Group and Galois Cohomology

56 Pages·2016·0.313 MB·English

by  Romyar Sharifi

G G C
ROUP AND ALOIS OHOMOLOGY
Romyar Sharifi

Contents
Chapter1. Groupcohomology 5
1.1. Grouprings 5
1.2. Groupcohomologyviacochains 6
1.3. Groupcohomologyviaprojectiveresolutions 10
1.4. Homologyofgroups 12
1.5. Inducedmodules 13
1.6. Tatecohomology 15
1.7. Dimensionshifting 19
1.8. Comparingcohomologygroups 21
1.9. Cupproducts 30
1.10. Tatecohomologyofcyclicgroups 36
1.11. Cohomologicaltriviality 39
1.12. Tate’stheorem 42
Chapter2. Galoiscohomology 47
2.1. Profinitegroups 47
2.2. Cohomologyofprofinitegroups 53
3

CHAPTER 1
Group cohomology
1.1. Grouprings
LetGbeagroup.
DEFINITION 1.1.1. The group ring (or, more specifically, Z-group ring) Z[G] of a group G
consistsofthesetoffiniteformalsumsofgroupelementswithcoefficientsinZ
(cid:40) (cid:41)
∑ a g|a ∈Zforallg∈G, almostalla =0 .
g g g
g∈G
withadditiongivenbyadditionofcoefficientsandmultiplicationinducedbythegrouplawonG
andZ-linearity. (Here,“almostall”meansallbutfinitelymany.)
Inotherwords,theoperationsare
∑ a g+ ∑ b g= ∑(a +b )g
g g g g
g∈G g∈G g∈G
and
(cid:32) (cid:33)(cid:32) (cid:33)
∑ a g ∑ b g = ∑(∑ a b )g.
g g k k−1g
g∈G g∈G g∈G k∈G
REMARK 1.1.2. Intheabove,wemayreplaceZbyanyringR,resultingintheR-groupring
R[G]ofG. However,weshallneedhereonlythecasethatR=Z.
DEFINITION 1.1.3.
i. Theaugmentationmapisthehomomorphismε: Z[G]→Zgivenby
(cid:32) (cid:33)
ε ∑ a g = ∑ a .
g g
g∈G g∈G
ii. TheaugmentationidealI isthekerneloftheaugmentationmapε.
G
LEMMA 1.1.4. The augmentation ideal IG is equal to the ideal of Z[G] generated by the set
{g−1|g∈G}.
PROOF. Clearlyg−1∈kerε forallg∈G. Ontheotherhand,if∑g∈Gag =0,then
∑ a g= ∑ a (g−1).
g g
g∈G g∈G
(cid:3)
5
6 1.GROUPCOHOMOLOGY
DEFINITION 1.1.5. If G is a finite group, we then define the norm element of Z[G] by NG =
∑ g.
g∈G
REMARK 1.1.6.
a. Wemayspeak,ofcourse,ofmodulesoverthegroupringZ[G]. Wewillreferheretosuch
Z[G]-modulesmoresimplyasG-modules. TogiveaG-moduleisequivalenttogivinganabelian
group A together with a G-action on A that is compatible with the structure of A as an abelian
group,i.e.,amap
G×A→A, (g,a)(cid:55)→g·a
satisfyingthefollowingproperties:
(i) 1·a=aforalla∈A,
(ii) g ·(g ·a)=(g g )·aforalla∈Aandg ,g ∈G,and
1 2 1 2 1 2
(iii) g·(a +a )=g·a +g·a foralla ,a ∈Aandg∈G.
1 2 1 2 1 2
b. Ahomomorphismκ: A→BofG-modulesisjustahomomorphismofabeliangroupsthat
satisfies κ(ga)=gκ(a) for all a∈A and g∈G. The group of such homomorphisms is denoted
byHom (A,B).
Z[G]
DEFINITION 1.1.7. WesaythataG-moduleAisatrivialifg·a=aforallg∈Ganda∈A.
DEFINITION 1.1.8. LetAbeaG-module.
i. ThegroupofG-invariantsAG ofAisgivenby
AG ={a∈A|g·a=aforallg∈G,a∈A},
whichistosaythelargestsubmoduleofAfixedbyG.
ii. ThegroupofG-coinvariantsA ofAisgivenby
G
A =A/I A,
G G
whichistosay(notingLemma1.1.4)thelargestquotientofAfixedbyG.
