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, more generally, of projective modules.

Throughout this chapter we let R and S denote rings, and unless otherwise specified, all rings are assumed to be unital and associative.

Most results are stated without proof, as they have been studied in the B.Sc. lecture Commutative Algebra. As further reference I recommend for example:

Reference: [Rot10] J. J. R￿￿￿￿￿, Advanced modern algebra. 2nd ed., Providence, RI: American Mathematical Society (AMS), 2010.

3 Modules, Submodules, Morphismsp q Definition 3.1 (Left R-module, right R-module,p R`q￿S -bimodule, homomorphism of modules) ¨ ˆ ›Ñ p qfiÑ ¨ (a) A left R-module is an abelian group M￿ endowed with a scalar multiplication (or external composition law) : R M ›ÑM￿ ￿￿￿p q￿ ￿ such that the map fiÑ p q “ ›Ñ fiÑ ¨ λ : R End M ￿ ￿ λ ￿ : λ : M M￿￿ ￿ ￿ , ¨ is a ring homomorphism. By convention, when no confusion is to be made, we will simply write "R-module" to mean "left R-module", and ￿￿ instead of ￿ ￿. ¨ ˆ ›Ñ p qfiÑ ¨ (a’) A right R-module is defined analogously using a scalar multiplication : M R M￿ ￿￿ ￿ ￿ ￿ on the right-handp side.q p `q (a”) If S is a second ring, then an R￿S -bimodule is an abelian group M￿ which is both a left R-module and a right¨p ¨S-module,q“p ¨ andq¨ which@ satisfiesP @ theP axiom@ P ￿ ￿ ￿ ￿ ￿ ￿ ￿ R￿ ￿ S￿ ￿ M￿

17 Skript zur Vorlesung: Cohomology of Groups SS 2018 18 § ¨ P P P (b) An R-submodule of an R-module M is a subgroup N M such that ￿ ￿ N for every ￿ R and every ￿ N. (Similarly for right modules and bimodules.) ›Ñ (c) A (homo)morphism of R-modules (or an R-linear map, or an R-homomorphism) is a map of R-modules ￿ : M N such that:

(i) ￿pis¨ a groupq“ ¨ homomorphism;p q@ P @ and P (ii) ￿ ￿ ￿ ￿ ￿ ￿ ￿ R, ￿ M. – A bijective homomorphism of R-modules is called an isomorphism (or an R-isomorphism), and ã we write M N if there exists an R-isomorphism between M and N. Ñ An injective (resp.⇣ surjective) homomorphism of R-modules is sometimes called a monomor- phism (resp. epimorphism) and we sometimes denote it with a hook arrow " " (resp. a two-head arrow " "). (Similarly for right modules and bimodules.)

R Notation: We let Mod denote the category of left R-modules (with R-linear maps as morphisms), we R p q p q R S let Mod denote the category of right R-modules (with R-linear maps as morphisms), and we let Mod denote the category of R￿S -bimodules (with R￿S -linear maps as morphisms). For the language of category theory, see the Appendix.

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

Example 5 Z (a) Vector spaces over a field K are K -modules, and conversely.

(b) Abelian groups are -modules, and conversely.

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

Notation 3.2 p q “t ›Ñ | u R Given R-modules M and N, we set Hom M￿N : ￿ : M N ￿ is an R-homomorphism . This is an` abelian groupp forqˆ the pointwisep q addition›Ñ of functions:p q R p qR fiÑ ` R ›Ñ fiÑ p q` p q : Hom M￿N Hom M￿N Hom M￿N “ ￿￿ ψ p q “ p ￿ qψ : M N￿￿ ￿ ￿ ψ ￿ . p q R R In case N M, we write End M : Hom M￿M for the set of endomorphisms of M and R Aut M for the set of automorphisms of M, i.e. the set of invertible endomorphisms of M. Skript zur Vorlesung: Cohomology of Groups SS 2018 19

Exercise [Exercise 1, Exercise Sheet 3] Let M￿N bep Rq-modules. Prove that: R (a) End M , endowed with the usual composition andp sumq of functions, is a ring. R (b) If R is commutative then the abelian group Hom M￿N is a left R-module.

Lemma-Definition 3.3 (Quotients of modules) { Let U be an R-submodule of an R-module M. The quotient group M U can be endowed with the structure of an R-module in a natural way:ˆ { ›Ñ { R `M fi›Ñ M¨ ` ` U˘ U ›Ñ { ￿￿￿fiÑ `U ￿ ￿ U

The canonical map π : M M U￿￿ ￿ U is R-linear.

