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