Lecture 10 Relative deformationproblemNov 8

1 Relativedeformationproblem

Let F be anumberfieldS afinitesetofplacesof F includingthosedividing lis

Fix f GF s GLnff absolutelyirreducible

a lift X Gts a 0 ofdel fWritefu PIG i GE GWF Xv XlGpPILn CNLo groups

A to G MnMA MAYThenwehavenaturaltransformationsoffunctors

Deff Deff forevery vmis x'f x'IXp X'ftp.ln xp i XII PGTn

Want Xp ItsXpTechnicalissue fr isoftenreduciblee.gwhenfuis anramifiedTo reconcilethis issue consider

Definition For a subset TES containingplansdividing lo define

Defft CNLo Sets

Def'p's A P GFS Gln A cont reph sit pmodena pp hide PIHA delp X L

say CpChief f Chihet if I xettMnMAs t p xpx hi hi fret

fp Gts GWA contrep and h frT.aEPILnfAs.t.pmodMa p

p phuthi ftp.fpXIaj fRuheT GEsGlnlAIcont.rep'n

T promodma FuanddelpvXuCheck xpx thx 1 t.hn xpx xhit.huphia Rift is Rft REE

TCaveat Oursetupslightlydiffersfrommostotherliteratures as ourRiotcontainsfactorat a

Wewillessentiallyusethecasewhen TES lateror rather we'll compare thecase T S and S SuQyetTES

TheoremWe canwriteRft as a quotientRPoofx x Kfi frwhere f dim Ij Gss Adopt and r dimHJG Adf

Here werecallthat It isthecohomologyofrelativeGaloiscohomology

RTfG Adopt Cone Rl Gas Adopt RTGtuAdopt f I

Proof For t wecomputetherelativetangentspareher tangentpperx tangent ifRic

DXv Elaher Def'fh Fla Deff

Defp Fla ZIGEs Adf AdoptT

Adof d sothisAdf isembedded

µ I hue 1 si g Ag hthroughthemap d id D

pigs In Ageptg IS

pa a f t.fgt.Ag xigx.huwefygowmweg.y

ftp.DefpkfFkI If Z Gr Adflitheplopfithuet gunsAguthuguck

IP Gr n g gGE

her Deftp fFIeD feTTDefp fFkD

T sends fo Chu

kerf faffAdopt Ad

It Ife Adopt to Ceodchite

Ontheotherhand Itf Gfs Adf canbecomputedasthecolonofi i

Adop dyi GEGES AdF I ContGssAdf

Ot X to

rest Adof contGr Adfi T VETIIt isnothardto seethat

I Es Adopt her 16 Adf a EHGEAdoptAdof VET

Combiningthiswithabove we deducetheresultfordinTT'TGasAdoptLeavetheproofforIT toexercise

RecalltheEulercharacteristicformulas X Gts Adf EsX Gr Adoptandwhen we SIT vfiniteaufe X GEAdopt 0

IfGes Adopt 0Note IF GtsAdopt HoGts Adopts o TtfGts Adopt o bk

H Gas Adopt HGgAdopt IT GfsAdf H GasAdopt

a oH J H

H Gts Adopt H Gr Adopt HIGasAdopt H GtsAdopt

Corollary Assumethat f is absolutelyirreducible

KrulldimRp 3 KrulldimRio or dimoXp 3 IsdinoXp

Caveatagain OntheRHS v o is allowed forwhichdimoXp isnegative

2 Localdeformationconditions

Example v amonarchplane fu GalFu Gln F an unramified residualrepresentation

is Defp CNLo SetsDef r

CA m l s fp Galen GhCA lifting fPip andRft they are related as followspin Gale Gh RIE

l s Huniv

gThen p R

Agunir In foreverygCIfv

Viewit intermsof schemes SpecRp aSpecRp is a closed subscheme

Then wehave a Cartesiandiagram

SpfRtpEiHXgg SpfyRIEiHXuru

SpfRp7s.X pot

IDXv SpfRp

XvSpfRpi

UpshotWecan imposeanylocalconditions wewantthe re ationsareuniversalm