Proof : Direct calculation.

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

Concretely, ￿ ￿ U ￿ ￿ ￿ U M U . “ p q (b) 1st isomorphism theorem: With the notation of (a), if U ker ￿ , then { p q ›Ñ p q ￿ : M ker ￿ Im ￿

is an isomorphism of R-modules. X ` 1 2 1 2 1 2 (c) 2nd isomorphism theorem: If U ￿U are R-submodules of M, then so are U U and U U , and there is an an isomorphism of R-modules p ` q{ – { X 1 2 1 U U 2 U 1 2 ￿ Ñ U U U 1 2 (d) 3rd isomorphism theorem: If U U are R-submodules of M, then there is an an isomor- { { – { phism of R-modules { ´ ¯ ´ ¯ M 1 2 1 M 2 ￿ U U U U

(e) Correspondence theorem: If U is an R-submodule of M, then there is a bijection Skript zur Vorlesung: Cohomology of Groups SS 2018 20 t | Ñ u–Ñt { u fiÑ { XR-submodule´ p ofqM U X –R-submodules of M U X1 X U π Z [ Z .

Proof : We assume it is known from the "Einführung in die Algebra" that these results hold for abelian groups and morphisms of abelian groups. Exercise: check that they carry over to the R-module structure.

DefinitionP 3.5 (Cokernel,p q coimage) { R { Let ￿ Hom M￿N . Then, the cokernel of ￿ is the quotient R-module N Im ￿ , and the coimage of ￿ is the quotient R-module M ker ￿ .

4t Directu P products and direct sums P t u P ￿ ￿ I ± ￿ ￿ ￿ I Let M be a family of R-modules. Then the abelian group ￿ I M , that is the product of M seen as a family of abelian groups, becomesˆ an R›-moduleÑ via the following external composition law: P P π ￿ π ￿ R M M p ￿ qI P fi›Ñ ￿ I¨ P ` ˘ ` ˘ ￿ ￿ I ￿ ￿ I P ￿￿ P￿ ›Ñ ￿ ￿ ￿ ￿ ± ￿ ￿ Furthermore, for each ￿ I, we let π : ￿ I M M denotes the ￿-th projection from the product to ￿ the module M .

Proposition 4.1›Ñ (Universalu P property of the direct product) ￿ ›￿Ñ￿ I P ˝ “ P If ￿ : L M is a collection of R-linear maps, then there exists a unique morphism of ± ￿ ￿ ￿ R-modules ￿ : L ￿ I M such that π ￿ ￿ for every ￿ I.

L

￿ ✏ ￿ ￿￿ ￿ P ±￿ I ￿ { M # ⌘ π￿ π￿

￿ ￿ M M

In other words ›Ñ p q ´ P ¯ P R π ￿ π R ￿ Hom L￿ M Hom L￿M ￿ I fi›Ñ ￿ I ˝ ` ￿ ˘ ￿ π ￿ ￿

is an isomorphism of abelian groups. Skript zur Vorlesung: Cohomology of Groups SS 2018 21

Proof : Exercise 2, Exercise Sheet 3.

P P p q P “ À ￿ “ ± ￿ P ￿ ￿ I ￿ Now let ￿ I M be the subgroup of ￿ I M consisting of the elements ￿ such that ￿ 0 al- ￿t u P P most everywhere (i.e. ￿ 0 exept for a finite subset of indices ￿ I). This subgroup is called the ›Ñ P ￿ ￿ I direct sum of the family M and is in fact an R-submodule of the product. For each ￿ I, we let ￿ ￿ À ￿ ￿ η : M ￿ I M denote the canonical injection of M in the direct sum.

Proposition 4.2› (UniversalÑ u P property of the direct sum) ￿ ￿ P￿ I ›Ñ ˝ “ P If ￿ : M L is a collection of R-linear maps, then there exists a unique morphism of À￿ I ￿ ￿ ￿ R-modules ￿ : M L such that ￿ η C￿O [ for every ￿ I.

L

￿ ￿ ￿￿ ￿ ; P c À ￿ ￿ I M η￿ η￿

￿ ￿ M M

In other words ›Ñ p q ´ P ¯ P R à ￿ π R ￿ Hom M ￿L Hom M ￿L ￿ I fi›Ñ ￿ I˝ ` ￿˘ ￿ ￿ η ￿

is an isomorphism of abelian groups.

Proof : Exercise 2, Exercise Sheet 3.

