Cohomology of Groups

Jun.-Prof. Dr. Caroline Lassueur TU Kaiserslautern

Lecture Notes, SS 2018

(4SWS Lecture + 2SWS Exercises)

Version: 20th of July 2018 Contents

Foreword 6

Chapter 1. Background Material: Group Theory7

1 Semi-direct Products...... 7 Chapter2 2.Presentations Background of Material: Groups...... Module Theory 1017

3 Modules, Submodules, Morphisms...... 17 4 Direct Products and Direct Sums...... 20 5 Exact Sequences...... 22 6 Free, Injective and Projective Modules...... 25 Chapter7 3.Tensor Homological Products...... Algebra 2731

8 Chain and Cochain Complexes...... 31 9 Projective Resolutions...... 37 Chapter10 4.Ext Cohomologyand Tor ...... of Groups 4841

11 Modules over the Group Algebra...... 48 12 (Co)homology of Groups...... 51 13 The Bar Resolution...... 52 Chapter14 5.Cocycles Easy Cohomology and Coboundaries ...... 5657

15 Low-degree Cohomology...... 57 Chapter16 6.Cohomology Cohomology of Cyclic and Group Groups Extensions...... 5962 H

17 GroupH1 Extensions...... 62 18 2 and Group Extensions...... 65 19 and Group Extensions...... 69

i Lecture Notes: Cohomology of Groups SS 2018

ii Chapter 7. Subgroups and Cohomology 74

20 Restriction in Cohomology...... 74 21 Transfer in Cohomology...... 75 Chapter22 8.Induction Finite Groupsand Coinduction in Cohomology...... 7780

23 Cohomology of Finite Groups...... 80 24 The Theorems of Schur and Zassenhaus...... 81 Chapter25 8.Burnside’s The Schur Transfer Multiplier Theorem ...... 8386

1 26 The Schurp Multiplier...... 86 27 The Projective Lifting Property and Universal Central Extensions...... 87 Appendix:28 Universal The Language-Central of Category Extensions Theory...... 8788

A Categories...... 88 BibliographyB Functors...... 9093 Index of Notation 94

Index 97 Foreword

Cohomology of Groups

This text constitutes a faithful transcript of the lecture held at the TU Kaiser- slautern during the Summer Semester 2018 (14 Weeks, 4SWS). ‚ group theory Together with the necessary theoretical foundations the main aims of this lecture are to: ‚ homological algebra cohomology theory provide students with a modern approach to ; ‚ universal properties language of category learn about and a specific ; theory consistently work with and get acquainted with the ‚ :

establish connections between the cohomology of groups and the theory of central extensions of groups as developed by Schur at the beginning of the 1900’s. Grundlagen der Mathematik Algebraische Strukturen WeEinführung assume in as die pre-requisites Algebra bachelor-levelKommutative Algebraalgebra courses dealing with linear algebra and elemen- tary group theory, such as the standard lectures , , , and at the TU Kaiserslautern. In order to complement these pre-requisites, the first chapter will deal formally with more advanced background material on group theory, namely semi-direct products and presentation of groups, while the second chapter will provide a short introduction to the theory of modules, where we will emphasise in particular definitions using universal properties but omit proofs.

I am grateful to Jacques Thévenaz who provided me with his lecture "Groupes & Cohomologie" (14 weeks, 2SWS) hold at the EPFL in the Autumn Semester 2011, which I used as a basis for the development of this text, and I am grateful to Rafaël Gugliellmetti who provided me with the .tex files of his lecture notes from 2011.

Finally, I am also grateful to the students who mentioned typos in the preliminary versions of these notes. Further comments, corrections and suggestions are of course more than welcome.

Kaiserslautern, 14th July 2018

6 Chapter 1. Background Material: Group Theory

semi-direct products presentations of groups The1st aim of this2nd chapter cohomology is to introduce group formally two constructions of the theory of groups: and . Later on in the lecture we will relate semi-direct products with a and a . Presentations describe groups by generators and relations in a References:concise way, they will be useful when considering concrete groups, for instance in examples. A course in group theory

[Hum96] J. F. HumphreysPresentations, of groups , Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. [Joh90] D. L. Johnson, , London Mathematical Society Student Texts, vol. 15, Cambridge University Press, Cambridge, 1990. 1 Semi-direct Products

The semi-direct product is a construction of the theory of groups, which allows us to build new groups fromDefinition old ones. 1.1 (ItSemi-direct is a natural product generalisation) of the direct product. G semi-direct product N IJ G H ď G A group is said to be the (internal or inner) of a normal subgroup G “ NH by a subgroup if the following conditions hold: N X H “ t u (a) ; Notation: G “ N ¸ H (b) 1 . Example 1 . G ˆ G N “ G ˆ t u H “ t u ˆ G 1 2 1 (1) A direct product of two groups is the semi-direct product of : 1 by G “ S 2 N “ C “ xp qy IJ S H “ C “ xp qy ď S : 1S – C . ¸ C 3 3 3 2 3 (2) is the semi-direct product of 1 2 3 and 1 2 . 3 3 2 Hence . Notice that, in particular, a semi-direct product of an abelian subgroup by an abelian subgroup need not be abelian.

7 Lecture Notes: Cohomology of Groups SS 2018

8 G “ Sn n ě N “ An IJ Sn H “ C “ xp qy

2 Remark(3) 1.2More generally ( 3) is a semi-direct product of by 1 2 .

G N H

(a) If is a semi-direct productG{N of“ HNby{N,– thenH{ theH X 2ndN “ IsomorphismH{t u – H Theorem yields

1

and this gives rise to a short exactÝÑ sequenceN ÝÑ G ÝÑ H ÝÑ :

1N H 1 extension of N by H G “ N ¸ H N H H N Hence a semi-direct product of by is a special case of an . @h P H (b) In a semi-direct product of by , the subgroup acts by conjugation on , namely , θh N ÝÑ N n ÞÑ hnh´

: 1 1 N θhh1 “ θh ˝ θh1 , h; h P H

is an automorphism of . In addition for every , so that we have a group homomorphism θ H ÝÑ pNq h ÞÑ θh : Aut . Proposition 1.3 N;H θ G

ProofWith : Step the 1 above notation,g P G and are sufficient to reconstructg “ nh then groupP N h lawP H on . g “ nh “ n1h1 n; n1 P N h; h1 P H . Each can be written in a unique way as where , : n´ n1 “ hph1q´ P N X H “ t u ; indeed by (a) and (b) of the Definition, if with , , then 1 1 n “ n1 h “ h1 1 Step 2 g “ n h ; g “ n h P G n ; n P N h ; h P H hence and . 1 1 1 2 2 2 ´ 1 2 1 2 . Group law: Letg g “ n h n h “ n h n phwithh qh “ rn θ,h pn qs ¨ rh ash s above.: Then 1 θh pn q 1 1 2 1 1 2 2 1 1 2 1 1 2 1 2 1 2

loooomoooon1 2

With the construction of the group law in the latter proof in mind, we now consider the problem of constructingProposition 1.4 an "external" (or outer) semi-direct product of groups.

N H θ H ÝÑ pNq; h ÞÑ θh G “ N ˆ H Let and be two arbitrary groups, and let : Aut be a group homomorphism. Set : as a set. Then the binary operation Lecture Notes: Cohomology of Groups SS 2018

¨ G ˆ G ÝÑ G 9 pn ; h q; pn ; h q ÞÑ pn ; h q ¨ pn ; h q “ pn θh pn q; h h q

: 1 ` 1 G 1 2 2 ˘ 1 G1 “ p 2N ; 2H q 1 2 1 2pn; hq P N ˆ H ´ ´ ´ : pn; hq “ pθh´ pn q; h q defines a1 groupG law1 on1 . The1 neutral element is 1 N “1 N1ˆ tandu – theN inverseH “ of t u ˆ H – H is . 0 0 ProofFurthermore : is an internal semi-direct product of : 1 by : 1 .

DefinitionExercise 1.5 1, Exercice Sheet 1. G external (or outer) semi-direct product N H θ G “ N ¸θ H In the context of Proposition 1.3 we say that is the of Exampleby w.r.t. 2 , and we write .

Here areH a few examples ofN very intuitiveθh “ semi-directN @ h P H productsN ¸ ofθ groups,H “ N whichˆ H you have very prob- ably already encountered in other lectures, without knowing that they were semi-direct products: K (1) If acts trivially on (i.e. Id ), then . ˆ npK q “ npK q ¸ pλ; ;:::; q P npK q | λ P K ; (2) Let be a field. Then pλ; ;:::; q ( λ; ;:::; GL SL diag 1 1 GL K where diag 1 1 is the diagonal matrix with (ordered) diagonal entries 1 1. ˚ ˚ (3) Let be a fieldB “ and let P npK q p “ q; #˜ ˚ ¸ + . .. ˚ : 0 GL upper triangular matrices U “ P npK q p “ q; #˜ 1 ¸ + . λ .. : 0 1 GL upper unitriangular matrices T “ P npK q p “ q: 1 0 #˜ λn ¸ + . B .. T U : 0 GL diagonal matrices

Cm “ xgy Cn “ xhy m; n P Zě Then is a semi-directk productP Z of by kn. ” p mq 1 (4) Let and ( ) be finite cyclic groups. θ Cn ÝÑ pCmq Assume moreover that is such thati 1 mod i and set h ÞÑ pθhq : Aut θ C ÝÑ C ; g ÞÑ gk h m m , n n´ k n´ k kn pθhq pgq “ pθhq pg q “ pθhq pg q “ ::: “ g “ g where : . Then 2 n 1 n 2 opgq “ m k ” p mq pθhq “ Cm θ Cm Cn θ since and 1 mod . Thus Id and is a group homomorphism. It follows that under these hypotheses there exists a semi-direct product of by w.r.t. to . m ě n “ k “ ´ D m m ´ ´ g m h θhpgq “ hgh “ g 2 Particular case: 1, 2 and 1 yield the dihedral group of1 order1 2 with generators (of order ) and (of order 2) and the relation . Lecture Notes: Cohomology of Groups SS 2018

10

The details of Examples (1)-(4) will be discussed during the Präsensübung on Wednesday, 11th of April.

2 Presentations of Groups

Idea:

p q C “ xgy “ xg | gm “ y g describe a group usingm a set of generators and a set of relations between these generators. gm “

Examples: p1q D m “ Cm ¸θ C 1 q 1 generator: g; h 1 relation: gm “1 ; h “ ; hgh´ “ g´ 2 2 p2q Z “ x Zy (see Ex.2(4) 2 generators: Z 2 1 1 3 relations: ù 1 1 3 1 1 generator: 1 no relation ( "free group")

ToDefinition begin with 2.1 we (Free examine group free / Universal groups and property generators. of free groups) X free group of basis X free group on X F X G Letf X ÝÑbe aG set. A (or f )F isÝÑ a groupG containingf|X “ f as a subset and satisfying the following universal property: For any group and for any (set-theoretic) map ˜ ˜ : , there exists a unique group homomorphismf : such that , or in other words such that the following diagram commutes:X G ö i “inc D f f|X “f˝i“f : F ! ˜ s.t. ˜ ˜ |X| rank F

PropositionMoreover, 2.2 is called the of . F F X

1 ProofIf : exists,F then is the unique free groupX of basis up to a unique isomorphism. i X ãÑ F X F i1 X ãÑ F 1 X 1 FAssume is another free group of basis . i1 LetX : F 1 be the canonical inclusion of in and let : be the canonical inclusion of D i1 1 1 1 1 in i . i F ÝÑ F i “ i ˝ i 1 1 D i i F ÝÑ F i “ i ˝ i F ! ˜ By the universal property of Definition 2.1, there exists: - a unique group homomorphism ˜ : s.t. ˜ ; and i !˜ 1 X F - a uniquepi˝i q|X group“ i homomorphism ˜ : Fs.t.|X “ i˜ . F 1 i i ˝ i “ F i i1 ˜ ˜ F Id ˝ Then , but obviously we also have Id . Therefore, by uniqueness, ˜ ˜ we have 1 Id . 1 ˜ ˜ i ˝ i “ F 1 F F i i1 A similar argument yields ˜ ˜ Id , hence and are isomorphic, up to a unique isomorphism, namely ˜ with inverse ˜. Lecture Notes: Cohomology of Groups SS 2018

Proposition 2.3 11 F X X F

ProofIf : is aH free“ x groupXy of basis , thenF generates X. jH “ X ãÑ H X H jH Let : jH ˝ i “bejH the subgroup of generated by , and let : denote the canonical inclusion of in . By the universal property of Definition 2.1, there exists a unique group homomorphism jH r such that r : X H ö i D jH F ! r κ H ãÑ F H F

Therefore, letting : denote the canonical inclusion of in , we have the following commutative jH κ diagram: X H F jH i F r F Id κ˝jH r κ ˝ jH “ F jH H ÝÑ F

F “ p q “ pκ ˝ j q “ pj q Ď H Thus by uniqueness r Id , implyingF that r : H is injective.H Thus F “ H Im Id Im r Im r

Theoremand it 2.4follows that . The claim follows. X F X

ProofFor : anyX set“ tx,α there| α P existsIu a freeI group with basis X. Y “ tyα | α P Iu X X X X Y “H Z “ X Y Y 8 n Set : E “ n“whereZ is a setZ in“ bijectiontp qu with , set :Z “ Z Z “ inZ ˆ bijectionZ ::: with E but disjoint from , i.e.Ť , and0 let : . 1 2 Furthermore, set : 0 , where : (i.e. a singleton), : , : , pz ; : : : ; z q ¨ pz1 ; : : : ; z1 q “ pz ; : : : ; z ; z1 ; : : : ; z1 q : Then becomes a monoid for the concatenationn m of sequences, thatn is n PZ n PZ m PZ n`m 1 1 1 1 looooomooooon looooomooooon : loooooooooooomoooooooooooon ¨ p q P Z

Elementary Operations E 0

The law is clearly associative by definition,pz ; : : : ; z andnq the neutral element is thexα empty; yα sequence . Define the followingpz ; : : : ; zk ; xα ; yα ; zk` ; : : : ;on znq the elements of : 1 Type (1): add in a sequence pz ; : : : ; znq two consecutive elements yα ; xα and obtain 1 1 pz ; : : : ; zm; yα ; xα ; zm` ; : : : ; znq 1 Type (1bis): add in a sequence pz ; : : :two ; znq consecutive elements xαand; yα obtain 1 1 pz ; : : : ; zr; xα ; yα ; zr` ; : : : ; znq 1 Type (2): remove from a sequence pz ; : : : ; znq two consecutive elements yα ; xα and obtain 1 1 pz ; : : : ; zs; ˇyα ;ˇxα ; zs` ; : : : ; znq 1 Type (2bis): remove from a sequence„ E two consecutive elements and obtain 1 1 ˇ ˇ Now define an equivalenceE relation on ðñas follows:

two sequences in are equivalent : the 2nd sequence can be obtain from the 1st sequence through a succession of Elementary Operations of type (1), (1bis), (2) and (2bis). It is indeed easily checked that this relation is: Lecture Notes: Cohomology of Groups SS 2018

12

– reflexive: simply use an empty sequence of Elementary Operations; – symmetric: since each Elementary Operation is invertible; – transitive:F “ sinceE{„ 2 consecutiverz sequences; : : : ; zns of Elementary Operations isp againz ; : : : a; z sequencenq F “ ofE Elementary{„ Operations. Claim 1: 1 E F 1 Now set : , and write for the1 equivalence1 class of 1 1 in . F rz ; : : : ; zns ¨ rz ; : : : ; zms “ rz ; : : : ; zn; z ; : : : ; zms 1 1 1 1 The above monoidpz ; :law : : ; zonnq „induces pt ; : : : ; atk q monoidpz law; : : on: ; zmq. „ pt ; : : : ; tl q 1 1 The induced law on is: 1 1 . 1 11 1 1 1 Itp isz ; well-defined: : : : ; znq ¨ pz ; : if : : ; zmq “ pz ; : : : ; zn; z ; : : : ;and zmq 1 1 , then 1 1 „ pt ; : : : ; tk ; z ; : : : ; zmq 1 1 1 1 1 1 „ pt ; : : : ; tk ; t ; : : : ; tl q 1 1 1 1 “ pt ; : : : ; tnq ¨ pt ; : : : ; t q via Elementary Operations on the 1st part 1 m 1 via Elementary Operations on the 2nd part 1 1 rp qs Claim 2: F The associativity is clear, and the neutral element is . The claim follows. rz ; : : : ; zns P F pz ; : : : ; zendowednq with the monoid law defined in Claimx 1α is a groupyα . yα xα 1 Inverses: the inverse„ of is the equivalence of the sequence class obtained from 1 Claim 3: F by reversing theX order and replacing each with and each with . (Obvious by definitionG of .) f X ÝÑ G is a free group on f . E ÝÑ G Let be a group and : be a map. Define pz ; : : : ; znq ÞÑ fpz q ¨ ¨ ¨ ¨ ¨ fpznq

p: ´ f Y fpyα q 1“ fpx q yα1 P Y α , pz ; : : : ; znq „ pt ; : : : ; tk q fpz1 ; : : : ; znq “ fpt ; : : : ; tk q f Y fwhere is defined on by : for every . 1 1 1 1 Thus, if , then p p by definition of on . Hence induces a map f F ÝÑ G rz ; : : : ; zns ÞÑ fpz q ¨ ¨ ¨ ¨ ¨ fpznq r p: 1 1 f f , F G f r r By construction p is a monoid homomorphism, therfore so is p, but since and are groups, p is X f G in fact a group homomorphism. Hence we have a commutative diagram ö i f F pr i X ÝÑ F; x ÞÑ rxs f f where : is the canonical inclusion. f r r Finally, notice that the definition of p is forced if we want p to be a group homorphism, hence we r have uniqueness of p, and the universal property of Definition 2.1 is satisfied.

¨ rx s P F x ry s x´ rz ; : : : ; z s Notation and Terminology α α α α n z ¨ ¨ ¨ zn F 1 1 To lighten notation, we identify with , hence with , and with ¨ 1 pz ; : : : ; znq P E zi ď i ď n xαi P X ´ in . xαi word txα | α P Iu F 1 A sequence1 with each letter (1 ) equal to an element or is called a in the generators . Each word defines an element of via: Lecture Notes: Cohomology of Groups SS 2018

13 pz ; : : : ; znq ÞÑ z ¨ ¨ ¨ zn P F z ¨ ¨ ¨ zn P F word

¨ 1 1 equivalent ðñ F 1 . By abuse of language, we then often also call a . ¨ pz ; : : : ; z q P Z Ď E n P Z n length pz ; : : : ; z q Two words aren calledn ě : they define the same element of . n ¨ 1 reduced 0 1 If ( ), then is called the of the word .

A word is said to be if it has minimal length amongst all the words which are equivalent to this word. Proposition 2.5 G

ProofEvery : groupS “ tgαisP isomorphicG | α P I tou a factor group of a free group.G I “ G X “ txα | α P Iu S F X i X ãÑ F Let : be a set of generators for (in the worst case, take ). Let : X f be aG set in bijection with , and let be the free group on . Let : denote f X ãÑ G; xα ÞÑ gα the canonicalD inclusion.f i f F ÝÑ G f ˝ i “ f ö F ! ˜ By the universalf property of free groups the mapG : pfq D f G –˜F{ pfq ˜ can: proj: induces a unique group homomorphism : such that . p ˜ ˜ ! ˜ Clearly is surjective since the generators of are all Im . Therefore F{ pfq the 1st Isomorphism Theorem yields ker ˜ .

ker ˜

We can now consider relations between the generators of groups: S “ tgα P G | α P Iu G X “ txα | α P Iu Notation and TerminologyS F X Let : be a set of generators for the group , let : be in G – F{N N “ pfq gα Ø xα “ xα N f bijection with , and let be the free group on . pz ; : : : ; z q x F N G n α p By the previous proof, , where : ker ˜ ( via the homomorphism ˜ ). 1 Any word in the ’s which defines an element of in is mapped in to an expression of thez form¨ ¨ ¨ zn “ G ; zi “ zi G

1 1 pz ; :where : : ; znq : imagerelation of in inunder the group the canonicalG for the homomorphism. set of generators S R “ trβ | β P Ju N F N 1 In this case, the word is called aR R set of defining relations. Nowof G let : be a set of generators of as normal subgroup of (this means that is generated by the setpX;R of allq conjugatespresentation of ). Such a setG is called a . Then the ordered pair isG called“ xX a| Ry “ xtxα uαPofI | tr,β anduβPJ y we: write G finitely presented G “ xX | Ry |X|; |R| ă 8 The group is said to be if it admits a presentation , where both . In this case, byG abuse“ xx of; : notation, : : ; x|X| | r we“ also; : : often : ; r|R| write“ y presentations: under the form

1 1 1 1 Lecture Notes: Cohomology of Groups SS 2018

Example 3 14 n´ Cn “ t ; g; : : : ; g u n P Zě S “ tgu

1 1 TheX cyclic“ tx groupu 1 of order generated by : . In this case, we n have:R “ tx u F “ xxy – pC8; ¨q f n n C8 ÝÑ Cn; x ÞÑ g x Cn “ xtxu | tx uy ˜ n n Cn “ xx | x y Cn “ xx | x “ y has a kernel generated by as a normal subgroup. Then . PropositionBy abuse 2.6 of notation,(Universal we property write simply of presentations) or also 1 . G S “ tsα | α P Iu F X “ txα | α P Iu S R “ trβ | β P Ju G Let Gbe a group generated by G “ xX | Ry , isomorphic toG a quotient of a free group on in bijection with . Let : be a set of relations in . f ThenX admitsH the presentation H if and only if satisfiesf theX ÝÑ followingH universal ö property:j fprβq “ H @ rβ P R f G ÝÑ H f f ˝ j “ f j X ÝÑ G; x ÞÑ s f G For every group , and for every set-theoreticα α map : such that ˜ f 1 ,F thereX exists a unique group homomorphism : such that , where : , and ˜ is the unique extension Proof : ñ of toG the“ xX free| R groupy on G. – F{N N R fprβq “ H @ rβ P R N Ď pfq " ": Suppose that . Therefore , where is generated by as normal ´ ´ subgroup. Thus thef conditionpzrβz q˜ “ fpzq f1prβq fpzq implies“ H that@ rβ P R;ker@ z˜P, sinceF:

1 “ H 1 ˜ ˜ ˜ ˜ loomoon 1 1 f f G – F{N ÝÑ H f ˝ π “ f π F ÝÑ F{N Therefore,i X byÝÑ theF universal property of the quotient, ˜ jinduces“ π ˝ i a unique group homomorphism f ˝: j “ f such that ˜, where : is the canonical epimorphism. ð Now, if : denotesG the canonical inclusion, then , and as a consequence we have X;F;R . N “ R R " ": Conversely, assume that satisfies' the universalF{N ÝÑ propertyG of the statement (i.e. relatively to ). Set : for the normal closure of . Then we have two group homomorphisms: xα ÞÑ sα : f F ÝÑ G ψ G ÝÑ F{N induced by ˜ : , and sα ÞÑ xα : ' ˝ ψpsα q “ 'pxα q “ sα α P I ' ˝ ψ “ G ψ ˝ ' “ F{R given by the universal property. Then clearly for each , so that Example 4 (TheId dihedraland similarly groups) Id . The claim follows.

D m m ď m ă 8 D m

2 2 Consider the finite dihedral group of order 2 with 2 . We can assume that is generated by π r “ s “ R : m 2 2 : rotation of angle and : symmetry through the origin in Lecture Notes: Cohomology of Groups SS 2018

15 xry – Cm Ď G xsy – C D m “ xry ¸ xsy rm “ s “ srs´ “ r´ 2 m 2 ´ ´ ThenClaim D2m , 1 and1 wexr; have s | r seen“ that; s “ ; srs “ r withy three obvious relations m ´ 1, 1, and . F 2 X1 “ tx;1 yu R “ tx ; y ; yxy xu 2 N IJ F: admits the presentation R 1 G 1“ F{N . 2 1 In order to prove the Claim, we let bem the free group on´ : , : , be the normal subgroupG “ generated xx; y | x by“ ; y, and“ ; y: x y x “so thaty : 2 1 1 1 1 f tx; yu ÝÑ D By the universal property of presentations the map m x ÞÑ r 2 : y ÞÑ s

f G ÝÑ D induces a group homomorphism m x ÞÑ r 2 : y ÞÑ s D “ xr; sy f G m , m G x; y; x´ ; y´ 2 ´ m´ ´ ´ ´ which is clearly surjective since x; y x. In order“ x to provey that“ yis injective, wey provexy that“1 x 1 ´ is a groupyx “ ofx ordery at most 2 . Recall that each1 element1 of is1 an expression in 1 1, hence actually an1 expressiona b in , since and . Moreover, implies , hencex wey are left withď expressionsa ď m ´ of the formď b ď :

|G| ď m with 0 f 1 and 0 1

Thus we have 2 , and it followsrm that“ is an isomorphism. infinite dihedral group D8

Notice that if we remove the relation 1, we can also´ formally´ define an via the following presentationD8 “ xr; s | s “ ; srs “ r y : 2 1 1 Theorem 2.7 : 1

G s t G – D m ď m ď 8 m st G 2 Let be a group generated by two distinct elements, andm , both of order 2. Then , where 2 . Moreover, G is“ thexs; t order| s “ of ; t in“ ,; p andstq “ y : 2 2 m “ 8 1 1 1

Proof( : r simply“ st meansm "no relation".) r m ě m “ ñ st “ ñ s “ t´ “ t t “ ´ ´ Set srs: “andr let be the order of . 1 2 Firstly, note1 that 1 2srs, since´ “ sps 1tqs´ “ ts1´ “ t´ s´ “ pstasq´ “ r´1.: Secondly, we have the relation , since 1 “ G 1 1 1 1 1 1 loomoon G r 1 s r “ st t “ sr

Clearly can be generated by and as and so . Lecture Notes: Cohomology of Groups SS 2018

16 H “ xry – Cm H IJ G

srs´ “ r´ P H rrr´ “ r P H |G H| “ : Now, : and since 1 1 1 C “ xsy – C and (or because : 2) Claim s R H 2 Set : s. P H s “ ri “ pstqi ď i ď m ´ : . “ s “ spstqi “ ptsqi´ t “ pts ¨ ¨ ¨ tq s pts ¨ ¨ ¨ tq ; Indeed, assuming yields for some 0 1. Hence 2 1 i´ i´ 1 looomooon looomooon t s ::: t “lengths 1 length 1 opsq “

so that conjugating by , then , thenG “ HC, then ,H weX getC “1 t u, contradictingG “ H the¸ assumptionC “ D m that 2. The claim follows. m 2 Therefore, we have provedG that and xs; t | s “1 ,; so t that“ ; pstq “ y as seen in the previous section. 2 2 Finally, to prove that admits the presentation 1 1 1 , we apply the universal f tx ; x u ÝÑ xs; t | s “ ; t “ ; pstqm “ y property of presentations twice tos ther maps xs ÞÑ s 2 2 : xr ÞÑ st 1 1 1

m ´ g tys; ytu ÝÑ G “ xr; s | r “ ; s “ ; srs “ y and ys ÞÑ s 2 1 : yt ÞÑ sr 1 1 1

. f G “ xr; s | rm “ ; s “ ; srs´ “ y ÝÑ xs; t | s “ ; t “ ; pstqm “ y This yields the existence of two group homomorphisms 2 1 2 2 : 1 1 1 1 1 1 g xs; t | s “ ; t “ ; pstqm “ y ÝÑ G “ xr; s | rm “ ; s “ ; srs´ “ y and gf “ f2g “ 2 2 1 : 1 1 1 1 1 1 such that Id and Id. (Here you should check the details for yourself!) Chapter 2. Background Material: Module Theory

The aim of this chapter is to recall the basics of the theory of modules, which we will use throughout. We review elementary constructions such as quotients, direct sum, direct products, exact sequences, free/projective/injective modules and tensor products, where we emphasise the approach via universal properties. Particularly important for the forthcoming homological algebra and cohomology of groups are the notions of free and, moreR generally,S of projective modules. unital associative Throughout this chapter we let and denote rings, and unless otherwise specified, all rings are assumed to be and . Commutative Algebra Most results are stated without proof, as they have been studied in the B.Sc. lecture Reference:. As further reference I recommend for example: Advanced modern algebra. 2nd ed.

