HongfIkMfk Hom Iea MGk MEDAK
So the isomorphism HierGkMt
H'sing GKMHD isgivenby
here Dm MYEE
M Frs M MKtrk DM 0
dualo qyxyfrr.it M t M
Exercise explainbyhandwhatHo tf meanswhenM is anramified
2Proofoflocaldualitytheorem
Inputs thatwewillnotprove
HiCox kneeKO
HilbertgoQ z i 2
o i33
when4k is afiniteextension H2 L ftp.x CoressH2fkksep.xy
By Kummertheory I pen K'EP pKseP y
HYGi.fm
aHYGk.pen sH4GrkseM sH4Ge.knM
HS HS
te Q z
H Gk.pe etn747L Hfp3Ge Eoea oTakinglimit OE H GKHoek Eloe
Step Provetheduality Hs HilGe M tfifGK.MN i o.i.z
andthevanishing H GKM o a finitenessofHi GemNote For an exactsequence 0 M M Ma o Bydevissage
as for Mi Ma M byfirelemmaSomayassumethat M is a he rectuspaceSpecial case K Kline and M ke kECDwithtrivial6kaction
reduceto M e feHolck Fe H GkFelis H Gr Felis FeH GK Fe H Game
11 lls KummertheoryHom Ge Ee K e i ye KM Kup'sgp11 lls K YK
Horn bite Hom ti Ite Tgp H'for.pe HGekM
isjustlocaldassfieldtheory0
Also H GK fee o by LEFT
Forthegeneralcase weneed an exercise for4K afiniteexthand Na G module
thefollowingdiagramcommutes
Journey agHi L N x H L N as H2 L EtonIs Shapirolemma IS fISCores
Hi K IndEiN x tf K Ind NCn Is HCK EtonNowgiven ageneral k 6KImodule M thereexists a finite exth LIK
s t G artstrivially on M L L pieConsiderthe tautologicalexactsequence o M IndEEMla Q so
o HoGK M HoKokInd M HoGKQ H GK M s
HolaM
tho maD H4G dEEma Hyo DEH'la maDHTam
is injective trueforAI MSo is injective
is injective trueforall M byfivelemma
Duality we cantake o I ind Mla a M o
HConMf HKok I si thGeind M H2GK M H Gr103 tolls to
HKokMEI HCG IE sH4Ge.indEEmEiD aHoCGK.MaD o
is injective atrueforall Mis surjective
is surjective trueforall MFromthis we deduceHi GKM Hai GKMK
vanishingH GK M 0 andfinitenessbydimensionshifting D
StepI Euler characteristicformulaNote Bydevissage wemayassumethat M is a kevectorspace or even anFe vectorspace
I g l I e lArtin'sTheoremLet Gbe afinitegroup Write R e G fortheGrothendieckgroupofcategoryoffinitedinil representationsofFEIG i e
R lot to a lb ft Ef Muttu'sthed VEIrrepfeeks VERefdineifo v t
Foreachsubgroup Hof G we have
Indf Rite H R e Gw l i findfew
theoremftp.RpfH
xOlQIndtF RiFeG is surjective
et HThis is Artin'stheorem
sketch R H Q R G QHaG e t 2 vHcyclicijss
RACH Q DR CG QHaG e lHcyclic
MoreoverthecontributiontoIofthosecharacterscomingfromegroupsHare trivial b c theonlyirreducibleEerephofsuchgroup isthetriumph
thusIIndftfef ITftndtffteh.IM ttTfIndh9FeI
Now Gk Gyk.GGeM WLOG HEEL
L casewhen M Indi.TN Hiscyclicoforderprime lol
c1 H
reducetehNow Hike M Hi GEN
F H HI Hila Nta Gye NHK bioHOCH Felt mod o
Now HoGaye p withthenaturalH action
reHYG.pe Fewiththetrivial H action
H G pie LY geI Oilcoege LYCL.ge 74ez io
suppose OE tyklxQ peklxop.CH
Then oiycoiye.myexflilifltpusme't OFkpHifl p
as Hmodule
Then X Gie M D.inefHoCGipexONfDHdimpefH'CG.ye xoNCiDHtdimpefH4G.ye NGD
H H Hdimeµe NtD
d.imefNytDto uexoNtHttOffFplHIxoNtiDHxoOpI dimpefNCDT p p acancels
ears if
f 0 if lipcancels
FQp dimp FpfH7
Nc.is HiflpFFaIiForanyfinitegrapH anyfinitedim'lrepvof.tt
HICH V FpfHJdiN
as Hrep'm
F p dimN CKQp dinM
Infact our assumption is harmless
zpQp OFfHIzpQp istrue
UI U
Yuk OHH So as Hmodules
0Eµk andOffit havethesamesemisimplification
Steps whenftp.showthatunderlocalduality we have
0 HI Gem H Gem thing Gen o
angIls Ils Ils
o thingGeMED HkaMMD HbrGrMID o
Leave as an exercise