Remark 4.3 | |†8 P “ P À ￿ ± ￿ It is clear that if I , then ￿ I M ￿ I M .

The direct sum as defined above is often called an external direct sum. This relates as follows with the usual notion of internal direct sum:

Definition 4.4 (“Internal” direct sums) “ ‘ P 1 2 “ ` P P 1 2 Let M be an R-module and N ￿N be two R-submodules of M. We write M N N if every 1 2 1 1 2 2 ￿ M“can be‘ written in a unique way as ￿ ￿ ￿ ,“ where`￿ N andX￿ N“t. u 1 2 1 2 1 2 In fact M N N (internal direct sum) if and only if M N N and N N 0 .

Proposition 4.5 “ ‘ 1 2 1 2 If N ￿N and M are as above and M N N then the homomorphism of R-modules Skript zur Vorlesung: Cohomology of Groups SS 2018 22 ›Ñ ˆ “ ‘ “ ` fiÑ p 1 q2 1 2 ￿ : M N N N N (external direct sum) 1 2 1 2 ￿ ￿ ￿ ￿ ￿￿ ,

is an isomorphism of R-modules. “ P À ￿ The above generalises to arbitrary internal direct sums M ￿ I N .

5 Exact Sequences Definition 5.1 (Exact›Ñ sequence›Ñ ) “ ￿ ψ A sequence L M N of R-modules and R-linear maps is called exact (at M) if Im ￿ ker ψ.

Remark 5.2 (Injectivity/surjectivity/short exact sequences) ›Ñ ñ ›Ñ ›Ñ ￿ ￿ (a) L ›ÑM is injective ñ0 ›ÑL ›MÑ is exact at L. ψ ψ (b) M›Ñ N›Ñis surjective›Ñ ›Ñ M N 0 is exact at N. ￿ ψ { ›Ñ (c) 0 L M N 0 is exact (i.e. at L, M and N) if and only if ￿ is injective, ψ is surjective and ψ induces an isomorphism ψ : M Im ￿ N. SuchP a sequencep q is called a short exact sequence (s.e.s. in short). R (d) If ￿ Hom L￿M is an injective›Ñ morphism,›Ñ › thenÑ therep isq› aÑ s.e.s. ￿ π 0 L M coker ￿ 0

whereP π is thep canonicalq projection. R (d) If ψ Hom M￿N is a surjective morphism, then there is a s.e.s. ›Ñ p q ›Ñ ›Ñ ›Ñ ￿ ψ 0 ker ￿ M N 0 ￿

where ￿ is the canonical injection.

Proposition 5.3

Let Q be anpR-module.´q Then›Ñ the following holds: ›ÑR ›Ñ R›Ñ ›Ñ (a) Hom Q￿￿ : Modψ Ab is a left exact covariant functor. In other words, if 0 L M N 0 is a s.e.s of R-modules, then the induced sequence / ˚ / ˚ / p q p q p q ￿ ψ R R R 0 Hom Q￿L Hom Q￿M Hom Q￿N ˚ “ p q ˚p q“ ˝ ˚ R is an exact sequence of abelian groups. (Here ￿ : Hom Q￿￿ , that is ￿ α ￿ α and similarly for ψ .) p´ q ›Ñ R R (b) Hom ￿Q : Mod Ab is a left exact contravariant functor. In other words, if Skript zur Vorlesung: Cohomology of Groups SS 2018 23 ›Ñ ›Ñ ›Ñ ›Ñ ￿ ψ ˚ ˚ 0 L M N / 0 is a s.e.s of R-modules,/ then the induced/ sequence p q p q p q ψ ￿ R R R 0 Hom N￿Q Hom˚ “M￿Q p Homq L￿Q ˚p q“ ˝ ˚ R is an exact sequence of abelian groups. (Here ￿ : Hom ￿￿ Q , that is ￿ α α ￿ and similarly for ψ .) p ´q p´ q R R Proof : One¨ easily checks thatpHomq Q￿ and Hom ￿Q are functors. ¨ R p q (a) Exactness at Hom Q￿L : Clear. R Exactness atP Hom˚ Qñ￿M :˝We“ have β ker ψ ñ ψ βÄ 0 ñ Im β Ä ker ψ ñ @Im βP ImD ￿ P p q“ p q ￿ ￿ ñ D ￿ Q￿ ! ￿ ›LÑsuch that β ￿ ￿ ￿ ˝ “ ￿ ñ D aP map λ :pQ q L which sends ￿ to ￿ and such that˝ ￿“λ β ￿ inj R ￿ ñ λP Hom˚ Q￿L which send ￿ to ￿ and such that ￿ λ β β Im ￿ ￿

(b) Exercise 5, Exercise Sheet 3.

