Lecture7 Framed deformations

October26

Let Ebe afiniteexthofQeE z O Of IF afiniteexthofIfe

Fix Ga profinitegroupsatisfyingthefollowingequivalentconditions

A forall finitelength0 modules M withcontinuous G action H GM isfinitefilthfor all openunpatsubgroupH cG HCH te isfinite ti

Example If K is a localfield allowingQebutnotFektD G GKIf F is anumberfieldand5 afinitesetofplaces G Gps

ConsiderCNLo i categoryofcompletenoetherianlocal 0 algebra A mforwhichthestructurehomomorphism 0 A inducesIsom F Ahn

Given f G Gln F GL V weconsiderallpossibleliftslike

p i G Gln O

gtyg HopeiGettheuniversalone sothatwe canget

allotherliftsfromtheuniversaloneConsiderthefunctorDefp CNLo Sets

contframeddefamation Ain l a P G GHA sitp modm pI

Defp CNlo i Setscont

usualdeformationAmy tip G Gln A sitpmodm pI

P p if 7 x c I Mnm xpCgsx p

ffreeA modulesMofrkn withcont AlinearGaction

togetherwith a G IsomMhm VTheorem Assumethat G is aprofinitegroupsatisfyingCA

a Deff is represented by someRIPECNLo

Tp I T pb If End CGI f F thenDeff is represented by someRpECMOMeaningofCa thereexists a universalrepn

puniv G GlnPipSt forany Am CCNLo everyrephP G Glen A lifting fcomesfrom aunigeringhomomorphism Rpf A

s t p G P Glenpip4 GWA

In otherwords Deff A HomomoPip AIor Def'f I HomaudRpe arethesamefunctor

Remade ApplyingthesetwofunctorstoRpf getDef Rtp HomaudRpfRpt

Ulpuniv id

b is similarexceptthat weget Gaitimon anREmodulefreeofranknProof a FirstassumethatG isfinite say G g gs r.Cgi gs i rtCg gp D

Then a rephofG assigning amatrixtoeachg satisfyingtherelationsDefine R Ofxih.gr i.j i n k 1 sJ

fr.fx xsj id rtfx xsl io

i e R parametrizesallpossiblerephsofG add detfxi.gl

Then F R IF is a homomorphismbutcouldalsoputthisinto

Xij Cijsthentryofpigsthesetofrelations

Let Ji Ker fClaim ThecompletionofRat J Ig representsthefunctorDeffProof pun G i Gln R GlnRF

8h I i high is agrouphomomorphismlifting f

ForCA m ECNLo anymorphismRI A in CNLo

gives an induedrepn G GlenRf Gln A lifting fConversely givenPAEDeffAthen J R is A is a Homo

AXijk I Cijithentryofpalgh

tape

As A iscomplete thishomomorphisminduces ahomomorphismpig A DIngeneral write G figGIHi overopennormalsubgpsHio G

wemayassumethatalltheseHi Eker pForeach i weget a universalpair Ri pi GlHi Gln Ri

GL Rimi

Butwhen Hj Hi therepn pi i GIHi Gln Ri induces

a reph GlHj 4th GlnRi

ByuniversalpropertyofRj is a uniquehomomorphismRj Ris t GlHj Pj GlenRj

teG Hi Pi GlnRi

Setpips pun k Ri pi

i ShowthatRpi belongstoCNLo needtoshowthatRpt isnoetherian

showthatPiprepresentsthefunctorDeftForCD asRpt is a completelocalringbydefn

it sufficestoshowthatmfhmpe2 co isfinite later

suffues p p fFor 2 weargueasfollows given A ECNLo

amorphismPip A clearlydefines a repn G Gln A liftingfConversely given a contrepn fa i G Gln A lifting fforeachk consider pack G IesGLn A Gln AlinkNotethat Afula is afinitering sopathfactorsthroughsome4hinByuniversalpropertyofRin 7 amorphismRin Afunk

Takinginverselimit over all k as allthesemorphismsmustbecompatible

Tff A Db LetPGTn denotethecompletionof PGIn Spec0 at t lspecThenPGTn outs on SpfRff byconjugationwhenEnd a f F thisaction isfreeso we can set specRp SpeePip willdiscussthis indetailsnext

time

Tangentspace Let FIE FIX1 62o IF I IFIe IF 0

Theneveryrepn p i G Gln Fla Gln Vcolifting fo V V leg V o y

meansF FEEnoffWe can see therelationbetweenDefp Fla and H G Adp

PropositionCDThere's a canonicalisomfmpymp.gs ftp.fpbfpf

FIT

The same istruefortheunframedversionifDeff is representable2 If Gsatisfies H thenDefpiffle3 Z G Adf

DefpCFlo HCG Adp

p pProof G Willonlyprovethefamedversion andtheunframedversion issimilar

Deff Fla Home.noR'p7 FId HomcmofRpEfmip.z.o sFk3

Home.ve rspadmFfmpe.so e

2 Provetheframedversionfirst givenP G Gln Fla lifting f

PG p h peggd Ag e pig

i Age pig I Ahe f h t Agh e fCgh

fth AgfIgfth e pigsAhplb.ee tAghpCgh e

Ag pigsAhpigs Agh

Aggo.GEZ G AdfWemayreverttheargumentto seethateverycouplein 2 GAdf defines aliftDefffitted Z GAdf

ForDefp Fla wepointoutthatif we conjugate by G ex E 1 teMn F1

pig ft Age ptg Hex i Age pig ite x

Ag pigs xpCg AgfIg peg x

Ag x pigsxpCgi'tAgSo theclassAggeo isdeferredby a coboundary

Deff Fla H GAdf