[Rot10] J. J. Rotman, , Providence, RI: American Mathematical Society (AMS), 2010. 3 Modules, Submodules, Morphisms

Definition 3.1 (Left R-module, right R-module, pR;Sq-bimodule, homomorphism of modules)

left R-module pM; `q scalar multiplication external composition law ¨ R ˆ M ÝÑ M; pr; mq ÞÑ r ¨ m (a)A is an abelian group endowed with a (or λ R ÝÑ pMq ) : such that the map r ÞÑ λprq “ λr M ÝÑ M; m ÞÑ r ¨ m : End R R : rm: r ¨ m , isright a ringR homomorphism.-module By convention, when no confusion is to be made,¨ M weˆ willR ÝÑ simplyM; pwritem; rq " ÞÑ-module"m ¨ r to mean "left -module", and instead of . (a’)A S is defined analogouslypR;Sq-bimodule using a scalar multiplicationpM; `q : R on the right-handS side. (a”) If is a second ring,r ¨ then pm ¨ s anq “ pr ¨ mq ¨ s @ risP anR; abelian@ s P S; group@ m P M: which is both a left -module and a right -module, and which satisfies the axiom

17 Lecture Notes: Cohomology of Groups SS 2018

R-submodule R M N ď M r ¨ n P N r P18R n P N (b) An of an -module is a subgroup such that for every (homo)morphism R R-linear map R-homomorphism and every . (Similarly for right modules and bimodules.) R ' M ÝÑ N (c)A of -modules (or an , or an ) is a map of ' -modules : such that: 'pr ¨ mq “ r ¨ 'pmq @ r P R @ m P M (i) is a group homomorphism; and R isomorphism R (ii) , . M – N R M N A bijective homomorphism of -modules is calledR an (or an -isomorphism),monomor- and wephism write if there exists an -isomorphism between andhook. arrow ãÑ Antwo-head injective arrow (resp. surjective) homomorphism of -modules is sometimes called a (resp. epimorphism) and we sometimes denote it with a " " (resp. a " "). (Similarly for right modules and bimodules.)

Notation RMod R R ModR R R RModS : We let pR;Sdenoteq the category ofp leftR;Sq-modules (with -linear maps as morphisms), we let denote the category of right -modules (with -linear maps as morphisms), and we let denote the category of -bimodules (with -linear maps as morphisms). For the language of categoryConvention theory, see the Appendix.

Example 5 : From now on, unless otherwise stated, we will always work with left modules.

K K

Z (a) Vector spaces over a field are -modules, and conversely. R (b) Abelian groups are -modules, and conversely.

(c) If the ring is commutative, then any right module can be made into a left module, and ' M ÝÑ N R p'q ' R conversely. M p'q “ 'pMq f R N (d) If : is a morphism of -modules, then the kernel ker of is an -submodule Notationof 3.2 and the image Im : of is an -submodule of .

R M N R pM;Nq “ t' M ÝÑ N | ' R u

Given -modules and , we set Hom : : is an -homomorphism . ` pM;Nq ˆ pM;Nq ÝÑ pM;Nq This is an abelianR group for the pointwiseR addition of functions:R p'; ψq ÞÑ ' ` ψ M ÝÑ N; m ÞÑ 'pmq ` ψpmq : Hom Hom Hom N “ M pMq “ pM;Mq M R R : . R pMq M M In case , we write End : Hom for the set of endomorphisms of and Aut for the set of automorphisms of , i.e. the set of invertible endomorphisms of . Lecture Notes: Cohomology of Groups SS 2018

19 M;N R Exercise [Exercise 1, Exercise Sheet 3] pMq Let Rbe -modules. Prove that: R pM;Nq R (a) End , endowed with the usual composition andR sum of functions, is a ring.

(b) If is commutative then the abelian group Hom is a left -module. Lemma-Definition 3.3 (Quotients of modules) U R R M M{U R Let be an -submodule of an -module . The quotient group can be endowed with the structure of an -module in a naturalR way:ˆ M{U ÝÑ M{U r; m ` U ÞÝÑ r ¨ m ` U ` ˘ π M ÝÑ M{U; m ÞÑ m ` U R

ProofThe : canonical map : is -linear.

TheoremDirect 3.4 calculation. Universal property of the quotient ' M ÝÑ N R U R M U Ď p'q R (a) ' M{U ÝÑ N : Let ' ˝:π “ ' be a homomorphism of -modules. If is an -submodule of such that ker , then there exists a unique -module homomorphism : such that , or in other words such that the following ' diagram commutes: M N ö π D ' M{ U !

'pm ` Uq “ 'pmq @ m ` U P M{U 1st isomorphism theorem U “ p'q Concretely, . ' M{ ÝÑ p'q (b) : With the notationp' ofq (a), if ker , then R : ker Im 2nd isomorphism theorem U ;U R M U XU U `U is an isomorphism of -modules. R 1 2 1 2 1 2 (c) : If are -submodules of , then so are and , pU ` U q{ – U { : and there is an an isomorphism of -modulesU U X U

1 2 2 1 1 2 3rd isomorphism theorem U Ď U R M R 1 2 (d) : If M{ are{ -submodules– M{ of: , then there is an an isomor- U U {U U phism of -modules ´ ¯ ´ ¯ 1 2 1 2 Lecture Notes: Cohomology of Groups SS 2018

Correspondence theorem U R M 20

(e) tXR : If is anM | U-submoduleĎ Xu ÐÑ of t,R then there is a bijectionM{Uu X ÞÑ X{U ´ -submoduleπ pZ ofq Ð Z -submodules of [ 1 . Proof : R We assume it is known from the "Einführung in die Algebra" that these results hold for abelian groups Definitionand morphisms 3.5 (Cokernel, of abelian coimage groups.) Exercise: check that they carry over to the -module structure. ' P R pM;Nq cokernel ' R N{ ' coimage ' R M{ ' Let Hom . Then, the of is the quotient -module Im , and the of is the quotient -module ker . 4 Direct Products and Direct Sums

tMiuiPI R iPI Mi tMiuiPI R ś Let be a family of -modules. Then the abelian group , that is the product of R ˆ M ÝÑ M seen as a family of abelian groups, becomes an i -module viai the following external composition law: iPI iPI ź ź r; pmiqiPI ÞÝÑ r ¨ mi iPI : ` ˘ ` ˘ j P I πj iPI Mi ÝÑ Mj j Mj ś Furthermore, for each , we let : denotes the -th projection from the product to theProposition module 4.1. (Universal property of the direct product)

'i L ÝÑ MiuiPI R R ' L ÝÑ M π ˝ ' “ ' j P I iPI i j j If : is a collection of -linear maps, then there exists a unique morphism of ś L -modules : such that for every .

'

'k 'j  iPI Mi

πkś πj { #  Mk Mj

R L; Mi ÝÑ R pL;Miq In other words iPI iPI ´ ź ¯ ź Hom f ÞÝÑ πi Hom˝ f i ` ˘

Proofis : an isomorphism of abelian groups.

Exercise 2, Exercise Sheet 3. Lecture Notes: Cohomology of Groups SS 2018

21 iPI Mi iPI Mi pmiqiPI mi “ mi “ i P I À ś directNow let sum be thet subgroupMiuiPI of consistingR of the elements such thatj P I 0 al- mostηj M everywherej ÝÑ iPI M (i.e.i 0 exept for a finite subsetMj of indices ). This subgroup is called the of the family and is in fact an -submodule of the product. For each , we let À Proposition: 4.2 (Universaldenote property the canonical of the injection direct sum) of in the direct sum.

fi Mi ÝÑ LuiPI R R ' M ÝÑ L f ˝ η “ f j P I iPI i j j If : is a collection of -linear maps, then there exists a unique morphism of À L -modules : such that C O [ for every .

f

fk fj

iPI Mi ; c

ηk À ηj

Mk Mj

In other words R Mi;L ÝÑ R pMi;Lq iPI iPI ´ à ¯ ź Hom f ÞÝÑ f ˝Homηi i ` ˘

Proofis : an isomorphism of abelian groups.

RemarkExercise 4.3 2, Exercise Sheet 3.

|I| ă 8 iPI Mi “ iPI Mi À ś It is clear that if , then external. internal The direct sum as defined above is often called an direct sum. This relates as follows with the usualDefinition notion 4.4 of (“Internal”direct direct sum: sums) M R N ;N R M M “ N ‘ N m P M m “ n ` n n P N n P N 1 2 1 2 Let be an -module and be two -submodules of . We write if every M “ N ‘ N 1 M2 “ N ` N1 1 N X 2N “2 t u can be written in a unique way as , where and .

1 2 1 2 1 2 InProposition fact 4.5 (internal direct sum) if and only if and 0 . N ;N M M “ N ‘ N R

1 2 ' M ÝÑ N1 ˆ N2 “ N ‘ N If and arem as“ aboven ` n and ÞÑ pn ; n q then the homomorphism of -modules 1 2 1 2 : R (external direct sum) 1 2 1 2 , M “ iPI Ni is an isomorphism of -modules. À The above generalises to arbitrary internal direct sums . Lecture Notes: Cohomology of Groups SS 2018

5 Exact Sequences 22

Definition 5.1 (Exact sequence) ' ψ L ÝÑ M ÝÑ N R R exact at M ' “ ψ

RemarkA sequence 5.2 (Injectivity/surjectivity/shortof -modules exact and sequences-linear) maps is called ( ) if Im ker .

' ' L ÝÑ M ðñ ÝÑ L ÝÑ M L

ψ ψ (a) M ÝÑ Nis injective ðñ0 M ÝÑ N ÝÑis exact at . N ' ψ (b) ÝÑ L ÝÑis surjectiveM ÝÑ N ÝÑ 0 isL exactM at N. ' ψ ψ ψ M{ ' ÝÑ N (c) 0 0shortis exact exact (i.e. sequence at , and ) if and only if is injective, is surjective and induces an isomorphism : Im . ' P pL;Mq Such a sequenceR is called a (s.e.s. in short). ' π (d) If Hom is an injectiveÝÑ morphism,L ÝÑ M ÝÑ then therep' isq ÝÑa s.e.s.

π 0 coker 0 ψ P pM;Nq where isR the canonical projection. i ψ (d) If Hom is a surjectiveÝÑ morphism,p'q ÝÑ thenM ÝÑ thereN isÝÑ a s.e.s.;

i 0 ker 0

Propositionwhere 5.3 is the canonical injection. Q R

R pQ; ´q R Mod ÝÑ Ab left Let be an -module.' ψ Then the following holds: ÝÑ L ÝÑ M ÝÑ N ÝÑ R (a) Hom : is a exact covariant functor. In other words, if '˚ ψ˚ 0 / 0 isR p aQ;L s.e.sq of / -modules,R pQ;M thenq the/ inducedR pQ;N sequenceq

0 Hom Hom'˚ “ R pQ;'Homq '˚pαq “ ' ˝ α ψ˚ is an exact sequence of abelian groups. (Here : Hom , that is and R p´;Qq R Mod ÝÑ Ab left similarly for' .)ψ ÝÑ L ÝÑ M ÝÑ N ÝÑ R (b) Hom : is a exact contravariant functor. In other words, if ψ˚ '˚ 0 / 0 isR p aN;Q s.e.sq of -modules,/ R pM;Q thenq the induced/ R pL;Q sequenceq

˚ ˚ 0 Hom Hom' “ R p';QHomq ' pαq “ α ˝ ' ψ˚ is an exact sequence of abelian groups. (Here : Hom , that is and similarly for .) Lecture Notes: Cohomology of Groups SS 2018

23 Proof : R pQ; ´q R p´;Qq

¨ Exactness at R pQ;Lq: One easily checks that Hom and Hom are functors. ¨ Exactness at R pQ;Mq: (a) Hom Clear. β P ψ˚ ðñ ψ ˝ β “ Hom We have ðñ β Ă ψ ker ðñ β Ă 0'

ðñ @Imq P Q;kerD lq P L βpqq “ 'plqq

ðñ DIm Imλ Q ÝÑ L q lq ' ˝ λ “ β ' ! such that ðñ D λ P pQ;Lq q l ' ˝ λ “ β a map R: which sends qto and such that inj ðñ β P '˚: Hom which send to and such that Im

Remark(b) 5.4Exercise 5, Exercise Sheet 3.

R pQ; ´q R p´;Qq right

LemmaNotice 5.5 that (TheHom snake lemmaand) Hom are not exact in general. See Exercise 5, Exercise Sheet 3. R R

' ψ Suppose we are given the following commutativeL / M diagram/ N of -modules/ and -module homomor- phisms with exact rows: f g h  '1  ψ1  0 / L1 / M1 / N1

0 Then the following hold:

' ψ δ '1 ψ1 (a) There exists anf exact sequence/ g / h / f / g / h;

'1 ψ1ker ker ker coker coker coker δpnq “ 1´ ´ πL ˝ ' ˝ g ˝ ψ pnq n P phq πL L ÝÑ pfq

where 1, are the1 morphismsδ induced byconnecting the universal homomorphism property of the quotient, and for every ker (here : coker is the canonical ' L ÝÑ M '| f ÝÑ g homomorphism). The map is calledf the . 1 1 1 1 ψ M ÝÑ N kerψ g ÝÑ h (b) If : is injective, then : ker ker is injective.

Proof :(c) If : is surjective,δ then : cokern P h coker is surjective. m ; m P M n ψ m ´ m P ψ “ ' l P L m “ 'plq ` m 1 2 (a) First, we check that is well-defined. Let ker and choose two preimages of 1 2 1 1 2 under . Hence gpm qker “ g ˝ 'pImlq `.g Thus,pm q “ there' ˝ existsfplq ` gpm qsuch: that . Then, wen P haveh i P t ; u 1 2 2 1 ψ ˝ gpmiq “ h ˝ ψpmiq “ hpnq “ ; Since ker , for 1 2 we have 1 1 1 1 1 1 gpmiq P ψ “ ' l P L ' pl q “ gpmiq i 0 i 1 1 1 1 1 so that ker Im .g Therefore,pm q “ ' p therel q “ exists' ˝ fplq ` ' suchpl q: that . It follows that 2 2 1 Lecture Notes: Cohomology of Groups SS 2018

24 '1 l1 “ fplq ` l1 l1 l1 f δ Since is injective, we obtain 2 h1. Hence,m P 1 andg 2 havegpm theq “ same imageδψ inpmcokerq “ . Therefore, ψis well-defined.h Ă δ m P δ 1 1 Wel “ now wantˇ tol check“ fp thelq exactnessl P atL ker . Let ker . Then 0, so that 0 and thus Im ˇker ker . Conversely, let ker . With the previous notation, this means that 1 1 1 1 0, and thus 1 ˜ for someg ˝ '˜plq “.' We˝ havefplq “ ' pl q “ gpm q:

m ´ 'plq P g ˜ ˜ 1 1 n ψ

1 ψ m ´ 'plq “ ψpm q ´ ψ ˝ 'plq “ ψpm q “ n: Hence, ˜ ker . It remains to check that this element is sent to by . We get

` 1 ˘ 1 1 ψ h “ δ ˜ ˜ ˇ δ R Hence Im ˇker ker . The fact that is an -homomorphism, and the exactness at the other points are checked in a similar fashion. (b) Is obvious. Remark(c) 5.6Is a a direct consequence of the universal property of the quotient.

The name of the lemma comes from the following diagram

0 ' 0 ψ 0 f / g / h

ker ' ker ψ ker L / M / N /

f δ g h    0 / L1 / M1 / N1 '1 ψ1

1 0  '1  ψ  / f / g / h

coker coker coker

0 0 0

If fact the snake lemma holds in any abelian category. In particular, it holds for the categories of Lemma-Definitionchain and cochain 5.7 complexes, which we will study in Chapter 3. ' ψ ÝÑ L ÝÑ M ÝÑ N ÝÑ R split

A s.e.s. 0 0 of -modules is called iff it satisfies the following equivalent conditions:R σ N ÝÑ M ψ ˝σ “ N σ section ψ

(a) There exists an R-linear map ρ: M ÝÑ Lsuch that ρ ˝ 'id“ ( L isρ called a retractionfor ). ' (b) There exists an -linear map : such that id ( is called a for ). Lecture Notes: Cohomology of Groups SS 2018

' “ ψ direct summand M M251 M M “ ' ‘ M1 (c) The submodule Im ker is a of , that is there exists a submodule Proof : of such that Im .

ExampleExercise. 6

' π / Z{ Z / Z{ Z ‘ Z{ Z / Z{ Z / The sequence 'pr sq “ pr s; r sq π ' 0 2 2 2 2 0 ' π defined by 1 1 0 and / Z{isZ the canonical/ Z{ Z projection/ Z{ Z into/ the cokernel of is split but the squence 'pr sq “ pr sq π ' 0 2 4 2 0 defined by 1 2 and is the canonical projection onto the cokernel of is not split. 6 Free, Injective and Projective Modules

Free modules

Definition 6.1 (Generating set / R-basis / free R-module) M R X Ď M

M generated X M R Let be an -module and be a subset. xPX λx x λx P R (a) is said to be by if every element of can be written as an -linear combi- X řR-basis basis X M M in a nation , that is with almost everywhere 0. unique way R xPX λx X λs P R (b) is an (or a ) if generates and if every element of can be written M free R ř as an -linear combination (i.e. with pXq almost everywhere 0). Notation: M “ xPX Rx – xPX R “ R (c) is called if it admits an -basis. À À Remark 6.2 In this case we write : .

xPX λx X λs X ř M X M S R (a) When we write the sum , we always assume that the are 0 almost everywhere.

(b) Let be a generating set for . Then, is a basis of if and only if is -linearly R R R independent.

Proposition(c) If 6.3is (aUniversal field, then property every -module of free modules is free.) ( -vector spaces.)  P R X i X /P R M map ' X ÝÑ M R Let be a free -module with basis and let : be the inclusion map. For every -module and for every (of sets) : , there exists a unique morphism of -modules Lecture Notes: Cohomology of Groups SS 2018

' P ÝÑ M 26

' X / M ˜ : such that the following diagram commutes> i  ' P ˜

Proof : P Q m “ xPX λx x 'pmq “ xPX λx 'pxq ' ř ř If (unique expression), then we set ˜ . It is then easy to check Proposition˜ has the 6.4 required (Properties properties. of free modules)

R M R

P R pP; ´q (a) Every -module is isomorphic toR a quotient of a free -module.

(b) If is a free -module, then Hom is an exact functor. Proof : txiuiPI M M

' R ÝÑ M (a) Choose a set of generators of (take all elements of if necessary). Then define iPI à :priqiPI ÞÝÑ rixi: iPI ÿ

M – p iPI Rq { ' pP;À´q R P ψ M ÝÑ N It follows that R ker .ψ˚ R pP;Mq ÝÑ R pP;Nq β ÝÑ ψ˚pβq “ ψ ˝ β (b) We know that Hom α Pis leftR exactpP;N forq any -module . It remains to prove that if : is a surjective -linear maps, then : Hom Hom : is also surjective. So let Hom . We haveP the following situation: D α ~  M ? / N / ψ

teiuiPI R P αpeiq P N 0 ψ i P I mi P M ψpmiq “ αpeiq β teiuiPI ÝÑ M; ei ÞÑ mi Let be an -basis of . Each isR in the imageβ of P ,ÝÑ so thatM for each βpeiqthere “ mi @existsi P I such that . Hence, there is a map : . By the universal property of free modules thisψ ˝ inducesβpeiq “ anψpmi-linearq “ αpe mapiq ; ˜ : such that ˜ . Thusψ ˝ β α te u β α “ ˜ i iPI ψ ˝ β “ ψ˚ β ˜ ˜ so that ` ˘and coincide on the basis . By the uniqueness of , we must have ˜ ˜ . Injective modules

Proposition-Definition 6.5 (Injective module) I R

p´;Iq Let be an -module.R Then the following are equivalent:

(a) The functor Hom is exact. Lecture Notes: Cohomology of Groups SS 2018

27 ' P R pL;Mq '˚ R pM;Iq ÝÑ R pL;Iq R α L ÝÑ I R β M ÝÑ I (b) βIf ˝ ' “Homα is a injective morphism, then : Hom Hom is surjective (hence, any -linear map : can be lifted to an -linear map : , i.e., η I ÝÑ M R η ρ M ÝÑ I ). ρ ˝ η “ I (c) If : is an injective -linear map, then splits, i.e., there exists : such I I injective that Id .

ProofIf : satisfies these equivalent conditions, then is called .

RemarkExercise. 6.6 p q L ď M '

Note that Condition b is particularly interesting when and is the inclusion. Projective modules

Proposition-Definition 6.7 (Projective module) P R

pP; ´q Let be an -module.R Then the following are equivalent: ψ P pM;Nq R (a) The functor RHom is exact. ψ˚ R pP;Mq ÝÑ R pP;Nq (b) If Hom is a surjective morphism of -modules, then the morphism of abelian π M ÝÑ P R π σ P ÝÑ M groups : Hom Hom is surjective. π ˝ σ “ P (c) If : is a surjective -linear map, then splits, i.e., there exists : such P R that Id . P P projective (d) is isomorphic to a direct summand of a free -module.

ExampleIf satisfies 7 these equivalent conditions, then is called . R “ Z Z Z

e R e “ e R – Re ‘ Rp ´ eq Re (a) If , then every submodule of a free -module is again free (main theorem on -modules). e ‰ ; 2 (b) Let be an idempotent in , that is . Then, 1 and is projective tI u I but not free if 0 1. j jPJ j tP u P (c) A product of modules i isiPI injective if and only if each is injective.i

(d) A direct sum of modules is projective if and only if each is projective. 7 Tensor Products Definition 7.1 (Tensor product of R-modules) M R N R F

Let be a right -module and let be a left -module. Let be the free abelian group (= free Lecture Notes: Cohomology of Groups SS 2018

Z M ˆ N G F 28

-module) with basispm ` m ; n.q Let ´ pm be; nq the ´ p subgroupm ; nq; of@m ;generated m P M; @n byP allN; the elements pm; n ` n q ´ pm; n q ´ pm; n q; @m P M; @n ; n P N; 1 2 1 2 1 2 pmr; nq ´ pm; rnq; @m P M; @n P N; @r P R: 1 2 1 2 1 2 and tensor product of M and N (balanced over R) M bR N “ F{G pm; nq P F M bR N m b n The , is the abelian group : . The Remarkclass 7.2 of in is denoted by .

M bR N “ xm b n | m P M; n P NyZ M b N (a) R .

(b) In , wepm have` m theqb relationsn “ m b n ` m b n; @m ; m P M; @n P N; m b pn ` n q “ m b n ` m b n ; @m P M; @n ; n P N; 1 2 1 2 1 2 mr b n “ m b rn; @m P M; @n P N; @r P R: 1 2 1 2 1 2 and m b “ “ b n @ m P M @ n P N p´mq b n “ ´pm b nq “ m b p´nq @ m P M @ n P N In particular, 0 0 0 , and Definition 7.3 (R-balanced, . map) M N A f M ˆ N ÝÑ A R-balanced Let and be as above and let be an abelian group. A map : is called if fpm ` m ; nq “ fpm ; nq ` fpm ; nq; @m ; m P M; @n P N; fpm; n ` n q “ fpm; n q ` fpm; n q; @m P M; @n ; n P N; 1 2 1 2 1 2 fpmr; nq “ fpm; rnq; @m P M; @n P N; @r P R: 1 2 1 2 1 2

Remark 7.4

t M ˆ N ÝÑ M bR N; pm; nq ÞÑ m b n R

PropositionThe canonical 7.5 (Universal map : property of the tensor product) is -balanced. M R N R A R f M ˆ N ÝÑ A Z f M bR N ÝÑ A Let be a right -module and let be a left -module.f For every abelian group and every M ˆ N /; A -balanced map : there exists a uniqueö -linear map : such that t the following diagram commutes:  f M bR N

Proof : { M ˆ N ÝÑ F π F ÝÑ F{G Z Z f F ÝÑ A Let :f ˝ { “ f fdenoteR the canonical inclusion,G andĎ letpfq : denote the canonical projection. By the universal property of the free -module, there exists a unique f -linearF{G ÝÑ map A˜ : such that ˜ . Since is -balanced, we have that ker ˜ . Therefore, the universal property of the quotient yields the existence of a unique homomorphism of abelian groups : such that Lecture Notes: Cohomology of Groups SS 2018

29 f ˝ π “ f f M ˆ N / A = K ˜:

{ f

 ˜ t F

f π

  M bR N – F{G t “ π ˝ { f ˝ t “ f ˝ π ˝ { “ f ˝ { “ f

RemarkClearly 7.6 , and hence ˜ .

tMiuiPI R M R N R tNj uiPJ R (a) Let be a collection of right -modules, be a right -module, be a left -module and be a collection of left M-modules.i bR N – Then,pM wei bR haveNq iPI iPI à à M bR Nj – pM bR Nj q: jPJ jPJ à à

R M R bR M – M r b m ÞÑ rm P R X M b P – M (b) For every -module , we have via R xP.X Q M pQ;Rq N À pR;Sq M b N (c) If be a free left -module with basis , then . R pQ;Sq (d) Let be a ring. Let be a -bimodule and let be an -module. Then can be endowed withqpm b thenq structures “ qm b ofns; a @q P-bimoduleQ; @s P S; via@m P M; @n P N:

R R pR;Rq M bR N R (e) If is commutative, then any -module can be viewed as an -bimodule. Then, in Tensor product of morphisms: f M ÝÑ M1 R particular, becomes an -module. g N ÝÑ N1 R 1 1 (f) Let Z: fbeb g a morphismM bR N ÝÑ of rightM bR-modulesN and pf b: gqpm b nq “befp am morphismq b gpnq of left -modules. Then, by the universal property of the tensor product, there exists a unique -linear map : such that Proposition 7.7 (Right exactness of the. tensor product)

N R ´ bR N ModR ÝÑ Ab M R M b ´ Mod ÝÑ Ab (a) Let be a left -module. Then R : R is a right exact covariant functor.