EXAMPLE 1.1.9.
a. IfAisatrivialG-module,thenAG =AandA ∼=A.
G
b. OnehasZ[G] ∼=Z. WehaveZ[G]G =(N )ifGisfiniteandZ[G]G =(0)otherwise.
G G
1.2. Groupcohomologyviacochains
ThesimplestwaytodefinetheithcohomologygroupHi(G,A)ofagroupGwithcoefficients
in a G-module A would be to let Hi(G,A) be the ith derived functor on A of the functor of G-
invariants. However,notwishingtoassumehomologicalalgebraatthispoint,wetakeadifferent
tack.
DEFINITION 1.2.1. LetAbeaG-module,andleti≥0.
i. Thegroupofi-cochainsofGwithcoefficientsinAisthesetoffunctionsfromGi toA:
Ci(G,A)={f : Gi →A}.
1.2.GROUPCOHOMOLOGYVIACOCHAINS 7
ii. Theithdifferentialdi =di : Ci(G,A)→Ci+1(G,A)isthemap
A
di(f)(g ,g ,...,g)=g · f(g ,...g)
0 1 i 0 1 i
i
+ ∑(−1)jf(g ,...,g ,g g ,g ,...,g)+(−1)i+1f(g ,...,g ).
0 j−2 j−1 j j+1 i 0 i−1
j=1
WewillcontinuetoletAdenoteaG-modulethroughoutthesection. WeremarkthatC0(G,A)
istakensimplytobeA,asG0 isasingletonset. Theproofofthefollowingislefttothereader.
LEMMA 1.2.2. Foranyi≥0,onehasdi+1◦di =0.
REMARK 1.2.3. Lemma1.2.2showsthatC·(G,A)=(Ci(G,A),di)isacochaincomplex.
WeconsiderthecohomologygroupsofC·(G,A).
DEFINITION 1.2.4. Leti≥0.
i. WesetZi(G,A)=kerdi,thegroupofi-cocyclesofGwithcoefficientsinA.
ii. We set B0(G,A)=0 and Bi(G,A)=imdi−1 for i≥1. We refer to Bi(G,A) as the group
ofi-coboundariesofGwithcoefficientsinA.
We remark that, since di◦di−1 =0 for all i≥1, we have Bi(G,A)⊆Zi(G,A) for all i≥0.
Hence,wemaymakethefollowingdefinition.
DEFINITION 1.2.5. WedefinetheithcohomologygroupofGwithcoefficientsinAtobe
Hi(G,A)=Zi(G,A)/Bi(G,A).
ThecohomologygroupsmeasurehowfarthecochaincomplexC·(G,A)isfrombeingexact.
Wegivesomeexamplesofcohomologygroupsinlowdegree.
LEMMA 1.2.6.
a. ThegroupH0(G,A)isequaltoAG,thegroupofG-invariantsofA.
b. Wehave
Z1(G,A)={f : G→A| f(gh)=gf(h)+ f(g)forallg,h∈G}
and B1(G,A) is the subgroup of f : G→A for which there exists a∈A such that f(g)=ga−a
forallg∈G.
c. IfAisatrivialG-module,thenH1(G,A)=Hom(G,A).
PROOF. Let a∈A. Then d0(a)(g)=ga−a for g∈G, so kerd0 =AG. That proves part a,
and part b is simply a rewriting of the definitions. Part c follows immediately, as the definition
ofZ1(G,A)reducestoHom(G,A),andB1(G,A)isclearly(0),inthiscase. (cid:3)
Weremarkthat,asAisabelian,wehaveHom(G,A)=Hom(Gab,A),whereGab isthemaxi-
malabelianquotientofG(i.e.,itsabelianization).
One of the most important uses of cohomology is that it converts short exact sequences of
G-modules to long exact sequences of abelian groups. For this, in homological language, we
needthefactthatCi(G,A)providesanexactfunctorinthemoduleA.
8 1.GROUPCOHOMOLOGY
LEMMA 1.2.7. If α: A→B is a G-module homomorphism, then for each i≥0, there is an
inducedhomomorphismofgroups
αi: Ci(G,A)→Ci(G,B)
taking f toα◦ f andcompatiblewiththedifferentialsinthesensethat
di ◦αi =αi+1◦di.