Remark 5.4 p ´q p´ q R R Notice that Hom Q￿ and Hom ￿Q are not right exact in general. See Exercise 5, Exercise Sheet 3.

Lemma 5.5 (The snake lemma)

Suppose we are given the following commutative/ diagram/ of R-modules/ and R-module homomor- phisms with exact rows: ￿ ψ

L✏ 1 M✏ 1 N✏ 0 / 1￿ / ￿1 / ￿1 ￿ ψ 0 L M N Then the following hold: (a) There exists an exact sequence/ / / 1 / 1 / ￿ ψ δ ￿ ψ 1 1ker ￿ ker ￿ ker ￿ coker ￿ coker ￿ coker ￿￿ p q“ ´ ˝ 1 ˝ ˝ ´ p q P p q ›Ñ p q where ￿ 1, ψ are the1 morphisms induced by the universal property of the quotient, and δ ￿ L L π ￿ ￿ ψ ￿ for every ￿ ker ￿ (here π : L coker ￿ is the canonical homomorphism).›Ñ The map δ is called| the connecting›Ñ homomomorphism. 1 1 1 ker ￿ (b) If ￿ : L ›ÑM is injective, then ￿ 1 : ker ￿ ›kerÑ ￿ is injective. (c) If ψ : M N is surjective, then ψ : coker ￿ coker ￿ is surjective. P P ´ P “ P “1 p2 q` Proof : (a) First, we check that δ is well-defined. Let ￿ ker ￿ and choose two preimages ￿ ￿￿ M of 1 2 1 2 ￿ under ψ. Hence ￿ ￿ ker ψ Im ￿. Thus, there exists ￿ L such that ￿ ￿ ￿ ￿ . Skript zur Vorlesung: Cohomology of Groups SS 2018 24

1 p q“ ˝ p q` p q“ ˝ p q` p q Then, we have 1 2 2 P Pt ￿ u￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ 1 Since ￿ ker ￿, for ￿ 1￿ 2 we have˝ p q“ ˝ p q“ p q“

1 1 ￿ ￿ 1 1 1 1 p qP “ ψ ￿ ￿ ￿ ψ ￿ ￿P ￿ 0￿ p q“ p q ￿ ￿ 1 1 1 ￿ 1 1 ￿ so that ￿ ￿ ker ψ Im ￿ . Therefore,p q“ p thereq“ exists˝ p￿q`L suchp q that ￿ ￿ ￿ ￿ . It follows that 1 1 2 1 1 1 ￿ ￿“ p ￿q`￿2 ￿ ￿ ￿ ￿ ￿1 ￿ 2 1 1 2 Since ￿ is injective, we obtain ￿ ￿ ￿ ￿ . Hence,P ￿ and ￿ havep theq“ same image inpcokerq“￿. Therefore, δ is well-defined.Ä P We1 now want to check1 the exactness at ker ￿. Let ￿ ker ￿. Then ￿ ￿ 0, so that δψ ￿ 0 “ ˇ “ p q P and thus Im ψˇker ￿ ker δ. Conversely, let ￿ ker δ. With the previous notation, this means that 1 1 ˜ ˜ 1 1 1 ￿ 0, and thus ￿ ￿ ￿ for some˝ ￿p q“L. We˝ havep q“ p q“ p q 1 1 ´ p qP ￿ ￿ ˜￿ ￿ ￿ ˜￿ ￿ ￿ ￿ ￿ ￿ 1 Hence, ￿ ￿ ˜￿ ker ￿. It remains´ p q to“ checkp thatq´ this˝ elementp q“ p is sentq“ to ￿ by ψ. We get ` ˘ 1 ˜ 1 ˜ 1 “ ψ ￿ ￿ ￿ ψ ￿ ψ ￿ ￿ ψ ￿ ￿￿ ˇ Hence Im ψˇker ￿ ker δ. The fact that δ is an R-homomorphism, and the exactness at the other points are checked in a similar fashion. (b) Is obvious. (c) Is a a direct consequence of the universal property of the quotient.