Remark(b) 7.8Let be a right -module. Then : is a right exact covariant functor.

´ bR N M bR ´

The functors and are not left exact in general. Lecture Notes: Cohomology of Groups SS 2018

Definition 7.9 (Flat module) 30

R N flat ´ bR N ModR ÝÑ Ab

Proposition 7.10 A left -module is called if the functor : is a left exact functor. R

ProofAny : projective -module is flat. R P “ xPX Rx To begin with, we note that a direct sum of modules is flat if and only if each module in the sum is À' ψ flat. Next, consider the free -module / M / M . If / M /

R 1 2 3 0 0

'b P ψb P is a short exact sequence/ M bR of rightxPX R -modules,/ M thenbR wex obtainPX R / M bR xPX R / Id Id 1 ` À– ˘ 2 ` À– ˘ 3 ` À– ˘ 0 0  p'qxPX  pψqxPX  / xPX M / xPX M / xPX M / :

À 1 À 2 À 3 0 0 ´ bR P P 1 1 1 Since theN original sequenceR is exact, so isN the‘ bottomN “ P sequence, and thereforeR soP is the top sequence.R 1 Hence, N is exact whenN is free. Now, if is a projective -module, then for some free -module and for some - module . It follows that is flat, by the initial remark. Chapter 3. Homological Algebra

The aim of this chapter is to introduce the fundamental results of homological algebra. Homological algebra appeared in the 1800’s and is nowadays a very useful tool in several branches of mathematics, such as algebraic topology, commutative algebra, algebraic geometry, and, of particular interest to us, group theory. R unital associative Throughout this chapter denotes a ring, and unless otherwise specified, all rings are assumed to be Reference:and . An introduction to homological algebra. Second ed.

[Rot09] J. J. Rotman, An introduction to homological algebra , Universitext, Springer, New York, 2009. [Wei94] C. A. Weibel, , Cambridge Studies in Advanced Math- ematics, vol. 38, Cambridge University Press, Cambridge, 1994. 8 Chain and Cochain Complexes

Definition 8.1 (Chain complex)

chain complex complex) R

dn` dn (a)A (orpC simply‚; d‚q “ a ¨ ¨ ¨ ÝÑ Cofn` -modulesÝÑ Cn ÝÑ is aC sequencen´ ÝÑ ¨ ¨ ¨ ; ˆ 1 ˙ 1 1 n P Z Cn R dn P R pCn;Cn´ q dn ˝ dn` “ C‚ pC‚; d‚q 1 1 where for each , is an -modules and Hom satisfies 0. n degree R C We often write simply instead of . n R d n P Z differential maps (b) The integer is calledn the of the -module . C non-negative positive C “ n P Z (c) The -linear‚ maps ( ) are called the n . ă n P Zď 0 (d) A complex is called (resp. ) if 0, for all (resp. for all 0 ). d

Notice that sometimes we will omit the indices and write for all differential maps, and thus the

31 Lecture Notes: Cohomology of Groups SS 2018

32 dn ˝ dn` “ d “ N Cn “ n ď N 2 1 condition 0 can be written as 0. If there is an integer such that 0 for all dN` dN` , then we omit to write the zero¨ ¨ ¨ ÝÑ modulesCN` andÝÑ zeroCN` mapsÝÑ onCN the right-hand side of the complex: 2 1

N C2n “ 1 n ě N

Similarly, if there is an integer such that 0 for all , then we omit to write the zero dN dN´ modules and zero maps on the left-handCN ÝÑ sideCN of´ theÝÑ complex:CN´ ÝÑ ¨ ¨ ¨ 1 1 2

Definition 8.2 (Morphism of complexes) 1 morphism of complexes chain map pC‚; d‚q pD‚; d‚q '‚ C‚ ÝÑ D‚ R 'n Cn ÝÑ Dn n P Z 'n ˝ dn` “ 1 dAn` ˝ 'n` n P Z (or a ) between two chain complexes and , 1 written : , is a familiy of -linear maps : ( ) such that 1 1 dn` dn` dn dn´ for each ¨, ¨ that ¨ is such/ Cn` that the/ C followingn / Cn diagram´ commutes:/ ¨ ¨ ¨ 2 1 1 ' ' ' n` 1 n n´ 1 d1 d1 1 d1 n` 1  n`  dn 1  n´ ¨ ¨ ¨ / Dn` / Dn / Dn´ / ¨ ¨ ¨ 2 1 1 1 1 Notation. R Ch(RMod) Chain complexes together with morphisms of chain complexes (and composition given by degreewiseDefinition 8.3 composition (Subcomplex of /-morphisms) quotient complex form a) category, which we will denote by .

1 1 subcomplex C‚ pC‚; d‚q R Cn ď Cn n P Z 1 1 dnpCnq Ă Cn´ n P Z 1 1 (a)A pC‚;of d‚ aq chain complex is a family of -modulesC‚ ãÑ C‚ ( ), such 1 1 that forCn every Cn . n P Z In this1 case, becomes a chain complex and the1 inclusion given by the1 C‚ C‚ quotient complex C‚{C‚ R Cn{Cn canonical inclusion of into for each is a chain1 map. 1 n P Z dn Cn{Cn ÝÑ Cn´ {Cn´ (b) If is a subcomplex of , then the is the familiy of -modules 1 1 1 ( ) together with the differentialπ‚ C‚ mapsÝÑ C‚{:C‚ n PuniquelyZ determined 1 by the universalπn Cn propertyÝÑ Cn{ ofCn the quotient. In this case, the quotient map : defined for each by the canonical Definitionprojection 8.4 (Kernel: / image / cokernelis a) chain map. 1 '‚ C‚ ÝÑ D‚ pC‚; d‚q pD‚; d‚q

kernel ' C ' “ pt ' u Z; d q Let : be‚ a morphism of chain complexes‚ between ‚ andn nP ‚ . Then, 1 image '‚ D‚ '‚ “ pt 'nunPZ; d q (a) the of is the subcomplex of defined by ker : ker ‚ ; cokernel '‚ '‚ “ D‚{ '‚ (b) the of is the subcomplex of defined by Im : Im ; and (c) the of is the quotient complex coker : Im . Lecture Notes: Cohomology of Groups SS 2018

33 Ch(RMod)

WithDefinition these 8.5 notions (Cycles, of kernel boundaries, and cokernel, homology one) can show that is in fact an abelian category.

pC‚; d‚q R n-cycle d “ Z pC q “ Z Let be a chain complex of -modules.n n ‚ n n-boundary d “ B pC q “ B (a) An is an element of ker n: ` n: ‚ . n dn ˝ dn` “ Ď Bn Ď Zn Ď Cn @n P Z 1 (b) An is an element of Im : : . n-th homology module1 group C H pC q “ Z {B [Clearly, since 0, we have 0 ‚ n ‚ .]n n

(c) The (or simply ) of is : . n P Z Hnp´q Ch(RMod) ÝÑ RMod

In fact, for each , : is a covariant additive functor (see Exercise 1, ExerciseLemma 8.6 Sheet 6), which we define on morphisms as follows: 1 '‚ C‚ ÝÑ D‚ pC‚; d‚q pD‚; d‚q '‚ R Let : be a morphism of chain complexes between and . Then H p' q H pC q ÝÑ H pD q induces an -linear map n ‚ n ‚ n ‚ zn ` BnpC‚q ÞÑ 'npznq ` BnpD‚q : n P Z '˚ Hnp'‚q

Z 1 Prooffor : eachn P . To simplify,πn Zn thispC‚q map ÝÑ Z isnp oftenC‚q{B denotednpC‚q by πn insteadZnpD‚q of ÝÑ ZnpD‚.q{BnpD‚q

1 Fix , and'n letZnpC‚:q Ă ZnpD‚q z P Z,n resp. dn: ˝ 'npzq “ 'n´ ˝ dnpzq “ , be the canonical' projections.npzq P ZnpDq ` ˘ 1 First, notice that 'n BnpC‚q Ă BnpD‚becauseq if b P B, thennpC‚q b “ dn` paq 0a.P Hence,Cn` 1 we have '‚ . 'npbq “ 'n ˝ dn` paq “ dn` ˝ 'n` paq P BnpD‚q ` ˘ 1 1 1 Similarly, we have . Indeed, if , then R for someπn ˝ 'n , 1 1 and because is a chain map we have 1 . Therefore, by the universal property of the quotient, there exists a unique -linear map such ' π1 that the following diagram commutes:Z pC q n Z pD q n Z pD q{ n ‚ / n ‚ / n ‚ BnpD‚q 3 πn π1 ˝'  n n Z pC q{ n ‚ BnpC‚q

1 Hnp'‚q “ πn ˝ 'n

Set : . The claim follows. HnpC‚q

dn` dn It should be thought that the homologyC modulen` /Cn measures/Cn´ : the "non-exactness" of the sequence 1

Hnp´q n P Z 1 1

Moroever,R the functors ( ) are neither left exact, nor right exact in general. As a matter of fact, using the Snake Lemma, we can use s.e.s. of complexes to produce so-called "long exact sequences" of -modules. Lecture Notes: Cohomology of Groups SS 2018

Theorem 8.7 (Long exact sequence in homology) 34

'‚ ψ‚ ‚ /C‚ /D‚ /E‚ / ‚

Let 0 0 be a s.e.s. of chain complexes. Then there is a long exact δn` '˚ ψ˚ δn '˚ ψ˚ sequence¨ ¨ ¨ / HnpC‚q / HnpD‚q / HnpE‚q / Hn´ pC‚q / Hn´ pD‚q / ¨ ¨ ¨ ; 1 1 1 n P Z δn HnpE‚q ÝÑ Hn´ pC‚q R connecting homomor- phism 1 where for each , : is an -linear map, called Note: zero complex . ‚

Here 0 simply denotes the ¨ ¨ ¨ ÝÑ ÝÑ, thatÝÑ is theÝÑ complex ¨ ¨ ¨ 0 0 0 0 0 ‚ consistingProof : of zero modules and zero morphisms. We often write simplyC‚ 0D‚insteadE‚ of 0 . d n P Z To simplify, we denote all differential maps of the three complexes , , with the same letter , 'n ψn and we fix . First, we apply the/ “non-snake”Cn / D partn of the/ SnakeEn Lemma/ to the commutative diagram

dn dn dn 0 0  'n´  ψn´  / Cn´ / Dn´ / En´ / ; 1 1 1 1 1 0 0

'n ψn and we obtain two exact sequences / ZnpC‚q / ZnpD‚q / ZnpE‚q ;

0 'n´ ψn´ C { D { E { : n´ dn / n´ dn / n´ dn / and 1 1 1 1 1 n ´ n Im Im Im 0 Shifting indices in both sequences we obtain similar sequences in degrees 1, and respectively. ' ψ Therefore, we have a commutativeC { diagramn withD exact{ rows ofn theE form:{ n dn` / n dn` / n dn` /

dn 1 dn 1 dn 1 Im Im Im 0  'n´  ψn´  / Zn´ pC‚q / Zn´ pD‚q / Zn´ pE‚q ; 1 1 1 1 1 dn Cn{ dn0` ÝÑ Zn´ pC‚q R dn Cn ÝÑ Cn´ dn` Ď dn 1 1 whereD‚ : E‚ Im is the unique -linear map induced by the universal property of 1 1 the quotient by : (as Im ker by definition of a chain complex), and similarly δn dnpE‚q ÝÑ dnpC‚q for and . Therefore, the Snake Lemma yields the existence of the connecting homomorphisms “HnpE‚q “Hn´ pC‚q : ker coker n P Z loooomoooon loooooomoooooon1

δn` '˚ ψ˚ δn '˚ ψ˚ for each¨ ¨ ¨ as/ H wellnpC‚q as the required/ HnpD‚q long exact/ HnpE sequence:‚q / Hn´ pC‚q / Hn´ pD‚q / ¨ ¨ ¨ 1 “ dn “ dn “ dn “ 1 dn “ 1 dn loomoon loomoon loomoon loooomoooon loooomoooon ker ker ker coker coker Lecture Notes: Cohomology of Groups SS 2018

35

We now describe some important properties of chain maps and how they relate with the induced mor- phismsDefinition in homology. 8.8 (Quasi-isomorphism)

'‚ C‚ ÝÑ D‚ quasi-isomorphism Hnp'‚q n P Z A chain map : is called a if is an isomorphism for all Warning: C‚ D‚ . A quasi-isomorphism does not imply that the complexes and are isomorphic as chain complexes. See Exercise 2, Sheet 5 for a counter-example.

In general complexes are not exact sequences, but if they are, then their homology vanishes, so that there is a quasi-isomorphism from the zero complex. C‚ R Exercise [Exercise 3, Exercise Sheet 5] C exact C n P Z Let be‚ a chain complex of -modules.n Prove that TFAE: C acyclic H pC q “ n P Z (a) ‚ is (i.e. exact atn ‚for each ); ÝÑ C (b) is ‚ , that‚ is, 0 for all ;

(c) The map 0 is a quasi-isomorphism. Definition 8.9 (Homotopic chain maps / homotopy equivalence) 1 '‚; ψ‚ C‚ ÝÑ D‚ pC‚; d‚q pD‚; d‚q (chain) homotopic R tsn Cn ÝÑ Dn` unPZ Two chain maps : between chain complexes and are called 1 1 if there exists' an familiy´ ψn “ ofdn`-linear˝ sn ` mapssn´ ˝ d:n such that

n P Z 1 1

dn` dn for each ¨ ¨. ¨ / Cn` / Cn / Cn´ / ¨ ¨ ¨ 1 sn 'n´ψn 'n´ ´ψn´ 'n` ´ψn` 1 1 sn´ 1 1 1 1  y   y 1 ¨ ¨ ¨ / Dn` 1 / Dn 1 / Dn´ / ¨ ¨ ¨ dn` dn

1 1 1 '‚ „ ψ‚ '‚ C‚ ÝÑ D‚ homotopy equivalence

In thisσ case,D‚ ÝÑ weC write‚ σ.‚ ˝ '‚ „ C‚ '‚ ˝ σ‚ „ D‚ Moreover, a chain map : is called a if there exists a chain Note „ map : such that id and id .

Proposition: One easily 8.10 checks that is an equivalence relation on the class of chain maps. '‚; ψ‚ C‚ ÝÑ D‚

If : are homotopic morphisms of chain complexes, then they induce the same morphisms in homology,H thatnp' is‚q “ Hnpψ‚q HnpC‚q ÝÑ HnpD‚q @n P Z:

: Lecture Notes: Cohomology of Groups SS 2018

36 Proof : n P Z z P ZnpC‚q

' ´ ψ pzq “ d1 s ` s d pzq “ d1 s pzq ` s d pzq P B pD q: Fix andn let n .n Then,` n withn´ then notationn of` Definitionn n ´8.9,n we haven ‚ PB pD q “ ` ˘ ` 1 1 ˘ 1n ‚ 1 loooomoooon loooomoooon 0 z ` BnpC‚q P HnpC‚q

Hence, for everypHnp'‚q ´ Hnpψ‚qq pz ` B, wenpC have‚qq “ Hnp'‚ ´ ψ‚qpz ` BnpC‚qq “ ` BnpD‚q :

Hnp'‚q ´ Hnpψ‚q ” Hnp'‚q “ Hnpψ‚q 0 RemarkIn other 8.11 words 0, so that .

homomotopy category R (Out of theHo( scopeRMod) of the lecture!) Homotopy of complexes leads to considering the so-called of -modules, denoted , which is very useful in algebraic topology or representation theory of finite ¨ Ho( Mod) “ Ch( Mod) groups for example. It is defined as follows: R R ¨ “ { The objects are the chain complexes,Ho(RMod) i.e. Ob Ch(RMod) „Ob .

The morphisms are given by Hom : Hom .

It is an additive category, but it is not abelian in general though. The isomorphisms in the homotopy category are exactly the classes of the homotopy equivalences.

Dualizing the objects and concepts we have defined above yields the so-called "cochain complexes" and theDefinition notion 8.12of "cohomology". (Cochain complex / cohomology)

cochain complex R

d n (a)A pCof‚; d-modules‚q “ ¨ ¨ ¨ is ÝÑ a sequenceC n´ ÝÑn´ C n ÝÑd C n` ÝÑ ¨ ¨ ¨ ; 1 ˆ 1 1 ˙ n n n n` n` n n P Z C R d P R pC ;C q d ˝ d “ ‚ ‚ ‚ C pC ; d q 1 1 where for each , is an -module and Hom satisfies 0. Z n “ Z npC ‚q “ dn n-cocycles We often write simply instead of . Bn “ BnpC ‚q “ dn´ n-coboundaries (b) The elements of : : ker are the . n-th cohomology module 1group C ‚ HnpC ‚q “ Z {B (c) The elements of : : Im are the . n n

(d) The (or simply ) of is : .

¨ Morphisms of cochain complexes cochain maps Similarly to the case of chain complexes, we can define: pC ‚; d‚q pD‚; d‚q '‚ C ‚ ÝÑ D‚ R 'n C n ÝÑ Dn n n´ n´ n´ n P Z ' ˝ d “ d ˝(or' simply n P Z ) between two cochain complexes and ˜ , written1 1: 1 , as a familiy of -linear maps : ( ) such that ˜ for each , that is such that the following diagram Lecture Notes: Cohomology of Groups SS 2018

37

dn´ dn´ dn dn` ¨ ¨ ¨ / C n´ / C n / C n` / ¨ ¨ ¨ commutes: 2 1 1 'n´ 1 'n 'n` 1 dn´ 1  dn´  dn 1  dn` ¨ ¨ ¨ / Dn´ / Dn / Dn` / ¨ ¨ ¨ 2 1 1 ˜ ˜ ˜ ˜ ¨ 1 1

¨ subcomplexes, quotient complexes; ¨ kernels, images, cokernels of morphisms of cochain complexes. R CoCh(CochainRMod) complexes together with morphisms of cochain complexes (and composition given by degreewise composition of -morphisms) form an abelian category, which we will denote by Exercise .

Theorem: 8.13 formulate (Long these exact definitions sequence in in a cohomology formal way.) '‚ ψ‚ ‚ /C ‚ /D‚ /E‚ / ‚ n P Z δn HnpE‚q ÝÑ Hn` pC ‚q

Let 0 0 be a s.e.s. of cochain1 complexes. Then, for each , there

exists a connecting homomorphism˚ : ˚ such that˚ the following˚ sequence is δn` ' ψ δn ' ψ exact: ¨ ¨ ¨ / HnpC ‚q / HnpD‚q / HnpE‚q / Hn` pC ‚q / Hn` pD‚q / ¨ ¨ ¨ 1 1 1

Proof :

Similar to the proof of the long exact sequence in homology (Theorem 8.7). Apply the Snake Lemma.

9 Projective Resolutions

Definition 9.1 (Projective resolution) M R projective resolution M R d d d Let be an -module.p AP‚; d‚q “ ¨ ¨ ¨ / P of / isP a non-negative/ P complex of projective -modules 3 2 1 Pn n ě ` 2H pP‚q “1 P { d0 ˘– M Pn R n ě P‚ free resolution M 0 0 1 which is exact at for every 1 and such that Im . Notation: ε P M augmented Moreover, if is a free -module for every 0, then is called a of . complex 0 d d d ε Letting : ¨ ¨ ¨ denote/P the quotient/P homomorphism,/P / /M we/ ; have a so-called 3 2 1 2 1 0  0 M P‚  M associated to the projective resolution, and this augmented complex is exact. Hence we will also denote projectiveExample 8 resolutions of by . ¨n Z M “ Z{nZ /Z /Z

The -module admits the following projective resolution: 0 . Lecture Notes: Cohomology of Groups SS 2018

38

WeProposition now prove 9.2 that projective resolutions do exist, and consider the question of how "unique" they are. R

ProofAny : -module has a projectiveR resolution. (It can even chosenR to be free.) P R ε P  M M – P { ε We useP the fact thatR every -module is a quotient of aR free -moduled (PropositionP  ε 6.4Ď).P Thus there 0 0 0 Pexists{ ad free– moduleε together with a surjective -linear map : such that ker . 1 1 1 0 Next, let be a free -module together withd a surjective -linearε map : ker such that 1 1 P / P / / M: ker ker : 1 < 1 0 d " " - 1 ε

R dn´ Pn´ ÝÑ Pn´ R Pn ker R dn Pn  dn´ Ď Pn´ 1 1 2 Inductively,Pn{ dn – assumingdn´ that the -homomorphism : has already been defined, then 1 1 there exists a free -module and a surjective -linear map : ker with 1 Theoremker 9.3 (Liftingker Theorem. The claim) follows. 1 pP‚; d‚q pQ‚; d‚q P R n ě Let n and be two non-negative chain complexes such that Q Q n ě H pQ q “ n ě 1. ‚ is a projectiven -module for every 0n; ‚ ε P H pP q ε1 Q H pQ q 2. is exact at‚ for every 1 (that‚ is 0, for all 1). f H pP‚q ÝÑ H pQ‚q R '‚ P‚ ÝÑ Q‚ 0 0 0 0 1 Let : f and : be the quotientH homomorphims.p'‚q “ f f ˝ε “ ε ˝' 0 0 If : '‚ is an -linear map, then there exists a chain map : inducing 0 0 the given map in degree-zero homology, that is such that and . Moreover, ' lifts f such a chain map is unique up to homotopy.‚

ProofIn the : situationExistence. of the Theorem,P it is said that ε1 . R ' P ÝÑ Q 0 Beacuse is projective and is surjective, by definition (Def. 6.7), there exists an 0 0 0 d ε -linear map : ¨such ¨ ¨ that the/ P following/ diagram/ H pP‚q commutes“ P { d 1 0D' ö 0 f 0 1 Im d1 1  0 ε  1 ¨ ¨ ¨ / Q / / H pQ‚q “ Q { d ; 1 1 1 0 0 0 1 1 f ˝ ε “ ε ˝ ' ε ˝ ' ˝ d “ f ˝ ε ˝ d “ Imp'1 ˝ d q Ď ε “ d “ 0 P 0 1 d1 1 0 1 R 1 that is . But then, loomoon 0, so that Im ker Im . Again ' P ÝÑ Q ' ˝ d “ d1 ˝ ' 0 1 by Definition 6.7, since is projective and 1 is surjective onto its image, there exists an -linear map 1 1 1 0 1 1 1 d : such that P : / P ' ˝d 1 D' 1 ' 0 ö 0 1 1   0  Q / / d1 “ ε1 / Q d1

1 0 1 Im 1 ker inc Lecture Notes: Cohomology of Groups SS 2018

39 'n Pn ÝÑ Qn n '‚ P‚ ÝÑ Q‚

Uniqueness.The morphisms : are constructed similarlyψ‚ P by‚ ÝÑ inductionQ‚ on . Hence the existencef of a chain map : '‚ „ ψ‚as required. '‚ ´ ψ‚

n ěFor the uniquenessσn “ 'n ´ statement,ψn supposeσ‚ P‚ ÝÑ: Q‚ also lifts the given morphism . We haveσ “ ' to prove´ ψ “ thatH p'‚q ´ H (orpψ‚ equivalentlyq “ f ´ f “ that iss´ homotopicÝÑ H top theQ‚q zero chains´ H map).pP‚q ÝÑ Q For each 0 set : , so that : is becomes a chain map. In particular 0 0 0 0 0 2 0 1 0 0 0. Then we let : 0 and : ε be the zero maps. Therefore, in degreeP zero, we/ / H havepP‚ theq following/ maps: 0 0 0 s´ s´ 0 x  x Q 1 / / H pQ0‚q 2 / ; ε1

0 0 1 0 “ s´ ˝ ` ε ˝ s´ 0 sn Pn ÝÑ Qn` n n ě si Pi ÝÑ Qi` 2 1 1 where clearly´ ď0i ď n ´ 0 . Thisdi` provides˝ si ` s usi´ with˝ di the“ σ startingi pointi ě for´ constructing a homotopy 1 1 : by induction on . So let 0 and suppose : 1 1 is already constructed1 P “ H pP q;Q “ H pQ q;P1 “ “ 1Q ; d “ ε; d “ ε ; d “ “ d : for each´ 2 ‚ 1 and´ satisfies‚ ´ ´ for each 1, and´ where we´ identify

1 0 1 0σ ´ s ˝2d 2 0 d1 “0 d1 1 1 n n´ n 0 n n` 0

1 1 1 1 1 Now, we check that thedn image˝ σn ´ ofsn´ ˝ dn “ dnis˝ σ containedn ´ dn ˝ s inn´ker˝ dn Im : 1 “ d ˝ σn ´ pσn´ ´ sn´ ˝ dn´ q ˝ dn ` 1 ˘ n 1 1 “ d ˝ σn ´ σn´ ˝ dn n 1 2 1 “ σn´ ˝ dn ´ σn´ ˝ dn “ ; 1

1 σ‚ 1 1 0 Pn dn` R sn Pn ÝÑ Qn` 1 where thed last-nut-onen` ˝ sn “ σ equalityn ´ sn´ holds˝ dn because both is a chain map. Therefore, again by Definition 6.7, 1 since is projective and 1 is surjective onto its image, there exists an -linear map : 1 1 such that : dn dn´ Pn / Pn´ / Pn´ / ¨ ¨ ¨ Dsn 1 sn´ sn´ σn´sn´ ˝dn σn´ 1 σn´ 2 1 2  1  w 1  w 2  x Qn` 1 / Qn 1 / Qn´ 1 / Qn´ / ¨ ¨ ¨ dn` dn dn´

1 1 1 1 2 1 'n ´ ψn “ σn “ dn` ˝ sn ` sn´ ˝ dn

1 Hence we have 1 , as required.

AsTheorem a corollary, 9.4 (Comparison we obtain the Theorem required) statement on the uniqueness of projective resolutions: ε ε1 P‚  M Q‚  M R M P‚ Q‚ '‚ P‚ ÝÑ Q‚ ψ‚ Q‚ ÝÑ P‚

Let and M be two projectiveψ‚ ˝ ' resolutions‚ „ P‚ of' an‚ ˝ ψ-module‚ „ Q‚ . Then and are homotopy equivalent. More precisely, there exist chain maps : and : Prooflifting : the identity on and suchM thatM ÝÑ M Id and Id . '‚ P‚ ÝÑ Q‚ 1 H p'Consider‚q “ M the identityM ˝ε morphism“ ε ˝ ' Id : . ψ‚ Q‚ ÝÑ P‚ 1 By the Lifting Theorem,H p' there‚q “ existsM a chainM ˝ε map“ ε ˝ ψ: , unique up to homotopy, such that 0 0 Id and Id . Likewise, there exists a chain map : , unique up to 0 0 homotopy, such that Id and Id . Lecture Notes: Cohomology of Groups SS 2018

40

dn d ε ¨ ¨ ¨ / Pn / ¨ ¨ ¨ / P / P / M / O O 1 O O ¨¨¨ Dψn D'n ¨¨¨ Dψ 1 D' ö Dψ 0 D' ö M M 0  1  1 0  0 ε1  Id Id ¨ ¨ ¨ / Qn 1 / ¨ ¨ ¨ / Q 1 / Q / M / dn d 1 0 1 0 ψ‚ ˝'‚ P‚ M H pP‚q ÝÑ H pP‚q

ψ‚ ˝ '‚ „ P‚ '‚ ˝ ψ‚ Q‚ 0 0 Now, and Id are both chain maps that liftM theH identitypQ‚q ÝÑ mapHIdpQ‚:q . Therefore, by the uniqueness'‚ ˝ ψ‚ „ statementQ‚ in the Lifting Theorem, we have Id . Likewise, and Id 0 0 are both chain maps that lift the identity map Id : , therefore they are homotopic, that is Id . Horse- shoe Lemma Another way to construct projective resolutions is given by the following Lemma, often called the Lemma 9.5 (,Horseshoe because itLemma requires) to fill in a horseshoe-shaped diagram: ε1 1 2 1 1 / M / M / M / R P‚  M ε2 1 2 2 2 M P‚  M projective M Let 0 0 be a short exact sequence of -modules. Let be a resolution of and be a resolution of .