B A
PROOF. Weneedonlycheckthecompatibility. Forthis,notethat
di(α◦ f)(g ,g ,...,g)=g α◦ f(g ,...g)
0 1 i 0 1 i
i
+∑(−1)jα◦ f(g ,...,g ,g g ,g ,...,g)+(−1)i+1α◦ f(g ,...,g )
0 j−2 j−1 j j+1 i 0 i−1
j=i
=α(di(f)(g ,g ,...,g)),
0 1 i
asα isaG-modulehomomorphism(thefactofwhichweuseonlytodealwiththefirstterm). (cid:3)
In other words, α induces a morphism of complexes α·: C·(G,A)→C·(G,B). As a conse-
quence,oneseeseasilythefollowing
NOTATION 1.2.8. Ifnothelpfulforclarity,wewillomitthesuperscriptsfromthenotationin
the morphisms of cochain complexes. Similarly, we will consistently omit them in the resulting
mapsoncohomology,describedbelow.
COROLLARY 1.2.9. AG-modulehomomorphismα: A→Binducesmaps
α∗: Hi(G,A)→Hi(G,B)
oncohomology.
Thekeyfactthatweneedaboutthemorphismoncochaincomplexesisthefollowing.
LEMMA 1.2.10. Supposethat
ι π
0→A→− B−→C→0
isashortexactsequenceofG-modules. Thentheresultingsequence
0→Ci(G,A)→−ι Ci(G,B)−→π Ci(G,C)→0
isexact.
PROOF. Let f ∈Ci(G,A), and suppose ι◦ f =0. As ι is injective, this clearly implies that
f =0, so the map ιi is injective. As π◦ι =0, the same is true for the maps on cochains. Next,
supposethat f(cid:48)∈Ci(G,B)issuchthatπ◦ f(cid:48)=0. Define f ∈Ci(G,A)byletting f(g ,...,g)∈A
1 i
betheuniqueelementsuchthat
ι(f(g ,...,g))= f(cid:48)(g ,...,g),
1 i 1 i
which we can do since imι = kerπ. Thus, imιi = kerπi. Finally, let f(cid:48)(cid:48) ∈Ci(G,C). As π is
surjective,wemaydefine f(cid:48) ∈Ci(G,B)bytaking f(cid:48)(g ,...,g)tobeanyelementwith
1 i
π(f(cid:48)(g ,...,g))= f(cid:48)(cid:48)(g ,...,g).
1 i 1 i
1.2.GROUPCOHOMOLOGYVIACOCHAINS 9
Wethereforehavethatπi issurjective. (cid:3)
Wenowprovethemaintheoremofthesection.
THEOREM 1.2.11. Supposethat
ι π
0→A→− B−→C→0
isashortexactsequenceofG-modules. Thenthereisalongexactsequenceofabeliangroups
0→H0(G,A)−→ι∗ H0(G,B)−π→∗ H0(G,C)−δ→0 H1(G,A)→···.
Moreover,thisconstructionisnaturalintheshortexactsequenceinthesensethatanymorphism
(cid:47)(cid:47) ι (cid:47)(cid:47) π (cid:47)(cid:47) (cid:47)(cid:47)
0 A B C 0
α β γ
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
0 (cid:47)(cid:47) A(cid:48) ι(cid:48) (cid:47)(cid:47) B(cid:48) π(cid:48) (cid:47)(cid:47)C(cid:48) (cid:47)(cid:47) 0,
givesrisetoamorphismoflongexactsequences,andinparticular,acommutativediagram
··· (cid:47)(cid:47) Hi(G,A) ι∗ (cid:47)(cid:47) Hi(G,B) π∗ (cid:47)(cid:47) Hi(G,C) δi (cid:47)(cid:47) Hi+1(G,A) (cid:47)(cid:47) ···
(cid:15)(cid:15) α∗ (cid:15)(cid:15) β∗ (cid:15)(cid:15) γ∗ (cid:15)(cid:15) α∗
··· (cid:47)(cid:47) Hi(G,A(cid:48)) (ι(cid:48))∗ (cid:47)(cid:47) Hi(G,B(cid:48)) (π(cid:48))∗(cid:47)(cid:47) Hi(G,C(cid:48)) δi (cid:47)(cid:47) Hi+1(G,A(cid:48)) (cid:47)(cid:47) ···.