. . P.1 P.2

 1 1 P1 P2 ε1 ε2  0 0 / M1 / M / M2 /

  0 0

ε 1 2 P‚  M 0M Pn 0– Pn ‘Pn n P Zě 1 2 1 i‚ π‚ 2 / M / M / M / ‚ / P‚ / P‚ 0 / P‚ / ‚ Then, there exists a resolution of such that for each ε1 and the s.e.s. 1 1 i‚ π‚ P‚  M 0 ε 0 lifts to a s.e.s. of chain complexes 0 0 P‚  M where and are the canonical injection and projection. Moreover, if is a projective resolution, then so is . Proof : n Exercise 3, Exercise Sheet 6. [Hint: Proceed by induction on , and use the Snake Lemma.]

Finally,Definition we 9.6 note (Injective that dual resolution to the notion) of a projective resolution is the notion of an injective resolution: M R injective resolution M

Let be an -module. An of is a non-negative cochain complex of injective Lecture Notes: Cohomology of Groups SS 2018

R 41 d d d pI‚; d‚q “ I /I /I / ¨ ¨ ¨ -modules 0 1 2 In n ě ` 0 1H pI‚q2 “ d {˘ – M

Notation: { M ãÑ I 0 0 augmented complex which is exact at for every 1 and such that ker 0 . 0 { d d Letting : denoteM the natural/I injection,/I /I we have/ ¨ ¨ ¨ a so-called 0 1 pI‚; d‚0q 1 2 { M M ãÑ I‚ associated to the injective resolution , and this augmented complex is exact. Hence we will also denote injective resolutions of by .

Similarly to projective resolutions, one can prove that an injective resolution always exists. There is also a Lifting Theorem and a Comparison Theorem for injective resolutions, so that they are unique up to homotopy (of cochain complexes). 10 and

Ext Tor We now introduce the Ext and Tor groups, which are cohomology and homology groups obtained from applying Hom and tensor product functors to projective/injective resolutions. We will see later that Ext groupsDefinition can 10.1 be used (Ext in-groups group) cohomology to classify abelian group extensions. ε M N R P‚  M M n P Zě n-th Ext-group M N 0 Let and be two left -modulesn and let n be a projective resolution of . For , the of and is R pM;Nq “ H R pP‚;Nq ; ` ˘ n Ext : Hom R pP‚;Nq Recipe: that is, the -th cohomology group of the cochain complex Hom . P‚ M

R p´;Nq projective resolution 1. Choose a projective resolution of . d d d 2. Apply the left exact contravariantP‚ “ functor¨ ¨ ¨ Hom/ P / Pto the / P 3 2 1 cochain ` 2 1 0 ˘

d˚ d˚ d˚ to obtain a complexR pP ;Nq / R pP ;Nq / R pP ;Nq / ¨ ¨ ¨ : 1 2 3 abelian groups 0 1 3 Hom Hom Hom

of (which is not exact in general). n 3. Compute the cohomology of this new complex. R pM;Nq M First of all, we have to check that the definition of the abelian groups Ext is independent from the choice of the projective resolution of . Lecture Notes: Cohomology of Groups SS 2018

Proposition 10.2 42

1 ε ε n P‚  M Q‚  M M H R pP‚;Nq Hn pQ ;Nq R ‚ ` ˘ P Q If ` and ‚ ˘ ‚ are two projective resolutions of , then the groups Hom M and Hom are (canonically) isomorphic, via the homomorphisms induced by the chain Proofmaps : between and given by the Comparison Theorem'‚ P‚ ÝÑ appliedQ‚ toψ the‚ Q identity‚ ÝÑ P morphism‚ Id . M ψ‚ ˝ '‚ „ P‚ '‚ ˝ ψ‚ „ Q‚ By the Comparison Theorem,R p´ there;Nq exist chain maps : and : lifting the identity on ˚ and such that Id and ˚ Id . ' R pQ‚;Nq ÝÑ R pP‚;Nq ψ R pP‚;Nq ÝÑ R pQ‚;Nq : Now, applying the functor Hom yields morphisms of cochain complexes ˚ ˚ ˚ ˚ '‚ ˝ψ‚ „ Q ψ‚ ˝'‚ „ P ' ˝ψ „ ψ ˝' „ : Hom‚ Hom‚ and : Hom R pP‚;Nq Hom R pQ‚;Nq '˚ Hom Hom Since Id and Id , it follows that Id and Id . ˚ n n ' H R pQ‚;Nq ÝÑ H R pP‚;Nq But then, passing to cohomology, induces a group homomorphism ` ' ˘ ` ˘ '˚ '˚ : Hom ‚ Hom

˚ n n (see Exercise 1, Exerciseψ SheetH 5).R SincepP‚;Nq isÝÑ uniqueH up toR pQ homotopy,‚;Nq so is , and hence is ψ '˚ ˝ ψ˚ „ ψ˚ ˝ '˚ „ '˚ ˝ ψ˚ “ ψ˚ ˝ '˚ “ '˚ unique‚ because homotopic chain` maps induce the˘ same morphisms` in cohomology.˘ Likewise, there is a uniqueψ˚ homomorphism : Hom Hom of abelian groups induced by . Finally, Id and Id imply that Id and Id. Therefore, n Propositionand are 10.3 canonically (Properties defined of Ext isomorphisms.R ) M;M ;M N;N ;N R n P Zą

1 pM;N2 q – 1 pM;N2 q 0 Let R and R be -modules and let be an integer. The following holds: 0 R α M ÝÑ M (a) Ext Hom . ˚ 1 n 2 n (b) Any morphism of -modulesα : R pM ;Ninducesq ÝÑ aR grouppM ;N homomorphismq :

2 1 R β :N ExtÝÑ N Ext

1 n 2 n (c) Any morphism of -modulesβ˚ : R pM;N inducesq ÝÑ aR grouppM;N homomorphismq :

n 1 2 P R : Ext R pP;Nq “ Ext n ě I R n pM;Iq “ n ě (d) If is a projective -module, then ExtR 0 for all 1.

(e) If is an injectiveε -module, then Ext 0 for all 1. Proof : P‚  M M R p´;Nq P‚ (a) Let be a projective resolution of . Applying the left exact functor Hom to d˚ d˚ d˚ the resolution yieldsR pP the;Nq cochain/ complexR pP ;Nq / R pP ;Nq / ¨ ¨ ¨ : 1 2 3 0 1 3 Hom Hom Hom ˚ ˚ R pM;Nq “ H R pP‚;Nq “ d { – d :

Therefore, d 0 ε 0 ε P M ` ˘ 1 P M1 ¨ ¨ ¨ ÝÑExt ÝÑ ÝÑ Hom ker 0 ‚ker R 1 0 Now, the tail 0 of the augmented complex ˚ is an exact sequence ε˚ d of -modules, so that the induced/ sequenceR pM;Nq / R pP ;Nq / ¨ ¨ ¨ 1 0 0 Hom Hom Lecture Notes: Cohomology of Groups SS 2018

43 R pM;Nq R pP ;Nq

pM;Nq –0 d˚ “ ε˚ – pM;Nq is exact at Hom and atR Hom and it follows thatR ˚ 0 ε 1 Ext ker Im Hom ε ε1 1 P‚  M M P‚  M M because is injective. 1 α '‚ P‚ ÝÑ P‚ '‚ 1 ˚ 1 1 2 ˚ 2 (b) Let be a projective' resolutionR pP‚;Nq of ÝÑ and R pP‚;Nq be a projective' resolution of . The Lifting Theorem implies that lifts to a chain map : . Then, induces a morphism ˚ n n of chain complexes : Hom ' R pM ;NHomq ÝÑ R pMand;N thenq induces a morphism in cohomology ˚ ˚ n ě α “ ' 2 1 ε : Ext Ext P‚  M M for‚ each 0 and we set : . β R pP‚;N q ÝÑ R pP‚;N q β (c) Let beβ˚ a projective resolution of . Then, there is a morphism of cochain complexes 1 2 : Homε Hom induced by , which, in turn, induces a homomorphism of P‚  M M ¨ ¨ ¨ / / / P abelian groups in cohomology. P P “ P P “ P P ÝÑ P (d) Let be a projective resolution of .Choose 0 0 as a projective 0 1 resolution of (i.e. : , 0, . . . ), augmented by the identity morphism Id : . Then the induced cochain complexR p isP;Nq / / / ¨ ¨ ¨ ; 0 0 0 n R pP;Nq “Hom n ě 0 0 ε P‚  M M I R p´;Iq so that clearly Ext 0 for each 1. (e) Let be a projective resolution of . Since is injective, the functor Hom is exact. d˚ d˚ d˚ Therefore the induced cochainR pP ;Iq complex/ R pP ;Iq / R pP ;Iq / ¨ ¨ ¨ 1 2 3 0 1 2 Hom Hom Hom

Remark 10.4is exact and its cohomology is zero. The claim follows. n n P Zě R p´;Nq R Mod ÝÑ Ab n R pM; ´q R Mod ÝÑ Ab 0 Using the proposition one can prove that for every , Ext : is a Theoremcontravariant 10.5 (Long additive exact functor, sequences and Ext of Ext-groups: ) is a covariant additive functor.

' ψ /N /N /N / R

1 2 3 (a) Any s.e.s. 0 0 of -modules induces a long exact sequence of '˚ ψ˚ δ abelian groups/ R pM;N q / R pM;N q / R pM;N q / R pM;N q / ::: 0 0 '˚ 0 ψ˚ 0 δn 1 n 1 n 2 n 3 n` 1 :::0 / ExtR pM;N q / ExtR pM;N q / ExtR pM;N q / ExtR pM;N q / :::: 1 1 2 3 1 Ext Ext Ext Ext α β /M /M /M / R

1 2 3 (b) Any s.e.s. 0 0 of -modules induces a long exact sequence of β˚ α˚ δ abelian groups/ R pM ;Nq / R pM ;Nq / R pM ;Nq / R pM ;Nq / ::: 1 0 β˚ 0 α˚ 0 δn 1 n 3 n 2 n 1 n` 3 :::0 / ExtR pM ;Nq / ExtR pM ;Nq / ExtR pM ;Nq / ExtR pM ;Nq / :::: 1 3 2 1 3 Ext Ext Ext Ext Lecture Notes: Cohomology of Groups SS 2018

44 Proof : P‚ M

(a) Let be a projective resolution of . Then there is an induced short exact sequence of cochain '‚ ψ‚ complexes / R pP‚;N q / R pP‚;N q / R pP‚;N q /

1 2 3 0 HomPn Hom nHomP Zě 0

'˚ ψ˚ 0 because each module/ RispP projective.n;N q Indeed,/ R atpP eachn;N degreeq / R pPthisn;N sequenceq / is

1 2 3 0 Hom R pPnHom; ´q HomPn 0

obtained by applying the functor Hom '˚ , which is exact as is projective. It is then easily checked that this gives a s.e.s. of cochain complexes, that is that the induced differential maps commuteP‚ with the induced homomorphismsM . Thus,Q‚ applying Theorem 8.13 yieldsM the required long exact sequence in cohomology. R‚ M 1 3 (b) Let be a projective resolution of and let be a projective resolution of . By the 2 Horseshoe Lemma (Lemma 9.5), there exists a projective resolution of and a short exact sequence of chain complexes / P‚ / R‚ / Q‚ / ;

R 0 Qn 0 n ě P R Q lifting the initial s.e.s. of -modules. Since/ n / isn projective/ n for each/ 0, the sequences n ě p´;Nq 0 R 0 pQ ;Nq pR ;Nq pP ;Nq are split exact for each/ R0. Thereforen applying/ R Homn / yieldsR n a split exact/ s.e.s. n ě 0 Hom Hom Hom 0

for each for each / 0. ItR followspQ‚;Nq that there/ isR apR s.e.s.‚;Nq of cochain/ complexesR pP‚;Nq / :

0 Hom Hom Hom 0

The associated long exact sequence in cohomology (Theorem 8.13) is the required long exact sequence.

R pM; ´q R p´;Nq The above results show that the Ext groups “measure” and "repair" the non-exactness of the functors Hom and Hom .

The next result is called “dimension-shifting” in the literature (however, it would be more appropriate toRemark call it 10.6 “degree-shifting”); (Dimension shifting it provides) us with a method to compute Ext-groups by induction. N R

α β Let be an -module and consider a s.e.s./L /P /M /

R P 0 M P 0 M L n R pP;Nq “ n ě of -modules,n whereě is projective (if is given, take e.g. free mapping onto , with kernel ). Then Ext 0 for all 1 and applying the long exact sequence of Ext-groups yields at ˚ each degree 1 an exact sequenceα˚ n of the formδn n` β / R pL;Nq / R pM;Nq / ; 1 0 Ext Ext 0 Lecture Notes: Cohomology of Groups SS 2018

δn 45

n` M;N n L;N : where the connecting homomorphism R isp thereforeq – forcedR p toq be an isomorphism: 1 Ext Ext

Note that the same method applies to the second variable with a short exact sequence whose middle term is injective.

A consequence of the dimension shifting argument is that it allows us to deal with direct sums and Lemmaproducts 10.7 of modules in each variable of the Ext-groups. For this we need the following lemma: R

α β Consider the following commutativeA diagram1 / A of -modules/ A2 with/ exact rows:

f g   0 B1 / B / B2 / ' ψ

2 2 h P R pA ;B q h ˝0β “ ψ ˝ g f g h Then there exists a morphism Hom such that . Moreover, if and are Proof : isomorphisms, then so is .

PropositionExercise 10.8 5, Exercise(Ext and Sheet direct 7. sums)

tMiuiPI R N R

(a) Let be a family ofn -modules and let ben an -module. Then R Mi;N – R pMi;Nq @ n ě : ˜ iPI ¸ iPI à ź Ext Ext 0 M R tNiuiPI R

(b) Let be an -module andn let be a familyn of -modules. Then R M; Ni – R pM;Niq @ n ě : ˜ iPI ¸ iPI ź ź Ext Ext 0

Proof : Case n “

(a) 0. By Proposition 10.3(a) and the universal property of the direct sum (Proposi- n tion 4.2), weR have Mi;N – R Mi;N – R pMi;Nq – R pMi;Nq : ˜ iPI ¸ ˜ iPI ¸ iPI iPI 0 à à ź ź Ext n ě Hom i P I Hom R Ext

/ Li / Pi / Mi / ; Now, suppose that 1 and choose for each a s.e.s. of -modules Pi Pi Mi Li 0 0 where is projective (e.g. choose free with quotient isomorphic to and kernel ). These / Li / Pi / Mi / : sequences induce a s.e.s. iPI iPI iPI À À À 0 0 Lecture Notes: Cohomology of Groups SS 2018

46 Case n ě : n

1 We proceed by induction on . δ First, for n=1,R p usingPi;N a longq exact/ R sequencep Li;N ofq Ext-groups,/ R p we obtainMi;Nq a commutative/ R p diagramPi;Nq iPI iPI 0 iPI iPI 1 1 À – ö À– À À Hom Hom Ext Ext   iPI R pPi;Nq / iPI R pLi;Nq / iPI R pMi;Nq / iPI R pPi;Nq ś ś ś 1 ś 1 Hom Hom Ext Ext ¨ with¨ the following properties: ¨ the morphisms of the bottom row are induced componentwise;n “ both rows are exact; and Pi i P I iPI Pi the two vertical isomorphisms are given by the case 0. À Since is projective for every , so is , thus Proposition 10.3 yields R Pi;N – – R pPi;Nq @ i P I: ˜ iPI ¸ iPI 1 à ź 1 Ext 0 Ext

Therefore Lemma 10.7 yields R Mi;N – R pMi;Nq : ˜ iPI ¸ iPI 1 à ź 1 n ě Ext Ext pn ´ q

Now assume that 2 and assumen´ that the claim holdsn for´ the 1 -th Ext-groups, that is R Li;N – R pLi;Nq : ˜ iPI ¸ iPI 1 à ź 1 Ext Ext

Then, applying the Dimension Shiftingn´ argument yieldsn R Li;N – R Mi;N : ˜ iPI ¸ ˜ iPI ¸ 1 à à Ext Ext n´ n R pLi;Nq – R pMi;Nq @i P I; and 1 Ext n´ Ext n R pLi;Nq – R pMi;Nq : iPI iPI so that ź 1 ź Ext Ext

Hence the required isomorphism n n R Mi;N – R pMi;Nq : ˜ iPI ¸ iPI à ź Ext Ext

(b) Similar to (a): proceed by induction and apply a dimension shift. (In this case, we use s.e.s.’s with injective middle terms.)

M bR ´ ´ bR N To end this chapter, we introduce the Tor-groups, which “measure” the non-exactness of the functors and . Lecture Notes: Cohomology of Groups SS 2018

Definition 10.9 (Tor-groups) 47

M R N R P‚ N n P Zě n-th Tor-group M N Let be a right -module and be a left -module. Let be a projective resolution of . For 0 R , the of andn pM;Nis q “ HnpM bR P‚q;

n Tor : M bR P‚

Propositionthat is, the 10.10-th homology group of the chain complex . M;M ;M ;M R N;N ;N ;N R

R 1 2 3 pM;Nq 1 2 3 N Let ben right -modules and let be left -modules. R pM;Nq – M b N (a) The group Tor R is independant of the choice of the projective resolution of . R p´;Nq (b) Torn0 . R pM; ´q (c) Torn is an additive covariant functor. R pM ‘ M ;Nq – R pM ;Nq ‘ R pM ;Nq (d) Torn is an additiven covariant functor.n R R R pM;N1 ‘2N q – pM;N1 q ‘ pM;N2 q (e) Torn Torn Torn . R M1 N2 1 2 M N pM;Nq “ (f) Tor Tor Tor . n n ě

(g) If either or is flat (soα in particularβ if either or is projective), then Tor 0 for all 1. /M /M /M / R

1 2 3 (h) Any s.e.s. 0 0 of right -modules induces a long exact se- R δn` R α˚ R β˚ R δn quence¨ ¨ ¨ / n` pM ;Nq / n pM ;Nq / n pM ;Nq / n pM ;Nq / ¨ ¨ ¨ 1 δ αb βb R 3 1 N 2 N 3 ¨ ¨ ¨ Tor/ p1M ;Nq Tor/ M bR N Tor/ M bR N Tor/ M bR N / 1 Id Id 3 1 2 3 Tor1 0

α β of abelian groups./N /N /N / R

R 1 δ2n` R3 α˚ R β˚ R δn (i) Any¨ ¨ ¨ s.e.s. /0 n` pM;N q / n pM;N 0q of left/ -modulesn pM;N q induces/ an longpM;N exactq sequence/ ¨ ¨ ¨ 1 δ bα bβ R 3 1 M 2 M 3 ¨ ¨ ¨ Tor/ p1M;N q Tor/ M bR N Tor/ M bR N Tor/ M bR N / 1 Id Id 3 1 2 3 Tor1 0

of abelian groups.

The proof of the above results are in essence similar to the proofs given for the Ext-groups. Chapter 4. Cohomology of groups

pG; ¨q KG G K From now oncohomology we assume thatgroups we areG given a group (in multiplicative notation) and consider modules over the group algebra of over a commutative ring . The main aim of this chapter is to introduce the of and describe concrete projective resolutions which shall allow References:use to compute them in some cases. Cohomology of groups

[Bro94] K. S. BrownThe, cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. [Eve91] L. Evens, An introduction to the theory, Oxford of Mathematical groups. Fourth Monographs, ed. The Clarendon Press, Oxford University Press, New York, 1991. [Rot09] J. J. Rotman, An introduction to homological algebra , Graduate Texts in Mathema- tics, vol. 148, Springer-Verlag, New York, 1995. [Wei94] C. A. Weibel, , Cambridge Studies in Advanced Mathe- matics, vol. 38, Cambridge University Press, Cambridge, 1994. 11 Modules over the Group Algebra

Lemma-Definition 11.1 (Group algebra) G K group ring KG

gPG λgg λg P K If is a group and is a commutative ring, we may form the whose elements are ř the formalλ lineargg ` combinationsµgg “ pλg ` µgqgwith , andλg additiong ¨ andµhh multiplication“ pλgµh areqgh given : by gPG gPG gPG gPG hPG g;hPG ÿ ÿ ÿ ` ÿ ˘ ` ÿ ˘ ÿ and KG K K G KG group algebra of G over K group ring Thus is a -algebra, which as a -module is free with basis . Hence we usually call the Proof : KG ratherK than simply G . G K KG ˆ KG ÝÑ KG KG By definitionK is a free -module with basis , and the multiplication in is extended by - bilinearity to the multiplication . It is then straightforward to check that this makes into a -algebra. K “ Z

In this lecture, we will mainly work with the following commutative rings: the ring of integers, and fields.

48 Lecture Notes: Cohomology of Groups SS 2018

Remark 11.2 49 KG-modules and representations K KG V K (a) V K : G V If is a field, then specifying a -module is the same thing as specifying a -vector G ÝÑ pV q “ GLpV q ; space together with a -linear action ofK on , i.e. a group homomorphism K -representation of G Aut : K “ Z ZG M orZ in otherM words a Z . G M Similarly, if , then specifying a -module is the same thing as specifying a G ÝÑ ZpMq -module together with a -linear action of on , i.e. a group homomorphism integral representation of G Aut Left and right KG-modules also called an . G KG ÝÑ KG g ÞÑ g´ g P G

(b) : left KG M 1 right KG ´ Since is aG group, them ¨ mapg “ g m such that KGfor each is an anti- automorphism. It follows that any 1 -module may be regarded as a -module via the right -action : . Thus the sidedness of -modules is not usually an The trivial KG-module issue. K KG G (c) : ¨ G ˆ K ÝÑ K The commutative ring itself can be seen as a -module via the -action pg; λq ÞÝÑ g ¨ λ “ λ : K KG the trivial KG : KG A G a trivial KG-module extended by -linearity to the whole of . This module is called -module. Tensor products of KG-modules An arbitrary -module on which acts trivially is also called . M N KG M bK N M N (d) K KG : diagonal action G If and are two -modules, then the tensor product of and balanced g ¨ pm b nq “ gm b gn @ g P G; @m P M; @ n P N: over can be made into a -module via the of , i.e.

Morphisms of KG-modules : M N KG K pM;Nq (e) KG conjugation: action G If and are two -modules, then the abelian group Hom can be made into a pg ¨ fqpmq “ g ¨ fpg´ ¨ mq @ g P G; @m P M: -module via the of , i.e. 1 The augmentation map and the: augmentation ideal ε KG ÝÑ K εpgq “ g P G K (f) KG augmentation map: TheK map : defined by : 1 for every and extended by -linearity to the whole of is called the pεq “ IG . This is a surjective homomorphism of -algebras whoseaugmentation kernel ideal KG KG{IG – K ε KG ker : IG KG KG is called the of , and . (Notice that is hence also a homomorphism of -modules, so that we can also see as a -submodule of .) Lecture Notes: Cohomology of Groups SS 2018

Lemma 11.3 50

IG K K tg ´ | g P Gzt uu

X G IG KG (a) is a free -module with -basis 1 1 . tx ´ | x P Xu : (b) If is a set of generators for the group , then is generated as a -module by the set M KG IG ¨ M “ xg ¨ m ´ m | g P G; m P My 1 K

Proof :(c) If is a -module,S then“ tg ´ | g P Gzt uu . ε ε S K (a) First of all, the set : 1 1 is clearly contained in ker by definition of . The set is -linearly independent“ since λgpg ´ q pλg P K q g G Pÿzt u 0 “ 1 λgg ´ 1 λg g G g G Pÿzt u P ÿzt G u 1 1 λg “ g P Gzt u G K KG S IG gPG λgg λg P K IG “ ε implies that 0 for every ř 1 , because is -linearly independent in . To prove that spans , let “(εp λ)g begq “an elementλg of ker . Hence gPG gPG ÿ ÿ 0

λ g “ λ g ´ “ λ g ´ λ “ λ pg ´ q “ λ pg ´ q : and it follows thatg g g g g g gPG gPG gPG gPG gPG g G ÿ ÿ ÿ ÿ ÿ Pÿzt u 0 1 1 g ; g P G 1

g g ´ “ g p1g 2´ q ` pg ´ q g´ ´ “ ´g´ pg ´ q (b) Clearly, for every elements we have: 1 1 tg ´ 1 | 2g P Gzt uu1 Ď2 xtx ´ | x1 P Xuy 1 1 1 1 1 1 KGand 1 1 IG “ xtg ´ | g P Gzt uy Ď xtx ´ | x P Xuy Ď IG Therefore 1 1 1 K , which implies thatKG

1 1 1

and hence equality holds. Definition(c) Follows 11.4 ( fromG-fixed (a). points and G-cofixed points) M KG

G-fixed points M MG “ m P M | g ¨ m “ m @ g P G Let be a -module. G-cofixed points M M “ M{pIG ¨ Mq ( (a) The of are by definition :G .

(b) The of are by definition : .

M N KG Exercise [ExerciseMG 1, Exercise Sheet 8] M G Let and be -modules. Prove that: MG M G (a) is the largest submodule of on which acts trivially; (b) is the largest quotient of on which acts trivially; Lecture Notes: Cohomology of Groups SS 2018

G 51 K pM;Nq “ KGpM;Nq pM b Nq – M b N (c) Hom K G HomKG ; G pKGqG “ x gy G pKGqG “ (d) ; gPG K ř (e) If is finite, then and if is infinite, then 0.

12 (Co)homology of Groups

G

DefinitionWe can eventually 12.1 (Homology define the and homology cohomology and cohomology of a group) groups of a given group .

A KG n P Zě

KG H pG;Aq “ pK;Aq n-th0 homology group of G with coefficients in A Let ben a -modulen and let . Define: HnpG;Aq “ n pK;Aq n-th cohomology group of G with coefficients in A (a) : Tor KG , the ; and

Remark(b) 12.2 : Ext , the . n HnpG;Aq H pG;Aq K A priori the definition of the homology and cohomology groups and seem to KG ZG depend on the base ring , but in factn itpK;A is notq – the case.n pZ Indeed,;Aq it can be proven that there are group isomorphisms Torn Torn KGpK;Aq – ZGpZ;Aq and n P Zě KG A ZG Ext Ext Z ÝÑ K Z K 0 for each and every -module , which can also be seen as a -module via the unique K Z ring homomorphism , mapping 1 to 1 . See Exercise 2, Exercise Sheet 8.

InProposition view of the 12.3 above (Long remark, exact from sequences now on,) unless otherwise stated, we specify the ring to . ' ψ / A / B / C / ZG

Let 0 0 be a short exact sequence of -modules. Then there are long

exact sequences of abelian groups inδ homologyn` and' cohomology:˚ ψ˚ ¨ ¨ ¨ / Hn` pG;Cq / HnpG;Aq / HnpG;Bq / HnpG;Cq / ¨ ¨ ¨ (a) 1 δ '˚ ψ˚ ¨ ¨ ¨ / H p1G;Cq / H pG;Aq / H pG;Bq / H pG;Cq / 1 1 0 0 0 0 '˚ ψ˚ δ / H pG;Aq / H pG;Bq / H pG;Cq / H pG;Aq / ¨ ¨ ¨ (b) 0 0 '˚ 0 ψ˚ 0 δn 1 ¨ ¨ ¨ / HnpG;Aq / HnpG;Bq / HnpG;Cq / Hn` pG;Aq / ¨ ¨ ¨ 0 1 Lecture Notes: Cohomology of Groups SS 2018

52 Proof : G

By definition of the homology and cohomology groups of , (a) is a special case of Proposition 10.5(a) and (b) is a special case of Theorem 10.10(i).

ToProposition start our investigation12.4 we characterise the (co)homology of groups in degree zero: A ZG

H pG;Aq – Z b A – A Let be a -module.ZG Then G G H0 pG;Aq – pZ;Aq – A (a) ZG , and 0 (b) Hom

Proofas : abelian groups. ZG H pG;Aq “ pZ;Aq – Z bZG A Z bZG A – pZ bZ AqG – AG 0 0 (a) By Proposition 10.10(a),H pG;Aq “ TorZGpZ;Aq – ZGpZ;A. Moreoverq by Exercise 1(d), Exercise G G Sheet 8, we have 0 ZGpZ;Aq – 0 ZpZ. ;Aq – A (b) We already know that Ext Hom . Moreover by Exercise 1(c), Exer- cise Sheet 8, we have Hom Hom . ZG Gab “ G{rG;Gs G The degree-one (co)homology groups with coefficients in a trivial -module can also be characterised using the augmentation ideal. For this we let : denote the abelianization of the group .

Exercise [Exercise 4, Exercise Sheet 8] pIG{pIGq ; `q – pG ; ¨q Prove that: ab A ZG 2 (a) There is an isomorphism of abelian groups . ¨ H pG;Aq – IG b A – IG{pIGq b A – IG{pIGq b A – G b A (b) If is a trivial -module,ZG then: ZG Z ab Z ¨ H pG;Aq – ZGpIG;Aq – 2 ZGpIG{pIGq ;Aq 2 1 ; 1 – ZpIG{pIGq ;A2 q – ZpGab;Aq – GrppG;Aq Hom Hom 2 Hom Hom Hom . Corollary 12.5

Z ZG H pG; Zq – IG{pIGq – Gab

2 1 ProofIf : denotes the trivial -module, then .

This is straightforward from Exercise 4, Exercise Sheet 8. 13 The Bar Resolution

concrete Z ZG In order to compute the (co)homology of groups, we need projective resolutions of as a -module. Lecture Notes: Cohomology of Groups SS 2018

Notation 13.1 53

n P Zě Fn Z Z pn ` q pg ; g ; : : : ; gnq G G Fn 0 Let be a non-negative integer. Let be the free -module with -basis consisting of all 0 1 1 -tuples g ¨ pofg elements; g ; : : : ; g ofnq “. Then pgg ; the gg group; : : : ; ggnactsq ; on via

Fn Z0G 1 ZG 0 1 Bn “ p ; g ; : : : ; gnq | gi P G ď i ď n 1 ( and it follows that is a free -module with -basis : 1 . First, B Gn` ÝÑ Gn for each 0 , define mapsi pg ; : : : ;1 gnq ÞÑ pg ;:::; gi; : : : ; gnq : 0 g 0 pn ` q i q , n Z Fn n ě where the check notation means that is deleted from the initial 1 -tuple in order to produce an -tuple, and extend them by -linearitydn Fn ÝÑ to theFn whole´ of . If 1, define n 1 i : x ÞÝÑ p´ q Bipxq: i“ ÿ 1 Bi ZG 0 dn

Z Z Since the maps are -linear byε definition,F “ G soÝÑ is . Finally consider the augmentation map g ÞÝÑ @g P G: 0 : 1 Proposition 13.2

dn` dn dn´ d ¨ ¨ ¨ / Fn / Fn´ / ¨ ¨ ¨ / F ZG ZG 1 1 1 1 0 The sequence is a free -resolution of the trivial - Proof : F “ Z d “ ε ε “ d d module. ´ n 1 0 0 Set : and : (note that is consistent with the definition of ). We have to ε dn` dn dn´ d d prove that thepF resulting‚; d‚q  sequenceZ “ ¨ ¨ ¨ / Fn / Fn´ / ¨ ¨ ¨ / F / F´ ˆ 1 1 1 0 ˙ 1 0 1 :

is an¨ exactClaim complex. 1 dn´ ˝ dn “ n ě n` pg ; : : : ; gnq P G 1 : 0 for every1 1. 0 n´ n Indeed: Let be a basis element. Theni j pdn´ ˝ dnqpg ; : : : ; gnq “ p´ q p´ q pBi ˝ Bj qpg ; : : : ; gnq : 1 i“ j“ 1 0 ÿ ÿ 0 1 1 ď i ă j ď n gj 0 0 gi

0 0 0 pB0 ˝ B qpg ; : : : ; g q “ p´ qi `j g ;:::; g ;:::; g ; : : : ; g : Now let 0 i .j If we removen first and then , wei get j n 0 0 0 0 0 0 0 g g ` 0 j ´ ˘ i 1 j q q p´ qj ´ d ˝ d 0 0 n´ n 0 0 On the other hand,1 if we remove first, then is shifted to position 1 and must be removed 1 with sign 1 . So both terms cancel and it follows that is the zero map. Lecture Notes: Cohomology of Groups SS 2018

ε 54 ¨ Claim 2 F‚  Z Fn n ě pF‚; d‚q Z

: is an exact complex. F‚ Indeed: by definition of then ě modules ( 0), we may view as a complex of -modules. By Exercise 3, Exercise Sheet 5, it suffices to prove that there exists a homotopy between Id and sn Fn ÝÑ Fn` the zero chain map. For 0, define pg ; : : : ; gnq ÞÑ p ; g ; : : : ; gnq 1 : s´ Z “ F´ ÝÑ F – ZG 0 Z 0 p q si “ n` 1 , i ď ´ n ě pg ; : : : ; gnq P G 1 1 0 let : be the -homomorphism1 sending 1 to 1 , and let : 0 for all 0 n` 2. For 0 and compute j dn` ˝ sn ` sn´ ˝ dn pg ; : : : ; gnq “ pg ; : : : ; gnq ` p´ q ; g ;:::; gj´ ; : : : ; gnq 1 j“ ÿ ` 1 1 ˘ 0 0 n ` 0 1 i 1 1 1 q ` p´ q ; g ;:::; gi; : : : ; gnq i“ ÿ ` 0 “ pg ; : : : ; gnq; 0 1 1 q

0 dn` ˝ sn ` sn´ ˝ dn “ Fn n ď ´

1 1 Notationand 13.3 it is(Bar clear notation that ) Id for every 1, as required.

n P Zě

0 n` Given , setrg |g | ::: |gns “ ; g ; g g ; g g g ; : : : ; g ¨ ::: ¨ gn P G : 1 1 2 ` 1 1 2 1 2 3 1 ˘ : 1 ; h ; : : : ; h h h´ h h´ h ::: h´ h : With this notation, we havep nq “ | | | | n´ n 1 1 1 Z 1 “ 1 2 3 ‰ n Fn 1 G 1 rg |2::: |gns | gi P1 G “ G Gn F ZG tr su 1 ( Hence becomes a freen ě -module withď basisi ď n : , which as a set is 0 in bijection with . In particular is the free -module with basis (empty symbol). With this notation, for every 1 andg every¨ g 0| ::: |gn , we have i “ ; Birg | ::: |gns “ g | ::: |gi´ |gigi` |gi` | ::: |gn ď i ď n ´ ; $ 1 “ 2 ‰ &’ g | ::: |gn´ i “ n:0 1 “ 1 1 1 2 ‰ 1 1 %’ “ 1 1‰ bar resolution

Because of this notation the resolution of Proposition 13.2 is known as the . normalised bar resolution In fact, it is possible to render computations easier, by considering a slight alteration of the bar reso- lutionNotation called 13.4 the (The normalised bar notation. )

n P Zě Fn Dn ZG Fn rg | ::: |gns Fn gi 0 Letp ; h ; : : : ;, hn andq P F letn be as above and let be the -submodule of generated by all 1 elements of such that at least one of the coefficients is equal to 1. In other words, 1 if 1 p ; h ; :, : then: : ; hnq P Dn ðñ D ď i ď n ´ hi “ hi` :

1 1 1 1 1 such that Lecture Notes: Cohomology of Groups SS 2018

Lemma 13.5 55

D‚ F‚ s pD q Ă D n ě (a) n isn a subcomplexn` of . 1 Proof :(b) for all 0.

n ě dnpDnq Ď Dn´ p ; h ; : : : ; hnq P Dn ď i ď n ´ hi “ hi` 1 1 (a) Let 1. We have to prove that . So let 1 , so that there is an B p ; h ; : : : ; h q P D 1 ď j ď n j ‰ i; i ` : index 1 j 1 such thatn n´ . Then, clearly

1 1 Bip ; h ; : : : ; hnq “ Bi` p ; h ; : : : ; hnq 1 n ifor each 0 such that 1 dnp ; h ; : : : ; hnq “ i“ p´ q Bip ; h ; : : : ; hnq Bi Bi` 1 1 1 On the other hand, we have the equality 1 j ‰ i; i ` 1 and in the alter- 1 ř 0 1 1 nating sum 1 sn 1 1 , the signs of and are opposite to each other. Therefore, we are left with a sum over 1. Corollary(b) Obvious 13.6 by definition of . F F n F ZG n “ n{Dn ě ‚

ProofSet : : D‚ for every F0‚. Then is a free -resolutionpF of‚; thed‚q trivial module. ZG snpDnq Ă Dn` n ě Dn sn Since is a subcomplex of , weFn can` ÝÑ formF then` { quotientDn` complexZ , whichsn consistsFn ÝÑ ofF freen` 1 -modules.Z Now by thesn Lemma,F n ÝÑ F n` for each 0, therefore is in the kernel of 1 1 1 1 post-composed with the quotient map sn | n P andZ each -linear map : 1 induces a -linear maps : via the Universal Property( of the quotient. Hence, similarly to the proof of Proposition 13.2, we getdn a homotopy d andε“d we conclude that the sequence ¨ ¨ ¨ / F n / F n´ / ¨ ¨ ¨ / F / Z / : 1 0 1 0 0

Definitionis exact, 13.7 as required. (Normalised bar resolution)

pF ‚; d‚q normalised bar resolution Z ZG

ExampleThe chain 9 (Bar complex resolution in lowis called degrees the) of as a -module.

d d ε In low degrees the bar¨ resolution ¨ ¨ has the/ F form / F / F / Z / 2 1 rg |2g s rg1s r0 s 0 1 2 bases elts: ¨ εpr sq “ with ¨ d prgsq “ B prgsq ´ B prgsq “ gr s ´ r s 1; ¨ d1rg |g s “0 B prg |g 1sq ´ B prg |g sq ` B prg |g sq “ g rg s ´ rg g s ` rg s ; 2 1 2 0 1 F2‚ 1 1 2 2 1 2 1 2 1 2 1 . Similar formulae hold for . (Exercise!) Lecture Notes: Cohomology of Groups SS 2018

14 Cocycles and Coboundaries 56

HnpG;Aq n ě A ZG We nowZGpF use‚;Aq the (normalised) bar resolution in order to compute the cohomology groups n n ( 0), where is an arbitrary C-module.pG;Aq To“ thisSet end,pG we;Aq need to consider the cochain complex Hom . Define Gn A Z : Hom – n to be the set of all maps from to , soZ thatGpFn clearly;Aq ÝÑ thereC pG;A is anq isomorphism of -modules f ÞÑ f| Gn Hom ˚ n´ n mapping . Using this isomorphism,dn C wepG;A seeq that ÝÑ theC p correspondingG;Aq differential maps are: ˚ 1 f ÞÝÑ dnpfq :

n´ where ˚ i dnpfq rg | ::: |gns “ f g rg | ::: |gns ` p´ q f rg | ::: |gigi` | ::: |gns 1 i“ ` 1 ˘ ` 1 2 ˘ ÿ n ` 1 1 ˘ ` p´ q f1 rg | ::: |gn´ s : 1

` 1 1 ˘ 1 Definition 14.1 (n-cochains, n-cocycles, n-coboundaries)

C npG;Aq n-cochains G With the above notation: f P C npG;Aq d˚ f “ f n-cocycle G (a) The elements of aren` called the of . n Z npG;Aq 1 (b) If n is such that 0˚, thenn´ is called an n of , and the the set of f P C pG;Aq dn C pG;Aq ÝÑ C pG;Aq f n- all -cocycles is denoted .n coboundary G B pG;Aq 1 n (c) If is in the image of : , then is called an Proposition 14.2 of . We denote by the set of all -coboundaries. A ZG n ě HnpG;Aq – Z npG;Aq{BnpG;Aq

ProofLet : be a -module and 0. Then Z ZG.pFn;Aq Z C npG;Aq Compute cohomology via the bar resolution and replace the -module Hom by its isomorphic Remark-module 14.3 . The claim follows. n n n rg | ::: |gns gi n n n ZGpF n;Aq Ă ZGpFn;Aq C pG;Aq C pG;Aq Ď C pG;Aq If we used the normalised bar resolution instead, the -cochainsn are replaced by the -cochains 1 n Z pG;Aq vanishing on -tuples n having (at least) one of coefficient equal to 1. (This is because Hom n Hom B pG;A). Weq denote these by , and thus . The set of resulting normalised -cocycles is denoted by , and the set of resulting nor- n n n n n malised -coboundariesH bypG;Aq – Z .pG;A It followsq{B pG;A thatq – Z pG;Aq{B pG;Aq : Chapter 5. Easy Cohomology

G

In this short chapter we consider some cases in which the cohomology groups of a group have an easy interpretation. This is for example the case in low degrees (zero, one, two). Next we consider families References:of groups whose cohomology groups are easy to compute with the methods we have so far at our disposal. Cohomology of groups

[Bro94] K. S. BrownThe, cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. [Eve91] L. Evens, An introduction to the theory, Oxford of Mathematical groups. Fourth Monographs, ed. The Clarendon Press, Oxford University Press, New York, 1991. [Rot09] J. J. Rotman, An introduction to homological algebra , Graduate Texts in Mathema- tics, vol. 148, Springer-Verlag, New York, 1995. [Wei94] C. A. Weibel, , Cambridge Studies in Advanced Mathe- matics, vol. 38, Cambridge University Press, Cambridge, 1994. 15 Low-degree Cohomology

A. Degree-zero cohomology. H pG;Aq – AG G A

A ZG H pG;A0 q “ A We have already proved in Proposition 12.4 that0 , the -fixed points of . In particular, if is a trivial -module, then . B. Degree-one cohomology. H pG;Aq H pG;Aq “ Z pG;Aq{B pG;Aq

1 1 1 1 1-cocyclesUsing the bar resolution to compute yields .

˚ Z :pG;A By definition,q “ tf P andSet thepG description;Aq | d pfq of “ theu differential maps of the bar resolution, we have 1 “ tf P SetpG1;Aq | “ fpg rg sq ´ fprg g sq ` fprg sq @ rg |g s P G u Hom 2 0 1 2 f G ÝÑ A 1 2 1 2 1 1 2 Hom 0

In other words,1-cocycle a map : identity: is a f1p-cocycleg ¨ g q “ if andg ¨ f onlypg q if ` itfp satisfiesg q @theg ; g P G :

1 2 1 2 1 1 2

57 Lecture Notes: Cohomology of Groups SS 2018

58

1-coboundaries C pG;Aq “ Setptr su;Aq “ tfa tr su ÝÑ A; r s ÞÑ a | a P Au ÐÑ A

0 bij. ˚ : Hom d C pG;Aq ÝÑ: C pG;Aq . It follows that the differential˚ map 0 1 d pfaqpgq “ fapgr sq ´ fa1pr sq “ ga ´ a g P G a P A f G ÝÑ A : a P A fpgq “ ga ´ a g P G is such that 1 for every and every . Therefore, Definition: 15.1is a (1Derivation,-coboundary principal if and only derivation if there) exists such that for every . A ZG f G ÝÑ A

f derivation G pG;Aq Let be a -module and let : be a map. G A (a) If satisfies the 1-cocycle identity, then it is called a of . We denote by Der a P A fpgq “ ga ´ a g P G f the set of all derivations of to . principal derivation inner derivation G pG;Aq (b) If, moreover, there existsG A such that for every , then is called a (or an ) of . We denote by Inn the set of all Remark 15.2inner derivations of to . H pG;Aq – Z pG;Aq{B pG;Aq “ pG;Aq{ pG;Aq

1 1 1 ExampleIt follows 10 from the above that Der Inn . A ZG

Let be a trivial -module. In thisf case,pg ¨ h theq “ 1-cocyclefpgq ` fph identityq ; becomes

Z pG;Aq “ Grp pG; ¨q; pA; `q B pG;Aq “ 1 ` ˘ 1 so that Hom H pG;Aq “. FurthermoreGrp pG; ¨q; pA; `q : 0. Therefore 1 ` ˘ Hom C. Degree-two cohomology. H pG;Aq H pG;Aq “ Z pG;Aq{B pG;Aq

2 2 2 2 Again2-cocycles using the bar resolution to compute yields .

˚ Z:p ByG;A definition,q “ tf P andSet thepG description;Aq | d pf ofq “ theu differential maps of the bar resolution, we have

2 “ tf P SetpG2;Aq | “ fpg rg |g sq ´ fprg g |g sq Hom 3 0 2 ` fprg |g g sq ´ fprg |g sq @ rg |g |g s P G u 1 2 3 1 2 3 Hom 0 3 f G ˆ G ÝÑ A 1 2 3 1 2 1 2 3

In other2-cocycle words, identity: a map : g fpg ; g q `isfp ag2-cocycle; g g q “ iff andpg g only; g q if ` itf satisfiespg ; g q the @ g ; g ; g P G :

1 2 3 1 2 3 1 2 3 1 2 1 2 3 Lecture Notes: Cohomology of Groups SS 2018

59 2-coboundaries φ P C pG;Aq

˚ 1 d: Ifpφqprg |g sq “,φ thenpg rg sq ´ φprg g sq ` φprg sq @ rg |g s P G : 2 2f G ˆ 1G ÝÑ2 A 1 2 1 2 1 1 2 c G ÝÑ A

Therefore a map : fpg ; g q “isg ac2pg-coboundaryq ´ cpg g q if ` andcpg onlyq if there @ g ; g existsP G: a map : such that 1 2 1 2 1 2 1 1 2

16 Cohomology of Cyclic Groups

Cyclic groups, finite and infinite, are a family of groups, for which cohomology is easy to compute. Of course, we could use the bar resolution, but it turns out that in this case, there is a more efficient resolutionNotation: toA be used,ZG made up of freex P modulesZG of rank 1.mx A ÝÑ A; x ÞÑ x ¨ a x A x A If is a -module and , then we let : denote the left action ofPropositionon (or 16.1 left external (Free resolution multiplication of finite by cyclicin ). groups) n´ i Cn n P Zą g t “ i“ g P ZCn 1 mt mg´ 0 mt mg´ ř 0 Let be a finite cyclic¨ group ¨ ¨ of order/ ZCn / ZCgeneratedn / Z byCn , and/ letZCn :; . Then 1 1

ZCn ZCn

Proofis : a freeG “ C-resolutionn of the trivial -module. IG “ xtgi ´ | ď i ď n ´ uy “ xg ´ y : Set : . By Lemma 11.3, Z ZG m IG ε ZG ÝÑ Z g´ 1 1 1 1 n´ i n´ x “ i“ λi g P ZG tx “ i“ λi t 1 Therefore, the image1 of is equal to , which1 is the kernel of the augmentation map : . ř 0 ř 0n´ n´ Now, let . Then, . Hencei pmtq “ λig | λi “ 1 1 i“ i“ ÿ ÿ ( ker 0 mg0´ 0 pmg´ q Ď pmtq n´ i n´ h “ i“ λig P pmtq i“ λi “ h P pεq “ IG “ pmg´ q 1 1 and we claim that this is equal1 to the image of . Indeed,1 the inclusion Im ker is clear, ř ř 1 and conversely, if 0 ker , then 0 0, so that ker Im , Lecture Notes: Cohomology of Groups SS 2018

60 pmtq Ď pmg´ q pmg´ q “ pmtq

n´ 1 n´ 1 whence ker Im i . Finally, we claim that ker i Im . We have λig P pmg´ q ðñ pg ´ q λig “ 1 1 i“ ˜ i“ ¸ ÿ ÿ 1 n´ n´ 0 ker 1 i` 0 i 0 ðñ λig ´ λig “ 1 1 i“ i“ ÿ 1 ÿ n´ n´ 0 j 0 i 0 ðñ λj´ g ´ λig “ ; λ “ λn´ 1 1 j“ i“ ÿ 1 ÿ 1 1 n´ 0 where : 0 0i ðñ pλi´ ´ λiqg “ 1 i“ ÿ 1 ðñ @ ď i ď n ´ ; λi´ “ λi “ λ 0 0 n´ n´ i 1 i ðñ 0 λig “ λt ðñ1 λig P: pmtq: 1 1 i“ i“ ÿ ÿ Im Theorem 16.2 (Cohomology of finite cyclic groups0 ) 0 n Cn “ xg | g “ y n P Zą A ZCn

ACn m0 ; Let 1 be a finite cyclic group of order “and let be a -module. Then HmpC ;Aq – ACn { pm q m ě ; m ; n $ t & pmtq{ pmg´ q if m ě 0; m ; Im if 2 even n´ i % 1 t “ i“ g P ZCn x P tkert; g ´ uImmx if 1 odd x A ř 1 Proofwhere : 0 and forZG Z 1 , denotes left external multiplication by in .

mt mg´ mt mg´ By Proposition 16.1 the::: trivial / Z-moduleCn / ZadmitsCn the/ Z projectiveCn / Z resolutionCn : 1 1

Cn m “ H pCn;Aq “ A m ą ZCn p´;Aq 0 For 0, we already know that . For 0, applying the functor Hom yields m˚ ˚ m˚ the cochain complex g´ mt g´ ZCn pZCn;Aq / ZCn pZCn;Aq // ZCn pZCn;Aq / ¨ ¨ ¨ ; 1 1 – Hom Hom ZCn pZCn;AqHomÝÑ A; f ÞÑ fp q x P tg´ ; tu

where in each degree there is an isomorphism Hom ˚ 1 . Hence for 1 , mx there are commutative diagrams of theZCn formpZCn;Aq / ZCn pZCn;Aq

– ö – Hom Hom  m  A x / A:

mg´ mt mg´ mt Hence, the initial cochain complex is isomorphicA to/ A the cochain/ A complex/ A / ¨ ¨ ¨ 1 1

(degree) 0 1 2 3

and the claim follows. Lecture Notes: Cohomology of Groups SS 2018

61

ForTheorem infinite 16.3 cyclic (Cohomology groups the of situation infinite is cyclic even groups simpler:)

mg´ G “ xgy / ZG / ZG ZG 1 G If is an infinite cyclic group, then 0A n “ is; a free resolution of the trivial HnpG;Aq “ A{ pm q n “ ; -module, and $ g´ & if n ě 0: 1 Im if 1 mg´ % g ´ A 0 if 2 1 Proofwhere : denotes the left external multiplication by 1 in .

Exercise 1, Exercise Sheet 9. Chapter 6. Cohomology and Group Extensions

ÝÑ A ÝÑ E ÝÑ G ÝÑ G InA this chapter we consider connections between the short exact sequencesH ofp groupsG;Aq of the form 1 1 with abelian kernel and the cohomology of the groupH p1G;Awithq coefficients in . If the sequence splits, then we shall proveÝÑ thatA ÝÑ theE 1stÝÑ cohomologyG ÝÑ group 2 parametrisesZG the splittings.A Moreover, we shall also proveH p thatG;Aq the 2nd cohomology group is in bijection Ewith the isomorphism classes ofA extensionsG 12 1 inducing the given -module structure on , and the neutral element of corresponds, under this bijection, to a s.e.s. where References:is a semi-direct product of by . Cohomology of groups

[Bro94] K. S. BrownThe, cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. [Eve91] L. Evens, An introduction to the theory, Oxford of Mathematical groups. Fourth Monographs, ed. The Clarendon Press, Oxford University Press, New York, 1991. [Rot09] J. J. Rotman, An introduction to homological algebra , Graduate Texts in Mathema- tics, vol. 148, Springer-Verlag, New York, 1995. [Wei94] C. A. Weibel, , Cambridge Studies in Advanced Mathe- matics, vol. 38, Cambridge University Press, Cambridge, 1994. 17 Group Extensions

G N H

In Chapter 1, we have seen that if a group is a semi-direct product of a subgroup by a subgroup , then this gives rise to a s.e.s. of the formÝÑ N ÝÑ G ÝÑ H ÝÑ :

1group extension of N by H1

DefinitionThis is a special 17.1 (Group case of extension a so-called) . group extension

i p A is a short exact sequence/ A of/ E groups/ (writtenG / multiplicatively); of the form

1 E extension1 of A by G Convention: A E i and, in this situation, we also say that the group is an . We shall always identify with a normal subgroup of and assume that is simply

62 Lecture Notes: Cohomology of Groups SS 2018

A E A kernel 63

Lemmathe canonical 17.2 inclusion of in . Moreover, we shall say that is the of the extension. i p / A / E / G / A A ZG Let 1 1 be a group extension, where is an abelian group. Then Proofis : naturally endowed with the structurepA; of¨q a -module. g P G g P E g p ppgq “ g G A First note that with the above notation is a group written multiplicatively. Next, for each , ˚ G ˆ A ÝÑ A choose a preimage r of under , that is r , and define a left -action on via: pg; aq ÞÑ g ˚ a “ ga “ g ¨ a ¨ g´

: 1 ˚ : : r r , g P E ppgq “ g First, we check that is well-defined, i.e. that this definition is independent of the choice of the preimages: p g ¨ g´ “ g ¨ g´ “ indeed, if p is such that p , then, we have G 1 1 ´ ` ˘ g ¨ g P ppq “ A r p a P A 1 g “ ag x P A 1 g ¨ x ¨ g´ “ a gxg´ a´ “ aa´ gxg´ “ gxg´ ; hence r p ker , and thus, there exists such that r p. Therefore, for every , 1 AIJE 1 1 1 1 1 r r p p p p p p loomoonA ˚ Z ZG pA; ¨; ˚q ZG where the last-but-one equality holds because is abelian. We extend by -linearity to the whole of , and finally one easily checks that is a -module. i p Convention:See Exercise 2, Exercise Sheet 10. / A / E / G / A A ZG G A˚ “ pA; ¨; ˚q From now on,A givenZG a group extension 1 1 with abelian, we always see as a -module via the -action of the proof of Lemma 17.2. We write : toLemma indicate 17.3 that we see as a -module in this way. i p / A / E / G / A A E A˚ ZG Let 1 1 be a group extension with abelian. Then, is central in g ´ Proof :ifA˚ and only if ZGis trivialðñ as a a “-module.a @ a P A; @ g P G ðñ g ¨ a ¨ g “ a @ a P A; @ g P E ðñ ga “ ag 1 @ a P A; @ g P E is a trivial -module ðñ Ar Ď ZprEq : r r r r

Definition 17.4 (Central extension)

i p / A / E / G / A central extension A G A group extension 1 1 be a group extension with abelian satisfying Definitionthe equivalent 17.5 (Split conditions extension of Lemma) 17.3 is called a of by . i p / A / E / G / splits s G ÝÑ E p ˝ s “ G s (group-theoretic) section p splittingA group extension 1 1 iff there exists a group homomorphism : such that Id . In this case is called a of , or a of the extension. Lecture Notes: Cohomology of Groups SS 2018

Proposition 17.6 64

i p / A / E / G /

Let 1 1 be a group extension. Then the following are equivalent: H E p H ÝÑ G (a) The extension splits. H H E Eˇ A H (b) There exists a subgroup of such that ˇ : is an isomorphism. H E e P E e “ ah (c) There exists a subgroup of such that is the internal semi-direct product of by . a P A h P H (d) There exists a subgroup of such that every element can be written uniquely Proof : with and .