PROOF. Firstconsiderthediagrams
0 (cid:47)(cid:47)Cj(G,A) ι (cid:47)(cid:47)Cj(G,B) π (cid:47)(cid:47)Cj(G,C) (cid:47)(cid:47) 0
dj dj dj
(cid:15)(cid:15) A (cid:15)(cid:15) B (cid:15)(cid:15) C
0 (cid:47)(cid:47)Cj+1(G,A) ι (cid:47)(cid:47)Cj+1(G,B) π (cid:47)(cid:47)Cj+1(G,C) (cid:47)(cid:47) 0
for j≥0. NotingLemma1.2.10,theexactsequencesofcokernels(for j=i−1)andkernels(for
j=i+1)canbeplacedinaseconddiagram
Ci(G,A) ι (cid:47)(cid:47) Ci(G,B) π (cid:47)(cid:47) Ci(G,C) (cid:47)(cid:47)
0
Bi(G,A) Bi(G,B) Bi(G,C)
di di di
(cid:15)(cid:15) A (cid:15)(cid:15) B (cid:15)(cid:15) C
0 (cid:47)(cid:47) Zi+1(G,A) ι (cid:47)(cid:47) Zi+1(G,B) π (cid:47)(cid:47) Zi+1(G,C)
(recalling that B0(G,A) = 0 for the case i = 0), and the snake lemma now provides the exact
sequence
Hi(G,A)−α→∗ Hi(G,B)−β→∗ Hi(G,C)−δ→i Hi+1(G,A)−α→∗ Hi+1(G,B)−β→∗ Hi+1(G,C).
Splicingthesetogethergivesthelongexactsequenceincohomology,exactnessof
0→H0(G,A)→H0(G,B)
10 1.GROUPCOHOMOLOGY
beingobvious. Weleavenaturalityofthelongexactsequenceasanexercise. (cid:3)
REMARK1.2.12. Themapsδi: Hi(G,C)→Hi+1(G,A)definedintheproofoftheorem1.2.11
areknownasconnectinghomomorphisms. Again,wewilloftenomitsuperscriptsandsimplyre-
fertoδ.
REMARK 1.2.13. A sequence of functors that take short exact sequences to long exact se-
quences(i.e.,whichalsogiverisetoconnectinghomomorphisms)andisnaturalinthesensethat
every morphism of short exact sequences gives rise to a morphism of long exact sequences is
known as a δ-functor. Group cohomology forms a (cohomological) δ-functor that is universal
inasenseweomitadiscussionofhere.
1.3. Groupcohomologyviaprojectiveresolutions
Inthissection,weassumeabitofhomologicalalgebra,andredefinetheG-cohomologyofA
intermsofprojectiveresolutions.
For i ≥ 0, let Gi+1 denote the direct product of i+1 copies of G. We view Z[Gi+1] as a
G-moduleviatheleftaction
g·(g ,g ,...,g)=(gg ,gg ,...,gg).
0 1 i 0 1 i
Wefirstintroducethestandardresolution.
DEFINITION1.3.1. The(augmented)standardresolutionofZbyG-modulesisthesequence
ofG-modulehomomorphisms
···→Z[Gi+1]−→di Z[Gi]→···→Z[G]→−ε Z,
where
i
d(g ,...,g)= ∑(−1)j(g ,...,g ,g ,...,g)
i 0 i 0 j−1 j+1 i
j=0
foreachi≥1,andε istheaugmentationmap.
At times, we may use (g ,...,g ,...,g)∈Gi to denote the i-tuple excluding g . To see that
0 (cid:98)j i j
thisdefinitionisactuallyreasonable,weneedthefollowinglemma.
PROPOSITION 1.3.2. Theaugmentedstandardresolutionisexact.
PROOF. Inthisproof,taked0 =ε. Foreachi≥0,compute
i+1i+1
d ◦d (g ,...,g )= ∑ ∑(−1)j+k−s(j,k)(g ,...,g ,...,g ,...,g ),
i i+1 0 i+1 0 (cid:98)j (cid:98)k i+1
j=0k=0
k(cid:54)=j
where s(j,k) is 0 if k < j and 1 if k > j. Each possible (i−1)-tuple appears twice in the sum,
withoppositesign. Therefore,wehaved ◦d =0.
i i+1
Next,defineθ : Z[Gi]→Z[Gi+1]by
i
θ(g ,...,g)=(1,g ,...,g).
i 1 i 1 i