ñ s G ÝÑ E p H “ s p H p ˝ s “ s h P H H G ˇ (a) (b): By (a)g thereP G exists a sectionh “ spg: q for . Define : Im . Then ˇ is an isomorphism ˇ ˇ since, on the one hand ps ˝ pId qphbyq “ definition ps ˝ pqps ofpgqq, “ andspg onq “ theh other hand for every , there exists such that H , so that s ˝ p “ ˇ H H ˇ

ñ ˇ H ď E p H H ÝÑ G and ˇ Id . t u “ ˇp “ ppq X H “ A X H: (b) (c): By (b) there is such that ˇ H: is an isomorphism. Hence ´ ´ e P E ppeq P G ñ p ` ˇ ˝˘ ppeq P H ppeq “ p p ˝ ppeq 1 kerH ˇ ker H 1 1 ` ˇ ˘ ´ ` ˇ ˘ ˇ ´ ˇ Now, let . Then e ¨ p H ˝ ppeq Pand p “ A: , so that 1 ´ 1 ¯ a P A ` ˇ ˘ ˇ ker ´ e “ a ¨ p ˝ ppeq P AH Therefore, there exists such that H 1 ´` ˇ ˘ ¯ ˇ PH looooooooomooooooooon ñ as required. ñ p H H ÝÑ G (c) (d): Was proven in Step 1 of the proof of Proposition 1.3. g P G ˇ p e P E g “ ppeq (d) (b): We have to prove that ˇ : is an isomorphism. e e “ ah a P A h P H p H Surjectivity: Let . Then by surjectivity of there exists such that , and by (d), g “ ppeq “ ppahq “ ppaqpphq “ ¨ pphq “ pphq : ˇ can be written in a unique way as with and . Hence ˇ is surjective since h P H p phq “ h P ppq “ A H 1 ˇ h “ ¨ h “ h ¨ P AH Injectivity: If is such that ˇ 1, then ker , therefore h “ pp q “ t u 1 1 H ñ p H H ÝÑ G ˇ s “ p H G ÝÑ E so that by uniqueness, we must have 1 and ker ˇ 1 . ˇ ` ˇ ˘ (b) (a): If ˇ : is an isomorphism, then we may define : ˇ : . This is obviously a group homomorphism and hence a splitting of the extension. H

If the equivalent conditions of the Proposition are satisfied, then there is a name for the subgroup , it is called a complement: Lecture Notes: Cohomology of Groups SS 2018

Definition 17.7 (Complement of a subgroup) 65 E A E H E complement A E E “ AH A X H “ E A H Let be a group and be a normal subgroup of . A subgroup of is called a of Remarkin 17.8if and 1, i.e. if is the internal semi-direct product of by . p i i UnlikeE short– A ˆ exactG sequences of modules, it is not true that admits a group-theoretic section if and only if admits a group-theoretic retraction. In fact, if admits a group-theoretic retraction, then . (See Exercise Sheet 10.) 18 H and Group Extensions

1

In order to understand the connexion between/ A the/ E group extensions/ G / of the form

H pG;A˚q1 1 E 1 Definitionwith abelian 18.1 kernel (Inner and automorphisms,, first automorphisms we need to investigate inducing the the automorphisms identity) of . E

x P E c E ÝÑ E; e ÞÑ xex´ E x Let be a group. x pEq “ t' P pEq | Dx P E '1“ c u (a) Given , write : forx the automorphism of of conjugation by . A ď G pEq “ t' P pEq | Dx P A ' “ c u (b) Set Inn : AutA with . x i p (c) If ,/ thenA set/ InnE / G: / Aut with .

(d) If 1 A;GpEq “ t' P pE1 qis | ' a|A group“ A extensionp ˝ ' withpeq “abelianppeq @ kernel,e P Eu then: set

Aut : ' Aut A;GpEq induceId theand identity on both A and G

We say that the elements of Aut .´ Einführung in die Algebra pEqE pEq '˝cx ˝' “ c'pxq x P E ' P pEq pEq{ pEq outer automorphism1 groupRecall (e.g.E from the pEq – E{ZpEq -lecture) that: Inn A;GAutpEq ď, as pEq for every and every Aut , and the quotient Aut Inn is called the Theoremof 18.2. Moreover, (H1 andInn automorphisms) . It is also obvious that Aut Aut .

i p / A / E / G /

H pG;A q – pEq{ pEq Let 1 ˚ A;G A 1 be a group extension with abelian kernel. Then: 1 (a) Aut Inn ; and

(b) if, moreover, the extension is a centralH pG;A extension˚q – thenA;GpEq : 1 Aut Lecture Notes: Cohomology of Groups SS 2018

66 Proof :

Claim 1: ApEq E A;GpEq a P A ca|A “ A A (a) Inn Aut . ´ ´ ´ p ˝ capeq “ p aea “ ppaqp ea e ppeq “ ppeq Indeed, clearly for each , Id because is abelian and, moreover, 1 1 1 e P E p ˝ ca “ p` ˘ A E` ď A;G˘pEq

´ for every , so that .' Therefore˝ ca ˝ ' Inn“ c'paq “Autca , and it is a normal subgroup,

because a P A ' P A;GpEq '|a “ 1A

Claim 2: A;GpEq – Z pG;A˚q for every , every 1 Aut as Id . Aut . α A;GpEq ÝÑ Z pG;A˚q We aim at defining a group isomorphism ¨ ' P A;GpEq 1 x P E 'pxq “ fpxqx fpxq P E : Aut . To begin with, we observe that givenf E AutÝÑ E and , we can write for some element . This defines a mapx (ofÞÑ sets)'pxqx´ pfq Ď A “ ppq : x P E 1 ´ , ´ ppfpxqq “ pp'pxqx q “ pp'pxqq ppx q “ G such that Im ker because for every , 1 “ppxq 1 1 ' G f looomooon E A fpxaq “ 'pxaq ¨ pxaq´ “ 'pxq ¨ 'paq ¨a´ ¨ x´ “ 'pxqx´ “ fpxq : since induces the identity on . Moreover, is constant on the cosets of modulo because 1 “a 1 1 1 f f G ÝÑ A; g ÞÑ loomoonfpgq “ fpgq g p´ pgq g; h P G gh P p´ pghq g P p´ pgq ´ Thereforeh1 P p inducesphq a mapgh:“ gh : r where we may choose1 r arbitrarily1 in . This is a 1-cocycle since for all , we may choose Ă , r , 1 fpghq “ f gh “ f ghq “ 'pgq ¨ 'phq ¨ h´ ¨ g´ and r such that Ă rr, and hence ´ ´ g ` ˘ ` “ 'pgq ¨ g ¨ g ¨ f1phq ¨ g1 “ fpgq fphq ; Ă rr r r r r 1 1 r r r r ¨ which is the 1-cocycle identity in multiplicative notation. αp'q “ f G ÝÑ A :

As a consequence, we set ` ' ˘;' P A;GpEq f ; f G ÝÑ A : : αp' q “ f αp' q “ f 1 2 To prove that this defines a group homomorphism, let Aut ´and respectively let 1 2 ' pgq “ f pgqg; ' pgq “ f pgqg @1 g P G1 g P2p pg2q ; : be the associated 1-cocycles, i.e. and . Then 1 1 1 A 2 2 r r r r with r αp' ˝ ' qpgq “ p' ˝ ' qpgqg´ “ ' f pgqg g´ and hence using the fact that is abelian yields 1 ´ 1 1 2 1 2 “ f 1p`gq2' pgq˘g r r r r ´1 “ f 2pgqf 1pgqgg r r 1 “ f 2pgqf 1pgq rr “ αp' qpgq ¨ αp' qpgq “ αp' q ¨ αp' q pgq ; 2 1

1 2 ` 1 2 ˘

as required. Lecture Notes: Cohomology of Groups SS 2018

67 ¨ α

β Z pG;A˚q ÝÑ A;GpEq In order to prove that is an1 c isomorphism,ÞÑ weβp definecq E ÝÑ E; g ÞÑ cpgq g ; g “ ppgq : Aut βpcq : r g; hr P E where r . r First, we check that is indeed a group homomorphism:g for r with the above notation, we have βpcq g ¨ h “ cpghqgh “ cpgq ¨ cphq ¨ gh 1-cocycle id. ` ˘ “ cpgqgcphqg´ gh r r rr rr “ cpgqgcphqh 1 r r rr “ βpcqpgq ¨ βpcqphq : r r g P A “ ppq g “ G r r βpcqpgq “ cp q ¨ g “ ¨ g “ g ; Next, if r ker , then 1 and therefore

cp Gq “ A p G q r 1 r 1 r r h P G cp G ¨ hq “ cp Gq ¨ cphq “ cp Gqcphq where we useβpc theq|A fact“ A that a1 1-cocycle is always normalised (indeed 1 1 , since for , 1 cpgq1 P A “ ppq ppc1pgqq “ Gby the 1-cocycle identity). Thus we have proved that Id . p ˝ βpcq pgq “ p cpgq ¨ g “ p cpgq ¨ppgq “ ppgq Furthermore, since ker , 1 and we get ` ˘ ` ˘ `“ G ˘ r r looomooon r r p ˝ βpcq “ p βpcq 1 G

EandÝÑ so E , or in other wordsA inducesG the identity on . Finally, usingβpc Exerciseq P A;G 2(c),pEq Exercisec SheetP Z pG;A 10, we˚q obtain that any group homomorphism ¨ inducing the identityα β on and on 1 must be an isomorphism. Therefore, we have proved that Aut for every . pα ˝ βqpcq pgq “ βpcqpgq ¨ g´ “ cpgqgg´ “ cpgq @ g P G; @ c P Z pG;A q ; It remains to prove that and are inverse to each other. Firstly, ˚ 1 1 1 `α β ˘ Z G;A ˝ r pr ˚q rr pβ ˝ αqp'q pgq “ pαp'qqpgq1 ¨ g´ “ 'pgqgg´ “ 'pgq @g P E; @ ' P pEq ; so that is the identity on . Secondly, A;G 1 1 ` β ˝ α ˘ pEq r A;Gr rr r Aut Claim 3: ApEq – B pG;A˚q so that is the identity on Aut . 1 ¨ a P A ca P ApEq g P G Inn . αpc qpgq “ c pgq ¨ g´ “ a ¨ g ¨ a´ ¨ g´ “ ga´ g´ ¨ a “ gpa´ qa “ d˚pa´ qpgq Let a and aInn . Then for every , 1 PAIJ1 E 1 1 1 1 1 1 r r loooooomoooooonr r r r αpcaq P B pG;A˚q α A E Ď B pG;A˚q a P A d˚paq P B pG;A q d˚paqpgq “ ga ¨ a´ 1 ` ˚ ˘ 1 and˚ therefore ˚ g, i.e.1 ´ Inn ´ ´ . 1 ´ ´ β d paq pgq “ d paqpgq ¨ g “ a ¨ a ¨ g “ g ¨ a ¨ g ¨ a ¨ g “ g ¨ g a g ¨ a “ c ´ pgq: Conversely, if and 1 , then 1 and a ` ˘ 1 1PAIJE1 “ 1 1 1 1 r 1 r r r r r r r r r ˚ loooooomoooooon loomoon β d paq “ ca´ P ApEq β B pG;A˚q Ď ApEq 1 ApEq pEq α β ` ˘ A;G 1 ` 1 ˘ Hence 1 Inn , and Inn . It follows that Inn H pG;A q “ Z pG;A q{B pG;A q – pEq{ pEq : corresponds to Aut ˚under the bijection˚ given˚ by andA;G , and weA obtain 1 1 1 Aut Inn Lecture Notes: Cohomology of Groups SS 2018

68 A E a P A a ´ ´ ca E ÝÑ E; e ÞÑ aea “ aa e “ e E (b) If is a central subgroup of1 , then1 for every the conjugation automorphism by is given pEq “ tc E ÝÑ E | a P Au “ t u by : A a, i.e. the identity on . ThusE

Inn : Id H pG;A q – pEq{ pEq “ pEq : and it follows from (a) that ˚ A;G A A;G 1 Aut Inn Aut

TheoremWe are now 18.3 ready (H1 toand parametrise splittings) the slpittings of split group extensions with abelian kernel:

i p E‚ “ p / A / E / G / q

Let : 1 1 be a split group extension with abelian kernel. Then H pG;A q S A the following holds: ˚ 1 (a) There is a bijection between and the set of -conjugacy classes of splittings of H pG;A q E the given extension. ˚ A E 1 (b) There is a bijection between and the set of -conjugacy classes of complements Proof : of in . s G ÝÑ E

0 α A;GpEq ÝÑ t E‚u (a) Choose a splitting : and define' a mapÞÑ ' ˝ s : Aut splittings of α '˝s 0 p's “ ps “ G . 0 0 0 It is obvious that is well-defined, i.e. that is a splitting of the extension as Id . β t E‚u ÝÑ A;GpEq Define a second map s ÞÑ ψs E ÝÑ E; as pgq ÞÑ aspgq : splittings of Aut x`P E 0 ˘ x “ as pgq : , a P A g P G β ψs 0 wherex by“ Propositiona s pg q; x17.6“ aans arbitrarypg q P E element can be written in a unique way as with and . We check that is well-defined. Firstly, is a group homomorphism: for ψ1px ¨ 1x q0 “1ψ 2a s p2g 0q ¨ a2 s pg q “ ψ a ¨ g a ¨ s pg g q every s s , we haves “ a ¨ g a ¨1 s g g 1 2 ` 1 0 1 2 0 2 ˘ ` 1 2 0 1 2 ˘ “ a spg1 q ¨ a spg q 1 2 ` 1 2˘ “ ψ a s pg q ¨ ψ a s pg q “ ψ px q ¨ ψ px q : 1s 1 2 2 s s s

ψs|A “ A pψs`“1p0 1 ˘ x`“2 as0 p2gq˘ P E 1 2

pp ˝ ψ qpxq “ pp ˝ ψ qpas pgqq “ ppaspgqq “ ppaq ¨ ppspgqq0 “ g “ ppas pgqq : Secondly, s Id by definition.s Thirdly, since for , we have “ “ pgq 0 G 0 loomoon loomoon ψs 1 Id ψs A G β Finally, the fact thatα isβ an isomorphism follows again from Exercise 2(c), Exercise Sheet 10 because induces the identity on both and . Whence is well-defined. pα ˝ βqpsq “ αpψ q “ ψ ˝ s @ s P t E u Next, we check that and are inverses to eachs other. On the one hand, ‚

0 splittings of Lecture Notes: Cohomology of Groups SS 2018

69 g P G pψs ˝ s qpgq “ ψsp A ¨ s pgqq “ Gspgq “ spgq α ˝ β E‚ ' P A;GpEq 0 0 but for every , 1 1 , hence is the identity on pβ ˝ αqp'q “ βp' ˝ s q “ ψ'˝s the set of splittings of . On the other hand, for every Aut , we have 0 x “ as pgq P E a P A g P G 0 '| “ ψ as0 pgq “ a ¨ p' ˝ s qpgq A“ A 'paq ¨ p' ˝ s qpgq “ ' a ¨ s pgq ; and for each '˝s (with and ), we have Id 0 β ˝ α ` 0 ˘ A;GpE0 q 0 ` 0 ˘

α / hence is the identity onA;GAutpEq o . t E‚u Therefore, β Aut splittingsA ofpEq 1 ' P A;GpEq cb P ApEq b P A ' “ cb ˝ '

are bijections (of sets). Finally,αp'q “ we' ˝ determines theα behaviourpcb ˝ 'q “ ofcbInn˝ ' ˝ s under: these bijections. Let Aut and Inn with . Let . Then ApEq 0 α 0 A and

Hence a coset modulo Inn A ˆis t mapped via E‚tou an ÝÑ equivalence t class forE‚u the action by conjugation of on splittings pb; sq ÞÑ cb ˝ s

splittings of splittings ofA;GpEq ApEq .

Thus passing to the quotient (group quotient„ on the left hand side Aut , and orbits of Inn pEq{ tA E u A;G ApEq / ‚ on the right hand side)O yields a bijection p : q – Aut -conjugacy classes of splittings of  Inn Thm. 18 2 H pG;A˚q 1

s spGq A E as required. ´ H A E p H G ÝÑ H (b) By PropositionA 17.6, a splittingH of the extensionE corresponds to a complementH 1 of einP E, and ` ˇ ˘ ´ ´ conversely, a complement ofe “inah correspondsa P A toh aP splittingH ˇeHe : “ aHa . Moreover, the -conjugacy class of is the same as the -conjugacy class of , because1 every1 may be written in a unique way as with and and so . The claim follows. 19 H and Group Extensions

Convention:2

Definition 19.1In this (Equivalent section all group group extensions extensions) are assumed to have abelian kernel. 1 i p i1 p / A / E / G / / A / E1 / G / equivalent ' E ÝÑ E1 Two group extensions 1 1 and 1 1 with abelian kernels are called if there existsp a group homomorphism : such A i E G that the following diagram commutes / / / /

A ö ' ö G 1  1 1  i 1 p  1 Id/ A / E / GId / :

1 1 Lecture Notes: Cohomology of Groups SS 2018

Remark 19.2 70

'

(a) In the context of Definition 19.1E , the homomorphism is necessarily bijective. However an isomorphismA of groups doesG not induce an equivalenceZG of extensions in general.A In other words, the same middle group can occur in non-equivalent group extensions with the same kernel , the same quotient and the same induced -module structure on .

Notation:(b) EquivalenceG of group extensionsA˚ “ pA; ¨ is; ˚q an equivalenceZG relation. EpG;A˚q

If is a group and : iis a p-module (which may see simply as an abelian group), then we let denote the set/ A of equivalence/ E / G classes/ of group extensions ZG A 1 1 inducingTheorem the 19.3 given -module structure on . G A˚ “ pA; ¨; ˚q ZG

„ Let be a group and let : H pG;Abe˚ aq fixedo -module/ EpG;A (written˚q : multiplicatively). Then, there

is a bijection 2 H pG;A˚q

2 ProofMoreover, : the neutral element ofEpG;A˚q ÝÑ HcorrespondspG;A˚q to the class of the split extension. ¨ 2 We want to define a bijection . i p To begin with, fix an extension / A / E / G / ˚ A s G ÝÑ E p 1 1 p ˝ s “ G s inducing the given action on , andspg weq ¨ choosesphq “ afp set-theoreticg; hq ¨ spghq section : for , i.e. such that Id . Possibly is not be a group homomorphism, but we may write fpg; hq P E f G ˆ G ÝÑ E for some element p.g; This hq defines ÞÑ afp mapg; hq “ spgq ¨ sphq ¨ spghq´ :fpg; hq P A “ ppq 1 : . ´ p fpg; hq “ p spgqsphqspghq´ “ p spgq ¨ p sphq ¨ p spghq “ ghh´ g´ “ Furthermore, notice that ker because G 1 1 1 1 ` g; h P˘G ` f P SetpG˘ ˆ G;A` q ˘ ` ˘ ` ˘ f 1 spgq ¨ sphq ¨ spkq “ fpg; hq ¨ spghq ¨ spkq “ fpg; hq ¨ fpgh; kq ¨ spghkq for every . Hence Hom , and as a matter of fact, is a 2-cocycle because: ` ˘ spgq ¨ sphq ¨ spkq “ spgq ¨ fph; kq ¨ sphkq “ spgq ¨ fph; kq ¨ spgq´ ¨ spgq ¨ sphkq and g ` ˘ “ fph; kq ¨ fpg; hkq ¨ sp1ghkq: E fpg; hq ¨ fpgh; kq “ gfph; kq ¨ fpg; hkq; Therefore, by associativity in , we obtain

s c G ÝÑ A which is precisely the 2-cocycle identity in multiplicative notation. Now, we note that if we modify by a 1-cochain : and define Lecture Notes: Cohomology of Groups SS 2018

s1 G ÝÑ E 71 g ÞÑ s1pgq “ cpgq ¨ spgq : : , f 1pg; hq “ s1pgq ¨ s1phq ¨ s1pghq´ then the corresponding 2-cocycle is given by “ cpgq ¨ spgq ¨ cphq ¨ sph1q ¨ spghq´ ¨ cpghq´ “ cpgq ¨ spgq ¨ cphq ¨ spgq´ spgq ¨ s1phq ¨ spgh1q´ ¨ cpghq´ “ cpgq ¨ spgq ¨ cphq ¨ spg´ 1q ¨ fpg; hq ¨ cpghq´ 1 1 “ cpgq ¨ gcphq ¨ cpghq´ ¨ f1pg; hq A 1 “ gcphq ¨ cpghq´ ¨ cpg1q ¨ fpg; hq A ˚ as is abelian “ pd pcqqpg; hq ¨1fpg; hq @ g; h P G:

as is˚ abelian 2 f d pcq rfs “ fB pG;A˚q f H pG;A˚q 2 To sum up, we have2 modified the 2-cocycle2 by the 2-coboundarys . Therefore, the cohomology

class : ξof in EpG;Ais˚q well-defined, ÝÑ dependingH pG;A on˚q the given extension, but does not depend on the choice of the set-theoretici p section . Hence, we may define a map r / A / E / G / s ÞÑ rfs2 : ¨ ξ 1 1 . p 1 p1 i i 1 We check that ris well-defined./ A / E / G Suppose/ s “ that r we/ A have/ E two equivalent/ G / s P extensionsEpG;A˚q ;

1 1 1 1 i p that is a commutative diagram of the form/ A / E / G /

A ö ' ö G 1 1 1  i1  p  Id/ A / E1 / GId / ' E ÝÑ E1 s G ÝÑ E 1 1 1 1 p ' ˝ s p p ˝ ' ˝ s “ p ˝ s “ G where is an isomorphism of . As above, we choose a set-theoretic section : of , andf 1pg; it h followsq “ p' ˝ thatsqpgq ¨ p'is˝ asqp set-theoretichq ¨ p' ˝ sqpgh sectionq´ “ for' sp,g sinceq ¨ sphq ¨ spghq´ Id . The corresponding 2-cocycle is given by 1 “ '`fpg; hq “ fpg; hq1˘ @ g; h P G

'|A “ A ξ ` ˘ ¨ s G ÝÑ E sp q “ as Id . Hence is well-defined.s c G ÝÑ A ˚ Remark:cp q “ We mayd choosepcq : is such that 1 1, and the associated 2-cocycle is normalised. Now if we modify by a normalised 1-cochain : (i.e. such that 2 ¨ 1 1), thenξ is a normaised 2-coboundary. Therefore, we may as well use normalised cocycles/cochains/coboundaries. α P H pG;A˚q f G ˆG ÝÑ A α “ rfs ESurjectivityf “ A ˆ2G of : Let pa;and gq choose ¨ pb; hq a “ normaliseda ¨ gb ¨ fpg;2-cocycle hq; g ¨ h : @ a; b P A; @suchg; h thatP G: . Construct : (as a set), which we endow with the product pEf ; ¨q ` p ; q˘

Then is a group whose neutral element is 1 1 . (Exercise, Exercise Sheet 11) Clearly there i A ÝÑ Ef ; a ÞÝÑ pa; q ; are group homomorphisms: p Ef ÝÑ G; pa; gq ÞÝÑ g : 1 : Lecture Notes: Cohomology of Groups SS 2018

72 ppq “ piq

i p such that ker Im , thus we get a/ A group extension/ Ef / G / :

1 1 rfs s G ÝÑ Ef ; g ÞÝÑ p ; gq We need to prove that theg; h cohomologyP G class of the 2-cocycle induced by this extension via the above construction is precisely . So consider the set-theoretic section : 1 spgq ¨ sphq ¨ spghq´ “ p ; gq ¨ p ; hq ¨ p ; ghq´ and compute that for all , we have ´ 1 “ ¨ g ¨ fpg; hq; gh ¨ p1ghq fpgh; pghq´ q´ ; pghq´ 1 1 1 ´1 “` pfpg; hqpghqpghq fpgh;˘ `pghq´ q; pghqpgh1q´ 1q 1˘ 1 1 1 “ fpg; hq; 1 1 ` ˘ 1

¨ as required. ξ

i p Injectivity of : r / A / E / G / s ;

Let i p r1 / A / E / G /1 s ˜ ˜ EpG;A q s G ÝÑ E s G ÝÑ E 1 ˚ ˜ 1 f f be two classes of group extensions in . Choose, respectively, : and ˜ : ˜ rfs “ f P H pG;A q : two set-theoretic section with corresponding 2-cocycles ˚and ˜ respectively. Now, assume that “ ‰ 2 f “ d˚pcq ˝ f c ˜ G ÝÑ A s s G ÝÑ E; g ÞÑ cpgq´ ¨ spgq f d˚pcq´ ˝ f ˚ ´˜ 2 Thend pcq ˝ f “ f for some1 1-cochain : .1 Changing the choice of ˜ by defining ˜ ˜ ˜ 2 ˜ ˜ : 1 ˜ modifiesE into E by the first part of thea proof.¨ spgq But a2P A ˜g P G, therfore,s weG mayÝÑ assumeE without lossp ofE generalityÝÑ G that the two 2-cocycles are the same. Compute the group law in : each element of can be written uniquely as for aspgq ¨ bsphq “ aspgqbspgq´ spgqsphq and because : is a section for : . Hence the product is g “ a bspgqsphq 1 “ agbfpg; hq spghq

PA loooomoooon Ef Ef – E pa; gq ÞÑ a ¨ spgq A G which isE exactely– Ef the group law in . Hence (viaξ ) as groups, but also ¨ as extensions, because the latter isomorphismξ induces the identity on both and . Similarly, we get that ˜ , as group extensions. The injectivity of follows. Finally notice that the image under / Aof the split/ A ¸ extensionG / G / G A ˚ 1 1 G swhereG theÝÑ actionA ¸ G; of g ÞÑon p ; gisq given by , and where the first map is the canonical inclusion and the secondf G mapˆ G theÝÑ projectionA; pg; hq ÞÑ onto , is trivial. This is because we can choose a section : 1 , which is a group homomorphism. Therefore the corresponding 2-cocycle is : 1. This proves the 2nd claim. Lecture Notes: Cohomology of Groups SS 2018

Remark 19.4 73

s G ÝÑ E sp q “ s G ÝÑ A c G ÝÑ A cp q “ (a) In the above proof, if we choosedc : such that 1 1, then we obtain a normalised 2-cocycle. If we modify : by a 1-cocycle : such that 1 1 (a normalized 1-cochain), then is a normalized 2-coboundary. So we see that we can use A H G;ZpAq normalized cochains, cocycles and coboundaries throughout. 3` ˘ (b) If the group is not abelian, then comes into play for the classification of the Exampleextensions. 11 This is more involved. n n ě central extensions G “ D n´

ForA example,“ C if we want to find all 2-groups of order 2 ( 3) with a central subgroup of order 21 and a corresponding dihedral quotient, then we have to classify the of : 2 2 by : . By Theorem 19.3 theÝÑ isomorphismC ÝÑ P classesÝÑ D ofn´ centralÝÑ extensions: of the form 1 2 Z2 H pG;A˚q 1 A˚ G 1 H pG;A˚q – pZ{ q 2 2 D n´ n´ xareρ; σ in|3 bijectionρ “ “ withσ ; pρσq “, wherey is theP trivial -module. Computations yield 1 2 , hence there are 8 isomorphism2 classes of such extensions. Since a presentation of 2 is 2 2 2 a b n´ c xr;1 s; t | rt “ tr; st “ ts;1 , t obviously“ ; r “ t admits; s “ t a; presentationprsq “ t ofy; the a; b; form c P t ; u : 2 2 2 2 2 a; b; c 1 P 0 1

a “ b “ c “ C ˆ D n´ Letting vary, we obtain the following groups : 1 a “ b “ c “ 2 D n (i) The case 0 gives the direct product 2 .

a “ c “ b “ b “ c “ a “ 2 pC n´ ˆ C q ¸ C (ii) The case 0, 1 gives the dihedral group . 2 a “ b “ c “ b “ a “ c “ 2 2 2 SD n (iii) The casesn 0, 1 and 0, 1 give the group . 2 (iv) The cases 0, 1 and 0, 1 both give the semi-dihedral group c “ a “ b “ C n´ ¸ C of order 2 . 2 a “ b “ c “ 4 Q n (v) The case 0, 1 gives the group 2 . 2 (vi)n ě The case 1 gives the generalised quaternion group n “. D If 4, theC groupsˆ C in cases (i)-(vi) are pairwise non-isomorphic. If 3 the above holds as 8 well, but the groups in (ii) and (iii) are all isomorphic to , and the groups in (iv) and (v) are all 2 4 isomorphic to . Chapter 7. Subgroups and Cohomology

G H ď G G ThroughoutG this chapter, unlessH otherwise stated, denotes a group in multiplicativerestriction, notation transfer and (or corestriction),a subgroup induction of . Thecoinduction aim of the chapter is to investigate relations between the cohomology of and the cohomology of . This can be done using four operations called H G , and . As our next aim in the lecture is to prove theorems about finite groups using cohomology, we will most of the time work under the mild assumption that has finiteReferences: index in . Cohomology of groups

[Bro94] K. S. BrownThe, cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. [Eve91] L. Evens, , Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1991. 20 Restriction in Cohomology

Notation 20.1 (Restriction of ZG-modules) M ZG ZH Ď ZG ZG M ZH M M ZH G G Let be a M-module.ÓH BecauseH pMq is a subring, by [Exerciserestriction 3(a), of ExerciseM from G Sheetto H 4] we may restrict the action of on to an action of on ZGand regardZH as a -module, which

we denote byG or Res . This operation is called the . (In H ZGMod ÝÑ ZH Mod other words, restriction just forgetsG about the elements of outside .) M ÓH – ZG bZG M ZG pZH; ZGq Notice that Res : is a covariant functor, which is a special case of a forgetful functor.Definition One 20.2 can (Left also transversal, prove that right transversal), where is seen as a -bimodule.

left transversal H G tgiuiPI H G G “ iPI giH right transversal H G A H G of in is a set of representatives of the left cosets of in . Thus š . Similarly, a of in is a set of representatives of the rightG cosetsH of in . We want to investigate how restriction of modules interacts with the cohomology the groups and . To this end, first we need to understand restriction of projective resolutions.

74 Lecture Notes: Cohomology of Groups SS 2018

Lemma 20.3 75 G P ZG P ÓH ZH Let be a free (resp. projective) right -module. Then is a free (resp. projective) right Proof : P ÓG ZH P “ ZG ZG -module. (Similarly for leftH modules.) ZG tgiuiPI H G It suffices to prove that is a free -module for , because an arbitrary free -module À ZG “ giZH is isomorphic to a direct sum . So, choose a leftiPI transversal of in . Then à ZG ZH P ZG P G ZG P ÓH and it follows that isZ aH free right -module. Now, if ZHis a projective right -module, then is a direct summand of a free -module by Proposition-Definition 6.7, therefore by the above is a Remarkdirect 20.4 summand (Restriction of a free in Cohomology-module, hence) is a projective -module. d d d P‚ “ ¨ ¨ ¨ / P / P / P ZG Z M 3 Z2 G 1 HnpG;Mq ` 2 1 0 ˘ Let ZGpP‚;Mq beH a projective resolution of the trivial -module and let be an arbitrary -module. Then is the cohomology of the cochain complex G d G d G d G Hom . By LemmaH p P20.3‚q “, restricting¨ ¨ ¨ to/ P ÓyieldsH a/ P projectiveÓH resolution/ P ÓH 3 2 1 G ` 2 1 0 ˘ Z “ ZÓH ResZH

G G of seen as a i‚-module.ZG Now,pP‚;M thereq ãÑ is anZ inclusionH p H pP map‚q;M ofÓ cochainH q complexes:

: Hom Hom Res G n n G which, by functoriality (i.e. [ExerciseH 1,H ExercisepG;MqSheet ÝÑ H 5]),pH;M inducesÓH q: a homomorphism in cohomology

restriction G H res : Remark 20.5 called from to . G H Z npG;Mq G (a) The map res need not be injective in general. H Gn Hn (b) If the bar resolution is used to compute cohomology, then on , the map res is given by ordinary restriction of cocycles from to . 21 Transfer in Cohomology

H G r “ |G H| tgiu ďiďr H G L M ZG Z 1 Assume that has finite index inG , say : : . Let be a finite left transversal for H ZH pL;Mq ÝÑ ZGpL;Mq in . If and are -modules, then there is a -linearr map ´ tr : Hom ' ÞÝÑ Homgi'gi i“ ÿ 1 ´ ´ gi gi P G L gi 1 gi P G M 1 1 where denotes the action of on and denotes the action of on . Lecture Notes: Cohomology of Groups SS 2018

Lemma 21.1 76 G H ZG

Proof : The map tr is well-definedG and -linear. H 1 1 tgiu ďiďr H G gi “ gihi ď i ď r (1) Thehi P definitionH ' P of tr ZdoesH pL;M notq depend on the choice ofZ theH transversal:' 1 Assume r is another leftr transversal for in r and write r (1 ) for some 1 1 ´ ´ ´ ZH ´ ´ ´ . If Homgi'pgiq “thengi makinghi'hi usegi of the“ -linearitygi'hihi ofgi ,“ we getgi'gi ; i“ i“ i“ i“ ÿ 1 ÿ 1 1 -lin. ÿ 1 1 ÿ 1 1 1 1 1 ZG as required. s P G ď i ď r sgi “ gσpiqhi σ P Sr ´ (2) hi-linearity:P H i j σpiq “ σpjq gi “ gj ¨ hj ¨ hi i “ j Lettgiu . Then for each 1 x P,L we may write , where 1 is a permutation and (if and are such thatr , thenr we find r and thus , since G ´ ´ ´ is a transversal).s ¨ H p'q Now,pxq “ let sgi' andgi x compute“ gσpiqhi' gi x “ gσpiq' higi x i“ 1 i“ 1 i“ 1 ` ˘ ÿ ` ˘ ÿ ` ˘ ÿr ` ˘ tr 1 1 1 ´ “ gσpiq' gσpiqsx i“ 1 ÿr ` ˘ 1 ´ “ gσpiq'gσpiqpsxq i“ ÿ 1 “ G p'q psxq ; 1 H ` ˘ tr as required. pP‚; d‚q ZG Z n ě

G Assuming is a projective resolution of theH trivial -module , then for each 1, we may ZH pPn ;Mq / ZGpPn ;Mq consider the diagram ´ ´ ˚ tr ˚ dn 1 dn 1 Hom  Hom  pP ;Mq / pP ;Mq ; ZH n G ZG n H

' P ZH pPn´ ;MqHom tr Hom r r r ˚ G ˚1 ´ ´ ZG ´ G ˚ wherepd forn ˝ H qpHom'q “ dn gi'gwei compute“ gi'gi ˝ dn “ gip' ˝ dnqgi “ H dnp'q i“ i“ i“ ÿ ` 1˘ ÿ ` 1˘ -lin ÿ 1 ` ˘ dn trZG tr 1 1 1 G H ZH pP‚;Mq ÝÑ ZGpP‚;Mq since is -linear. Hence we have proved that n ě tr : HomG n Homn H H pH;Mq ÝÑ H pG;Mq : is in fact a cochain map, and therefore, for each 0, it induces a homomorphism in cohomology Definition 21.2 (Transfer) G n trn : H H pH;Mq ÝÑ H pG;Mq transfer H G relative trace map corestriction The map tr : is called from to (or the , or ). Lecture Notes: Cohomology of Groups SS 2018

Proposition 21.3 77 H G r “ |G H| ă 8

G G n n Suppose has finite index in H, say˝ H: H p:G;Mq ÝÑ. ThenH pG;M the compositeq map

|G H|tr res n: ě

Proofis : the multiplicationP‚ by ZG: for each Z 0. m P Zě

0 G H Let be a projective ZGpP-resolutionm;Mq of/ . ForZH pPm;Mq, the composition/ ZGpPm;Mq inc tr ' P ZGpPm;MHomq Hom Hom r r r maps Hom G to ´ ZG ´ H p'q “ gi'gi “ gigi ' “ ' “ |G H|': i“ i“ i“ ÿ 1 -lin. ÿ 1 ÿ tr 1 1G G 1 : H H r r These are maps of cochain complexes and induce res and tr in cohomology. Moreover, multiplication by induces multiplication by in cohomology. The claim follows. 22 Induction and Coinduction in Cohomology

Definition 22.1 (Induction) G M ZH H pMq “ ZG bZH M induction M H G

RemarkIf 22.2is a -module, we define Ind : , the of from to . G H pMq ZG ZG ZG extension of scalars Ind becomes a -module via the leftG H-module structure on . This coincides with the we studied in [Exercise 3, Exercise Sheet 4]. Hence we have a universal Propositionproperty 22.3 for the (Universal induction property of modules of the from inductionto as) follows. G M ZH { M ÝÑ H pMq; m ÞÑ b m G ZG N ZH ' M ÝÑ H pNq G LetZG be a '-moduleH pM andq ÝÑ letN : Ind 1 be the canonical map. Then for every -module and for every -linear map : Res , there exists a unique ' M / N -linear map ˜ : Ind such that the following; diagram commutes: {  ' G H pMq ˜

Ind G G In other words, there is an isomorphismZH M; ofH p abelianNq – groupsZG H pMq;N : ` ˘ ` ˘ Hom Res Hom Ind Proof :

This universal property was proven in [Exercise 3(d), Exercise Sheet 4]. Lecture Notes: Cohomology of Groups SS 2018

Remark 22.4 (Out of the scope of the lecture) 78 G G H H

DefinitionIn fact, 22.5 one can (Coinduction prove that) the functor Ind is left adjoint to the functor Res . G M ZH H pMq “ ZH pZG;Mq coinduction

RemarkIn 22.6is a -module, then we define Coind : Hom , the G H pMq ZG ZG ZG g P G ' P ZH pZG;Mq x P ZG We immediately see that Coind becomes a left -module, using the right -module struc- ture on . Explicitly, for , Hompg ¨ 'qpxq “ 'pxgandq : , we have

Proposition 22.7 (Universal property of the coinduction) G M ZH p H pMq ÝÑ M;' ÞÑ 'p q ZG N ZH ψ N ÝÑ M G ZLetG be a ψ-module.N ÝÑ Let H:p CoindMq 1 be the canonical evaluation map. Then for every -module and every -linear map : , there exists a unique G -linear map ˜ : Coind such that the followingH pMq diagram commutes: ψ : p ˜ Coind N / M ψ

Proof :

TheoremExercise. 22.8 (The Eckmann-Shapiro Lemma)

M ZH n P Zě

G 0 n G n Let be aHnpG;-module.H pMqq Then – H fornpH;M eachq there areH p groupG; isomorphismsH pMqq – H pH;Mq :

Ind and Coind Proof : n P Zě P‚ Z ZG ZH

0 G Fix and let PnbebZ aH projectiveM – Pn b resolutionZG ZG bZ ofH Mas– aPn b-moduleZG H p (henceMq : also as a -module).

Then HnpH;Mq G G Ind HnpG; H pMqq HnpG; H pMqq – HnpH;Mq Now, the left-hand side gives the homology group , while the right-hand side gives the homology G group Ind , hence ZHIndpPn;Mq – ZGpPn;. H pMqq ; Similarly HnpH;Mq n HomG Hom Coind H pG; H pMqq where the left-hand side gives the cohomology group while the right-hand side gives the Lemmacohomology 22.9 group Coind . G G M ZH H G H pMq – H pMq

If is a -module and has finite index in , then Coind Ind . Lecture Notes: Cohomology of Groups SS 2018

79 Proof :

α ZG b M ÝÑ pZG;Mq Define ZH ZH g b m ÞÝÑ 'g;m ZG ÝÑ M; : Hom s P G : sgm sg P H 'g;mpsq “ where for , # sg R H if ZG 0 if β pZG;Mq ÝÑ ZG b M and we easily check that this a -linearZH map. Now, definingZH r ´ : Hom ψ ÞÝÑ gi b ψpgi q; i“ ÿ 1

tg ; : : : ; gru H G 1 α ˝ β “ β ˝ α “

1 where is a left transversal of in , one easily checks that Id and Id. CorollaryHence the 22.10 claim follows. n G n M ZH H G H pG; H pMqq – H pH;Mq n P Zě If is a -module and has finite index in , then Ind for each 0 G G Proof : . H pMq – H pMq

By the previous lemma Coind Ind . Hence the claim follows from the Eckmann-Shapiro Lemma. Chapter 8. Finite Groups

p The aim of this chapter is to prove several central results of the theory of finite groups: Theorems of Schur and Zassenhaus and Burnside’s transfer theoremG (aslo knownfinite as Burnside’s normal -complement theorem). References:Throughout this chapter, unless otherwise stated, denotes a group in multiplicative notation. Cohomology of groups

[Bro94] K. S. Brown,An introduction to the theory, Graduate of groups. Texts in Fourth Mathematics, ed. vol. 87, Springer-Verlag, New York, 1994. [Rot09] J. J. Rotman, , Graduate Texts in Mathema- tics, vol. 148, Springer-Verlag, New York, 1995. 23 Cohomology of Finite Groups

ToLemma begin 23.1 with, we collect in this section a few general results about the cohomology of finite groups. G |G| ¨ HnpG;Mq “ n ě

ProofIf : is a1 finite group, then 1 0 for every 1. Hnp1;Mq – n ě H p1;Mq – M

Let denote the trivial group.0 Because is a cyclic group of order one Theorem 16.2 yields G G 0 if 1 (whereasHnpG;Mq 1 / Hnp1).;M Now,q the1 / H compositionnpG;Mq of the restriction with the transfer res tr – ně loooomoooon |G 1| “ |G|0 if 1 n ě |G| HnpG;Mq n ě equals multiplication by the index of : by Proposition 21.3 and factors through 0 if 1 by Propositionthe above. 23.2 Therefore multiplication by is zero in if 1. G M ZG H pG;Mq n H pG;Mq 0 |G| n ě If is a finite group and is a finitely generated -module then is a finitely generated Proofabelian : n groupP Zě and is a finite abelian group of exponent dividing for all 1. Claim 1 HnpG;Mq 0 Fix . : is a finitely generated abelian group.

80 Lecture Notes: Cohomology of Groups SS 2018

81 ZG G ZG P‚ Z Indeed: UsingZGp´ the;M factq thatP‚ is a noetherian ring as is finite, we may construct a projective - G resolutionm ě of ZGinpP whichm;Mq all “ the modulesZpPm;Mq areĎ finitelyZp generatedPm;Mq abelian groups. Now, applying the functor Hom Pm Mto we again obtain complexes of finitely generated abelian groups since for eachClaim 2 H0n,pHomG;Mq Hom n ě Hom , which is a finitely generated abelian group if both and are. The cohomology groups of this complex is again finitely generated. HnpG;Mq |G| ¨ HnpG;Mq “ : HnpG;Mis a finiteq group if 1. Indeed: Since is a finitely generated abelian group by the first claim and 0 by Lemma 23.1, must be torsion, hence finite.

M ZG HnpG;Mq “ n ě Exercise [ExerciseHn 1,pG;M Exerciseq “ Sheet 12]n ě M ZG If is a -module which is induced from the trivial subgroup, then 0 for all 1. Deduce that 0 for all 1 if is a projective -module. p G p P Exercisep [ExerciseG 2, ExerciseM SheetFpG 12] Let be a prime number, let be a finite group of order divisible by , and let be a Sylow G n n G -subgroup of . If is an -module,P H pG;M thenq ÝÑ the restrictionH pP; P mappMqq

n ě res : Res

is injective for all 0. 24 The Theorems of Schur and Zassenhaus

Schur-Zassenhaus Theorem In this section we prove two main results of the theory of finite groups, which are often considered as one Theorem and called the . Beacause of the methods we have developed, weTheorem differentiate 24.1 (Schur, between 1904 the) abelian and the non-abelian case.

G A “ pA; ¨; ˚q ZG m P Zě am “ a P A |G|; m “ 1 Let be a finite group and let be a -module such that there exists with ` i ˘ p 1 for all . Suppose that / A / E1. Then/ G the following/ hold: G A

(a) Every group extension 1 1 inducing the given -action on A E E splits.

Proof :(b) Any two complementsHnpG;Aq of in are n-conjugate.ě A m ¨ A “ |G| ¨ HnpG;Aq “ n ě n n Wem prove¨ A “ that the m ¨isC trivialpG;A forq “ all 1. Form ¨ convenience,H pG;Aq “ write additively in this proof. Thus by assumptionu; v P Z we have 0. By Lemma 23.1, we know that 0 for all 1. Since 0, we also have u ¨ |0G,| and ` v thus¨ m “ ; 0. Now, by the Bézout identity there exists such that HnpG;Aq “ u ¨ |G| ¨ HnpG;Aqq `v 1¨ m ¨ HnpG;Aq “ :

and hence “ “ loooooooomoooooooon loooooomoooooon 0 0 0 Lecture Notes: Cohomology of Groups SS 2018

82 H pG;Aq H pG;Aq A

E 2 1 Since vanishes any extension splits and since vanishes all complements of are Theorem-conjugate, 24.2 (Zassenhaus, by Theorem 19.3 1937 and) Theorem 18.3(b) respectively.

i p / A / E / G / A |A|; |G| “ Let 1 ` ˘ 1 be an extension of finite groups (where is not necessarily Proofabelian). : If 1, then|G the| ě extension splits. A ¨ |A| “ |A| A a|A| “ a P A W.l.o.g. we may assume that 1. Then we proceed by induction on the size of .

¨ If 1 or is prime,|A| P thenZě zPis abelian.q P Moreover,P 1 for all .|A Thus| Schur’sP Theorem qapplies and yieldsA the result.N “ NE pPq P E 2 SupposeClaim 1: nowE “ thatAN . Let be a prime number dividing , let be a Sylow -subgroupe P ofE , andP set :ePe´ for the normaliserq of inA . A ´ ´ ´ ´ ´ ´ a P A . ePe “ aPa1 a eqP a e “ P a e P NE pPq “ N ´ Indeed, if e,“ thena a ande P AN are Sylow -subgroups of ,1 hence -conjugate, so that there 1 1 ` 1 ` 1 ˘ 1 Claimexists 2: A such that` 1 ˘ E . Thus , i.e. , and therefore . Case 1: N ‰hasE a complement in . We split the proof of thisp claimN in two cases: . i p|N In this case, restricting to yields/ A theX N group extension/ N / G /

AXN Ĺ A G – N{pAXNq – AN{A “ E{A 1 1 H A X N N where|H| “ |G| becauseH X A “ t u H . Thus, by theA inductionE hypothesis, this Caseextension 2: N splits.“ E Hence, by Proposition 17.6, there exists a complement of in . Since Z “ ,Z wepPq have 1 ,P and therefore is a complement of Pin . P q Z . P P P LetN : be theZ centre of , whichE is a non-trivial subgroup of because is ap -group. Since is characteristicp inE{Z(i.e.ÝÑ invariantG; eZ ÝÑ underppeq all automorphismsA of{Z ), and since is normal in , we deduce that is normal in . Thus, by the universal property of the quotient, induces a group homomorphism : , whose kernel is . In other words, there is a group extension of the form / A{Z / E{Z / G / :

|Z| ‰ |A1{Z| ă |A| 1 F A{Z E{Z Now F 1ďimpliesE that Z , hence,F “ F{ byZ the induction hypothesis again, this extension splits. So let be a complement of in . By the Correspondence Theorem, there exists a subgroup r containing such that/ Z r/ F . In other/ F words,/ : there is a group extension F – G Z ď P ď A |Z|; |F| “ 1 r 1 `H Z ˘ F |H| “ |G| H X A “ H Since and A ,E we have 1 and therefore this extension splits by Schur’s Theorem. Thus, there is a complement of in r. But implies 1, so that is also a complement of in . The second claim is proved. We conclude that the extension splits using Proposition 17.6. Remark 24.3

Notice that both Schur’s and Zassenhaus’ Theorems can be stated in terms not involving cohomology, but their proofs rely on cohomological methods. Lecture Notes: Cohomology of Groups SS 2018

25 Burnside’s Transfer Theorem 83

H G |G H| “ r A trivial ZG H pG;Aq

ThroughoutH thispG;A section,q “ Z wepG;A letq “be aGrp subgrouppG;Aq of of index : : and be1 a - module. Our1 first aim is to1 understand the action of the transfer homomorphism on . So first G recall that H Hom GrppH;Aqby ÝÑ ExampleGrp 10pG;A, andq : hence we see the transfer as a Lemmahomomorphism 25.1 n` tr : Hom Hom n P Zě ZG n Z ZG

Z1 ZF S “ ts ; : : : ; sru 0 LetH G and let be the -th term of the bar resolution of as a -module.Z ZH View it as 1 a projective resolution of as a -module by restriction. Fix a right transversal of in . Then the comparison maps between this and the bar resolution of as a -module Z n` Z n` are given by the canonical inclusion in H ÝÑ G 1 1 : ' ZGn` ÝÑ ZHn` and by the map n pg ; : : : ;1 gnq ÞÑ ph ; : :1 : ; hnq : ď i ď n gi “ hisi 0 hi P H 0 si P S , Proofwhere, : for every 0 , for some and some .

i i Using the definition of theZHn differential` n / Z mapsGn` of the bar resolution,ZH we/ Z seeG; that there are commutative diagrams 0 1 1 dn ö dn ε ö ε     ZHn / ZGn Z / Z in´

1 Id 'n ' ZGn` / ZHn` ZG / ZH; 0 and 1 1 dn ö dn ε ö ε     ZGn / ZHn Z / Z 'n´

1 n ě Id

Proposition(where 25.21). Thus the Comparison Theorem yields the result.

S “ ts ; : : : ; sru H G H

1 1 Fix a right transversal G of in . Then the transfer for is described as follows: H GrppH;Aq ÝÑ GrppG;Aq r G tr : Hom f ÞÝÑ HomH pfq g ÞÝÑ fphiq ; ˜ i“ ¸ ÿ tr : sig “ his hi P H ď i ď r 1

where with for all 1 . Lecture Notes: Cohomology of Groups SS 2018

84 Proof : GrppH;Aq “ H pH;Aq – H ZH pZH ;Aq ; On the one hand 1 1` 2 ˘ Hom Hom pH;Aq “ H pH;Aq – H pZG ;Aq ; via the bar resolution. On the otherGrp hand, ZH 1 1 2 G H ` ˘ Hom Hom

via the bar resolution for restricted to . Now, transfer is defined using the second resolution, therefore, H pH;Aq “ Z pH;Aq; we need to compare these two resolutions. But 1 1 B pH;Aq “ H A

1 Z pH;Aq Ď C pH;Aq – ZH ;A : because 0 since acts trivially on , and ZH 1 1 ` 2 ˘ f P GrppH;Aq f Homf ZH ZH ;A h P H ` 2 ˘ If for a given Hom , ˜ denotes thef imageZH ofÝÑinA Hom , then for set p ; hq “ rh2s ÞÝÑ fphq ˜ : k P H 1 f pk; khq “ f k ¨ p ; hq “ k ¨ fphq “ fphq ; and thusH for each , A ` ˘ ` ˘ Z ZH ˜ ˜ 1 ' ZG ÝÑ ZH f ˝ ' ZG ÝÑ A 2 ZH because acts trivially2 on , and2 we extend this map by -linearityx P ZG to the2 whole of . Using the 1 ˜ 1 comparison map : of ther previous lemma yields r : 2 , which is -linear. Now, computing the transferG using its definition´ yields for every : H f ˝ ' pxq “ si f ˝ ' psixq ““ f ˝ ' psixq i“ i“ ` ˘ ÿ 1` ˘ ÿ ` ˘ ˜ 1 ˜ 1 ˜ 1 ´ ´ tr 1 1 s ; : : : ; sr H G A G 1 1 rgs “ p ; gq P G Ă ZG 1 ( because is a left transversal of in rand 2 is trivial.2 We wantr to view this as a 1-cocycle for , that is evaluateG this on anG element 1 : H pfqpgq “ H f ˝ ' p ; gq “ f ˝ ' psi; sigq “ f ; hi i“ i“ ` 1˘ ÿ ` 1˘ ÿ ` ˘ tr tr ˜ 1 ˜ ˜ 1 si “ ¨ si sig “ hisσpiq 1 1

r r because 1 and G. So we obtain H pfqpgq “ fp ; hiq “ fphiq ; i“ i“ ÿ ÿ ˜ tr 1 1 1

Lemmaas required. 25.3 (Choice of transversal for a fixed g P G) g P G H G Fix . m ´ m ´ ms´ S “ t ; t g; : : : ; t g ; t ; t g; : : : ; t g ; : : : ; ts; tsg; : : : ; tsg (a) There exists a right transversal of1 in of the form2 1 mi ´ 1 1 m ` ::: ` m1s “1 |G H|1 “ r 2tig2 ti P 2H ď i ď s (

G 1 s mi ´ f P 1 GrppH;Aq g P G pfqpgq “ f tig t with : and H i“for all 1 i . ř ` 1˘ (b) If Hom and , then tr 1 . Lecture Notes: Cohomology of Groups SS 2018

85 Proof : g Hs Hs ÞÝÑ Hsg g r g Ht ; : : : ; Htr (a) gThe element acts on the right on rightH cosets , via g . DecomposeHti the set of right mi´ 1 cosetsHt intoig -orbits.g “ Ht Leti be them numberi of -orbits and let be representatives of the -orbits. We1 get all the right cosets of by applying powers of to each and we suppose that (that is,p q is the cardinalityď i ď s of the orbit). With this choice, we obtain a right transversal with the required properties. t g; t g ; : : : ; t gmi´ (b) Proposition 25.2 together with a yieldi for i1 i: 2k 1 k` S tig ¨ g “ ¨ tig P H ¨ S ď k ď mi ´ t gmi´ g “ t gmi t´ t P H ¨ S i i i i ` ˘ ` 1˘ belong to the right transversal , so that 1 for 0 2 and ` 1 ˘ ` 1˘ s . ThereforeG mi ´ H pfqpgq “ f tig ti ; i“ ÿ ` 1˘ Htr 1 fp q “

Theorembecause 25.4 (Burnside’s all the other transfer elements theorem of appearing (or Burnside’s are 1 and normal1 p0-complement. theorem), 1911) G p p | |G| P p G P CGpPq “ NGpPq N P G GLet“ Nbe¸ aP finite group, let be a prime number such that and let be a Sylow -subgroup of . If is abelian and , then there exists a normal complement to in , i.e. Proof : . Claim: g P G u P P u; gug´ P P gug´ “ u u P g´ Pg 1 1 P g´ Pg If there exist and such that ´ , then . p CGpuq P 1 g Pg CGpuq 1 ´ ´ ´ cIndeed:P CGpu usingq the assumptions,cPc “ g wePg have gcPpgc,1 whichq “ isP abelian, andgc thereforeP NGpPq both “ CGpPandq ´ ´ gugare Sylow“ pgc-subgroupsqupgcq “ ofu1 1 . Thusu P P and 1 are conjugate in , so that there exists 1 such that 1 , that is G and hence . Finally, P P GrppP;Pq P p P q P GrppG;Pq u P P because , as required. s Now consider the identity map Id GHom and tr mIdi ´ Hom . The previous lemma P p P qpuq “ P tiu ti : yields for a fixed , i“ ź ` PP 1 ˘ tr Id Id mi ´ mi 1 looomooon tiu ti “ u ď i ď s

1 s G mi |G P| Now using the claim yields P p Pforqpu eachq “ 1 u “ u, hence: i“ ź : G tr Id P p P q G  P 1 v P P u P P u|G P| “ v b |G P| b In particular,a; b P Z this provesa| thatP| `trb|GId P:|: “ isv a“ surjectivev “ pv groupq homomorphism,u because“ v for each , there exists such that by the Bézout1 identity.: (Indeed, Bézout implies that there N “ Gp q exist such that P : 1, hence and we choose .) ´ ¯ G Finally, set : ker tr Id , so that we have a groupP p extensionP q / N / G / P / tr Id ¨ { { P ÝÑ G N |G P| 1 1 P 1G with a section given by : , where : is the canonical inclusion. It follows that is a normal complement of in . Chapter 9. The Schur Multiplier and Universal Central Extensions

G The Schur multiplier is a very important tool of finite group theory and representation theory of finite groups; it can be defined as certain cohomology group associated to a given group , but as we will immediately see, it can also be defined as a homology group, which point of view makes it a very important tool of algebraic topology as well. We will also show that it has very natural connections with central extensions and projective representations.G K K ˆ K Throughout this chapter, unless otherwise stated, denotes a group in multiplicative notation, a fieldReferences: and denotes the multiplicative group of units of . Methods of representation theory. Vol. I

[CR90] C. W. CurtisAn and introduction I. Reiner, to the theory of groups. Fourth ed. , Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990. 1 [Rot09] J. J. Rotman, Universal p -central extensions , Graduate Texts35 in Mathema- tics, vol. 148, Springer-Verlag, New York, 1995. [LT17] C. Lassueur and J. Thévenaz, , Expo. Math. (2017), no. 3, 237–251. 26 The Schur Multiplier

Definition 26.1 (Schur multiplier) Schur multiplier multiplicator G MpGq “ H pG; Cˆq ˆ C ZG 2 RemarkThe 26.2 (Uniquely divisible(or groups) ) of a group is the abelian group : , where is seen as a trivial -module.

¨ pA; `q uniquely divisible by an integer n P Z Recall from group theory that: ě a P A b P A a “ nb 1 An abelian groupn misn saidA ÝÑ toA; be b ÞÑ nb Aif for all p|Athere|; nq “ exists aA unique with n , or equivalently iff the homomorphism of multiplication by , i.e. : is an isomorphism. For Example, if is finite ¨ A uniquely divisible and 1 then is uniquely divisible by . n P Zě Moreover, isQ said toR be ifQ it{ isZ uniquely divisible by every positive integer 1 . For example, and are uniquely divisible; is not uniquely divisible.

86 Lecture Notes: Cohomology of Groups SS 2018

Lemma 26.3 87

G M ZG |G| HnpG;Mq – n ě (a) If is a finite group and is a -module which is uniquely divisible by (i.e. as an HnpG; Cˆq – Hn` pG; Zq – HnpG; Q{Zq n ě abelian group), then 0 for every 1. 1 Proof :(b) for all 1.

n ě |G| m|G| M ÝÑ M; m ÞÑ |G|m n n |G| m|G| H pG;Mq ÝÑ H pG;Mq n (a) Let 1 be fixed. Since multiplication by H ,pG;Mq: – , is an isomorphism, so ˆ ˆ is multiplication by , : byC functoriality– Rą ˆ S of;cohomology. z ÞÑ p|z|; p Butzqq this πit map is the zero map by LemmaR ÝÑ 23.1S ,; hence t ÞÑ e 0. 1 R{Z – S 0 (b) Recall that polar coordinates induce1 a group2 isomorphism arg1 and the exponential map exp : induce a group homomorphism . Hence ˆ ˆ ˆ there is a group extension / Z / Rą ˆ R / Rą ˆ S / : Id exp Cˆ 1 1 0 0 1 ˆ loooomoooon pRą ; ¨q ÝÑ pR; `q R ˆ R Because the natural logarithm ln : 0 is a group isomorphism the term in the middle is isomorphic to , and hence is uniquely divisible. Now the long exact sequence in ˆ ˆ ˆ cohomology¨ ¨ ¨ / H associatedpG; Rą ˆ toR thisq exact/ H pG; sequenceC q has/ H thepG; formZq / H pG; Rą ˆ Rq / ¨ ¨ ¨ 1 – 1 2 2 – 0 0 loooooooomoooooooon loooooooomoooooooon n 0ˆ n ˆ n` n` 0 ˆ ¨ ¨ ¨ / H pG; Rą ˆ Rq / H pG; C q / H pG; Zq / H pG; Rą ˆ Rq / ¨ ¨ ¨ – 1 1 – 0 0 loooooooomoooooooon loooooooooomoooooooooon 0 HnpG; Cˆq – Hn` pG; Zq n ě 0 inc can 1 / Z / Q / Q{Z / Hence, we conclude that for each 1. HnpG; Q{Zq – Hn` pG; Zq n ě

A similar argument using1 the s.e.s. 1 1 yields the isomor- Propositionphism 26.4 (Alternative descriptionsfor all of the1 Schur. multiplier) G H pG; Cˆq – H pG; Z{Qq – H pG; Zq – H pG; Zq

2 2 3 2 ProofIf : is a finite group, then H pG;. Zq – H pG; Zq the integral duality theorem 3 G n` 2 H ThepG; firstZq – twoHnp isomorphismsG; Zq aren giveně by Lemma 26.3(b). The isomorphism is a consequence1 of a result known as which states that for a finite group , whenever 1. We won’t prove it in these notes. 27 The Projective Lifting Property and Universal Central Extensions

28For this Universal section, we followedp1-Central faithfully Extensions [CR90, §11E].

This section followed [LT17]. Appendix: The Language of Category Theory

This appendix provides a short introduction to some of the basic notions of category theory used in this References:lecture. Categories for the working mathematician

[McL98] S. Mac Lane,An introduction to homological algebra , second ed., Graduate Texts in Math- ematics, vol. 5, Springer-Verlag, New York, 1998. [Wei94] C. A. Weibel, , Cambridge Studies in Advanced Math- ematics, vol. 38, Cambridge University Press, Cambridge, 1994. A Categories

Definition A.1 (Category) category C ‚ C objects A consists of: ‚ pA;Bq morphisms pA;Bq a class ObC of , ‚ composition function a set Hom of for every ordered pair of objects, and

a CpA;Bq ˆ CpB;Cq ÝÑ CpA;Cq pf; gq ÞÑ g ˝ f Hom Hom Hom pA;B;Cq

for each ordered triple of objects, (C1) Unit axiom A P C identity morphism A P CpA;Aq satisfying the following axioms: f P CpA;Bq B P C : for each object Ob , there exists an 1 Hom f ˝ “ f “ ˝ f : such that for every Hom for allA Ob B,

(C2) Associativity axiom f P 1CpA;Bq g1 P CpB;Cq h P CpC;Dq A;B;C;D P C : for every hHom˝ pg ˝ fq “ p, h ˝ gqHom ˝ f : and Hom with Ob ,

88 Lecture Notes: Cohomology of Groups SS 2018

89

Remark A.2 Let us start with some remarks and examples to enlighthen this definition: C

pA;Bq (a) Ob need not be a set! C f (b) The only requirement onf HomA ÝÑ B isA ÝÑ thatB it be a set, andf P it is allowedCpA;Bq to be empty. arrows morphisms A P C A P C (c) It is common to write : or instead of Hom , and to talk about P pA;Aq f P pA;Aq instead of A . It isC also common to write " " instead of " A Ob ". C fA “ fA ˝ A “ A (d) The identity morphism 1 Hom is uniquely determined: indeed, if Hom Examplewere A.3 a second identity morphisms, then we would have 1 1 .

C “ 1

(a) : category with one object and one morphism‚ (the identity morphism): ‚ 1

C “ 2

(b) : category with two objects and three morphism, where two of them are identity morphisms and the third one goes from one object to the other: A B A B 1 1

G CpGq CpGq “ t‚u CpGqp‚; ‚q “ G

(c) A group can be seen as a category with one object: Ob , Hom n ˆ m k n; m (notice that this is a set) and composition is given by multiplication in the group. Matk Matk “ Zą n ÝÑ m n m m ˆ n (d) The -matrices with entries in a field for ranging over the positive integers form 0 a category : Ob , morphisms from to are the -matrices, Exampleand A.4 (compositionsCategories and are algebraic given by the structures ordinary) matrix multiplication.

C “ Set category of sets

(a) , the : objects are sets, morphisms are maps of sets, and composition C “ Veck category of vector spaces over the field k k is the usual composition of functions. k (b) , the : objects are -vector spaces, mor- C “ Top category of topological spaces phisms are -linear maps, and composition is the usual composition of functions. (c) , the : objects are topological spaces, morphisms are C “ Grp category of groups continous maps, and composition is the usual composition of functions. (d) , the : objects are groups, morphisms are homomorphisms of groups, and composition is the usual composition of functions. Lecture Notes: Cohomology of Groups SS 2018

C “ Ab category of abelian groups 90

(e) , the : objects are abelian groups, morphisms are homomor- C “ Rng category of rings phisms of groups, and composition is the usual composition of functions.

(f) , the : objects are rings, morphisms are homomorphisms of rings, C “ Mod category of left R-modules left R andR composition is the usual composition of functions. R (g) , the : objects are modules over the ring , morphisms C “ Mod category of left R-modules right R are -homomorphisms,R and composition is the usual composition of functions. R (g’) , the : objects are modules over the ring , C “ Mod category of pR;Sq-bimodules pR;Sq morphismsR S are -homomorphisms, and composition is the usual composition of functions. R S pR;Sq (g”) , the : objects are -bimodules over the rings and , morphisms are -homomorphisms, and composition is the usual composition of ::: functions.

Definition(h) Examples A.5 (Monomorphism/epimorphism of your own )

C f P CpA;Bq f monomorphism g ; g C ÝÑ A Let be a category and let Hom be a morphism. Then is called 1 2 (a)a iff for all morphismsf ˝ g “ f ˝ g: ùñ g ,“ g :

epimorphism 1g ; g B2 ÝÑ C1 2

1 2 (b) an iff for all morphismsg ˝ f “ g ˝: f ùñ g , “ g :

1 2 1 2 Remark A.6

In categories, where morphismsRMod Mod areR set-theoreticRModS maps, then injective morphisms are monomorphisms, andWarning surjective morphisms are epimorphisms. In module categories ( , , , . . . ), the converse holds as well, but: : It is notRng true in general, that all monomorphisms{ Z ÝÑ Q must be injective, and all epimorphismsC mustg be; g surjective.P RngpQ;Cq g For example in , the canonical injection { : / is an epimorphism. Indeed, if is a ring 1 2 Z / Q / C and Hom g2 g ˝ { “ g ˝ { g “1 g { 1 2 1 2 are such that , then we must have by the universal property of the field of fractions. However, is clearly not surjective. B Functors Definition B.1 (Covariant functor) C D covariant functor F C ÝÑ D

Let and be categories. A : is a collection of maps: Lecture Notes: Cohomology of Groups SS 2018

‚ F C ÝÑ D ;X ÞÑ FpXq 91

‚ F pA;Bq ÞÑ pFpAq;FpBqq A;B: Ob C Ob D , and

: Hom Hom , f g satisfying:A ÝÑ B ÝÑ C C Fpg ˝ fq “ Fpgq ˝ Fpfq Fp q “ A P C (a) If A FpAq are morphisms in , then ; and

(b) 1 1 for every Ob . Definition B.2 (Contravariant functor) C D contravariant functor F C ÝÑ D

‚ F C ÝÑ D ;X ÞÑ FpXq Let and be categories. A : is a collection of maps: ‚ F pA;Bq ÞÑ pFpBq;FpAqq A;B: Ob C Ob D , and

: Hom Hom , f g satisfying:A ÝÑ B ÝÑ C C Fpg ˝ fq “ Fpfq ˝ Fpgq Fp q “ A P C (a) If A FpAq are morphisms in , then ; and

(b) 1 1 for every Ob . Remark B.3

Often in the literature functors are defined only on objects of categories. When no confusion is to be made and the action of functors on the morphism sets are implicitely obvious, we will also adopt Example B.4 this convention. Q P pRModq pQ; ´q Mod ÝÑ Ab Let Ob . Then R R M ÞÑ R pQ;Mq Hom : Hom , R p´;Qq RMod ÝÑ Ab is a covariant functor, and M ÞÑ R pM;Qq Hom : Hom , is a contravariant functor. Exact Functors.

We are nowRMod interested in the relationsabelian between categories functors and exact sequences in categories where it makes sense to define exact sequences, that is categories that behave essentially like module categories such as . These are the so-called . It is not the aim, to go into these details, ‚ Mod but roughly speaking abelian categoriesR are categories satisfying the following properties:

they have a zero object (in : the zero module) Lecture Notes: Cohomology of Groups SS 2018

92 ‚ RMod ‚ Mod R they have products and coproductsR (in : products and direct sums) ‚ Mod they have kernels and cokernels (in : the usual kernels and cokernelsR of -linear maps)

Definitionmonomorphisms B.5 (Pre-additive are kernels categories/additive and epimorphisms functors are cokernels) (in : satisfied)

C pre-additive

F C ÝÑ D additive F (a) A category in which all sets of morphisms are abelian groups is called . A;B A;B P C (b) A functor : between pre-additive categories is called iff the maps Definitionare B.6 homomorphisms (Left exact/right of exact/exact groups for all functors)Ob . F C ÝÑ D g ÝÑ A ÝÑf B ÝÑ C ÝÑ C F Let : be a covariant (resp. contravariant) additive functor between two abelian categories, Fpfq Fpgq Fpgq Fpfq and letleft0 exact ÝÑ FpAq ÝÑ0FbepBq aÝÑ s.e.s.Fp ofC objectsq andÝÑ morphismsFpCq ÝÑ in F.pB Thenq ÝÑ FispA called:qq

(a) if 0 Fpfq Fpgq (resp. 0 Fpgq Fpfq ) is an exactright exact sequence.FpAq ÝÑ FpBq ÝÑ FpCq ÝÑ FpCq ÝÑ FpBq ÝÑ FpAqq ÝÑ

(b) if Fpfq Fpgq 0 (resp. Fpgq Fpfq 0) is an exact sequence.ÝÑ FpAq ÝÑ FpBq ÝÑ FpCq ÝÑ ÝÑ FpCq ÝÑ FpBq ÝÑ FpAqq ÝÑ

(c) if 0 0 (resp. 0 Example0 B.7) is a short exact sequence.

R pQ; ´q R p´;Qq R pQ; ´q Q R p´;Qq Q The functors Hom and Hom of Example B.4 are both left exact functors. Moreover Hom is exact if and only if is projective, and Hom is exact if and only if is injective. Bibliography

Cohomology of groups

[Bro94] K. S. Brown, , Graduate Texts in Mathematics, vol. 87, Springer-Verlag, Methods of representation theory. Vol. I New York, 1994, Corrected reprint of the 1982 original.

[CR90] C. W. Curtis and I. Reiner, , Wiley Classics Library, John Wiley & Sons, Inc., New York, 1990, With applications to finite groups and orders, Reprint The cohomology of groups of the 1981 original, A Wiley-Interscience Publication.

[Eve91] L. Evens, , Oxford Mathematical Monographs, The Clarendon Press, A course in group theory Oxford University Press, New York, 1991.

[Hum96] J. F. Humphreys, , Oxford Science Publications, The Clarendon Press, Presentations of groups Oxford University Press, New York, 1996.

[Joh90] D. L. Johnson, , London Mathematical Society Student Texts, vol. 15, Universal p1-central extensions 35 Cambridge University Press, Cambridge, 1990.

[LT17] C. Lassueur and J. Thévenaz, , Expo. Math. (2017), no. 3, Categories for the working mathematician 237–251.

[ML98] S. Mac Lane, , second ed., Graduate Texts in Math- An introduction to the theory of groups ematics, vol. 5, Springer-Verlag, New York, 1998.

[Rot95] J. J. Rotman, , fourth ed., Graduate Texts in Mathematics, An introduction to homological algebra vol. 148, Springer-Verlag, New York, 1995.

[Rot09] Joseph J. Rotman, , second ed., Universitext, Springer, Advanced modern algebra. 2nd ed. New York, 2009.

[Rot10] J. J. Rotman, , Providence, RI: American Mathematical Society An introduction to homological algebra (AMS), 2010.

[Wei94] C. A. Weibel, , Cambridge Studies in Advanced Math- ematics, vol. 38, Cambridge University Press, Cambridge, 1994.

93 Index of Notation

General symbols C Fq q M field of complex numbersM pfq finite field with felements Id p'q identity map on the set' ImN image of the map kerN kernel of the morphism P the natural numbers withoutZ 0 0 Q the natural numbers with 0 R the prime numbers in Z field of rational numbers Zěa; Ząa; Zďa; Zăa tfieldm P ofZ | realm ě numbersa m ą a; m ě a; m ă aqu |X| ring of integer numbersX δij (resp. cardinality of the set Kronecker’s delta Ť union š disjoint union Ş ˆ intersection ř ¸ summation symbol ś ‘ , cartesian/direct product b semi-direct product H direct sum @ tensor product D empty set – for all a | b a - bthere a existsb a b pa; bq isomorphisma b f|S , divides , does notf divide S ãÑ gcd of and  restriction of the map to the subset injective map surjective map Lecture Notes: Cohomology of Groups SS 2018

Group theory 95 pGq G A;GpEq G A G AAutn automorphism group ofn the group CAutm automorphism the groupm inducing the identity on and CGpxq alternating group on lettersx G CGpHq cyclic group of order in multiplicativeH G notation D n centraliser of the elementn in pAq centraliser of the subgroup in A 2 EpG;A˚q dihedral group of order 2 i p End endomorphism/ A ring/ E of the/ G abelian/ group G ˚ G{N set of equivalenceG classesN of group extensions npK q 1 K 1 inducing the -action npK q quotient group modulo K GLH ď G H ă GHgeneral linear groupG over PGLN IJ GNprojective general linearG group over NGpHq, is a subgroupH ofG , resp. a proper subgroup N ¸θ H is a normal subgroupN H θ npK q normaliser of in K Q semi-direct product of in w.r.t. PGLQ n projective linear group over 8 Sn quaternion group ofn order 8 2 n SD n generalised quaternion group of order 8 npK q symmetric group on lettersK 2 Z{mZ semi-dihedral group ofm order 2 x ´ SLg special linearg groupx overgxg xgy Ď G cyclic groupG of order in additiveg1 notation G “ xX | Ry conjugate of by , i.e. G |G H| subgroup of generatedH byG x P G{N presentationx P G for the group G{N t u: 1 index of the subgroup in class of in the quotient group Module1 , 1, theory trivial group R pM;Nq R M N R pMq R R M KGHom -homomorphisms from toG K Endε KG ÝÑ K -endomorphism ring of the -module IG group algebra of the group over the ring G M: Gaugmentation map M MG Gaugmentation ideal M G G M ÓH H pMq -fixed pointsM of theG moduleH G H pMq -cofixed pointsM of theH moduleG G H , Res restriction of Gfrom H to G IndH induction of H fromG to res restriction from to in cohomology tr transfer from to Lecture Notes: Cohomology of Groups SS 2018

Homological algebra 96 BnpC‚q n C‚ BnpC ‚q n C ‚ n B pG;Aq n-boundaries of A pC‚; d‚q C‚ -coboundaries of ‚ ‚ ‚ pC ; d q C -coboundaries with coeff. in rel. to the bar resolution n C pG;A,q chainn complex A n R pM;N, q cochainn complex M N HnpC‚q n-cochains with coeff. in rel. toC‚ the bar resolution ExtHnpG;Mq n-th Ext-group of with coefficientsG in M n ‚ ‚ H pC q n-th homology group/module of C n H pG;Mq n-th homology group of the group Gwith coeff. in M P‚  M -th cohomology group/module of M R n pM;Nq n-th cohomology groupM of the group withN coeff. in ZnpC‚q projectiven resolutionC‚ of the module n ‚ ‚ TorZ pC q n-th Tor-groupC of with coefficients in n Z pG;Aq n-cycles of A rg |g | ::: |gns -cocycles of -cocycles with coeff. in rel. to the bar resolution 1 2 Category Theory bar notation C C CpA;Bq A B ObSet objects of the category HomVeck morphisms from to k Top the category of sets Grp the category of vector spaces over the field Ab the category of topological spaces Rng the category of groups RMod the category of abelianR groups ModR the category of ringsR RModS the category of leftpR;Sq-modules the category of left -modules the category of -bimodules Index

G G n -cofixed points, 50 of sets, 89 n -fixed points, 50 of topological spaces, 89 n-boundary, 33 of vector spaces, 89 n-coboundary, 36, 56 pre-additive, 92 n-cochain, 56 coimage, 20 n-cocycle, 36, 56 coinduction, 78 n-cycle, 33 cokernel -th cohomology group, 36, 51 of a morphism, 20 -th homology group, 33, 51 of a morphism of complexes, 32 complement, 65 action complex conjugation, 49 acyclic, 35 diagonal, 49 augmented, 37 arrow, 89 chain, 31 augmentation cochain, 36 ideal, 49 exact, 35 map, 49 non-negative, 31 axiom positive, 31 associativity axiom, 88 quotient complex, 32 unit axion, 88 zero complex, 34 composition function, 88 basis composition law of a free group, 10 external, 17 of a module, 25 connecting homomorphism, 23, 34 bimodule, 17 corestriction, 76 category degree, 31 abelian, 91 derivation homotopy category, 36 inner, 58 of abelian groups, 90 principal, 58 of bimodules, 90 dimension shifting, 44 of groups, 89 direct sum of left modules, 90 external, 21 of right modules, 90 internal, 21 of rings, 90

97 Lecture Notes: Cohomology of Groups SS 2018

98 epimorphism, 90 long exact sequence exact sequence, 22 in cohomology, 37, 51 short, 22 in homology, 34, 51 R split s.e.s., 24 Ext-group, 41 map extension -balanced, 28 central, 63 chain map, 32 1 equivalent,p 69 cochain map, 36 split, 63 differential map, 31 universal -extension, 87 relative trace, 76 module functor flat, 30 additive, 92 free, 25 contravariant, 91 homomorphism, 18 covariant, 90 injective, 27 exact, 92 left, 17 left exact, 92 projective, 27 right exact, 92 right, 17 trivial, 49 generating set, 25 monomorphism, 90 group morphism extension, 62 identity morphism, 88 finitely presented, 13 in a category, 88 free, 10 of chain complexes, 32 presentation, 13 of cochain complexes, 36 uniquely divisible, 86 group algebra, 48 object, 88 group ring, 48 projective lifting property, 87 homotopy equivalence, 35 quasi-homomorphism, 35 identity 1-cocycle, 58 rank 2-cocycle, 58 of a free group, 10 image relation, 13 of a morphism, 18 representation, 49 of a morphism of complexes, 32 resolution induction bar, 54 of a module, 77 free, 37 injective, 40 kernel normalised bar, 55 of a morphism, 18 projective, 37 of a morphism of complexes, 32 restriction of an extension, 63 in cohomology, 75 of a module, 74 lemma retraction, 24 Eckmann-Shapiro lemma, 78 horseshoe lemma, 40 scalar multiplication, 17 snake lemma, 23 Schur multiplier, 86 Lecture Notes: Cohomology of Groups SS 2018

99 section, 24 semi-direct product external,9 internal,7 subcomplex, 32 submodule, 18 tensor product of modules, 28 of morphisms, 29 theorem 1st isomorphism theorem, 19 2nd isomorphism theorem, 19 3rd isomorphism theorem,p 19 lifting theorem, 38 Burnside’s normal -complement theorem, 85 Burnside’s transfer theorem, 85 comparison theorem, 39 correspondence theorem, 20 Schur, 81 Zassenhaus, 82 Tor-group, 47 transfer, 76 transversal, 74 universal property of free groups, 10 of free modules, 25 of presentations, 14 of the coinduction, 78 of the direct sum, 21 of the induction, 77 of the product, 20 of the quotient, 19 of the tensor product, 28 word, 12 equivalent, 13 length of, 13 reduced, 13