Remark 5.6 The name of the lemma comes from the following diagram

0✏ 0✏ 0✏ / / ￿ ψ ker✏ ￿ ker✏ ￿ ker✏ ￿ / / / ￿ ψ L✏ M✏ N✏ 0 /￿ 1 / ￿1 / ￿1 δ 1 1

￿ ψ 0 L✏ 1 M✏ 1 N✏ / / / ￿ ψ coker✏ ￿ coker✏ ￿ coker✏ ￿

000

If fact the snake lemma holds in any abelian category. In particular, it holds for the categories of chain and cochain complexes, which we will study in Chapter 3. Skript zur Vorlesung: Cohomology of Groups SS 2018 25

Lemma-Definition›Ñ 5.7›Ñ ›Ñ ›Ñ ￿ ψ A s.e.s. 0 L M N 0 of R-modules is called split iffit satisfies the following equivalent conditions: ›Ñ ˝ “ N (a) There exists an R-linear map σ : N ›ÑM such that ψ σ˝ id“ (σ is called a section for ψ). L (b) There exists an R-linear map ρ : M L such that ρ ￿ id (ρ is called a retraction for ￿). “ 1 “ ‘ 1 (c) The submodule Im ￿ ker ψ is a direct summand of M, that is there exists a submodule M of M such that M Im ￿ M .

Proof : Exercise.

Example 6 / Z / Z Z / Z / { Z { Z ‘ { Z { Z The sequence ￿ π pr sq “ pr 0s r sq 2 2 2 2 0 defined by ￿ 1 1 ￿ 0 and /πZ is the canonical/ Z projection/ Z into/ the cokernel of ￿ is split but { Z { Z { Z the squence ￿ π pr sq “ pr sq 0 2 4 2 0 defined by ￿ 1 2 and π is the canonical projection onto the cokernel of ￿ is not split.

6 Free, Injective and Projective Modules Free modules

Definition 6.1 (Generating set / RÑ-basis / free R-module) Let M be an R-module and X M be a subset. P P (a) M is said to be generated by X if every element of M can be written as an R-linear combi- ∞ ￿ ￿ nation ￿ X λ ￿, that is with λ R almost everywhere 0. P P (b) X is an R-basis (or a basis) if X generates M and if every element of M can be written in a ∞ ￿ ￿ unique way as an R-linear combination ￿ X λ X (i.e. with λ R almost everywhere 0). p q “ P – P “ (c) M is called free if it admits an R-basis. À À X Notation: In this case we write M ￿ X R￿ ￿ X R : R .

Remark 6.2 P ∞ ￿ ￿ (a) When we write the sum ￿ X λ X, we always assume that the λ are 0 almost everywhere.

(b) Let X be a generating set for M. Then, X is a basis of M if and only if S is R-linearly independent. Skript zur Vorlesung: Cohomology of Groups SS 2018 26

(c) If R is a field, then every R-module is free. (R-vector spaces.)  / Proposition 6.3 (Universal property of free modules) ›Ñ Let P ›beÑ a free R-module with basis X and let ￿ : X P be the inclusion map. For every R- module M and for every map (of sets) ￿ : X M, there exists a unique morphism of R-modules / ￿˜ : P M such that the following diagram commutes> ￿ X✏ M ￿ ￿˜ P Q “ P p q“ P p q ∞ ￿ ∞ ￿ Proof : If P ￿ ￿ X λ ￿ (unique expression), then we set ￿˜ ￿ ￿ X λ ￿ ￿ . It is then easy to check ￿˜ has the required properties.

Proposition 6.4 (Properties of free modules)

(a) Every R-module M is isomorphic top a quotient´q of a free R-module. R (b) If P is a free R-module, then Hom P￿ is an exact functor. t u P ￿ ￿ I Proof : (a) Choose a set ￿ of generators of M (take›Ñ all elements of M if necessary). Then define àP ￿ : R M p￿ Iq P fi›Ñ P ￿ ￿ I ÿ ￿ ￿ ￿ ￿ I ￿ ￿ ￿ –p P q{ À p ´q￿ I ›Ñ It follows that M R ker ￿ . ˚ p q›Ñ p q ›Ñ ˚p q“ ˝ (b) We know that Hom P￿ is left exact for any R-module P. It remains to prove that if ψ : M N P p q R R is a surjective R-linear maps, then ψ : Hom P￿M Hom P￿N : β ψ β ψ β is R also surjective. So let α Hom P￿N . We have the following situation: D P ~ ? ✏ / α /

M ψ N 0 t u P p qP P ￿ ￿ IP p q“ p q ￿ t u P ›Ñ fiÑ Let ￿ be an R-basis of P. Each α ￿ N is in the image of ψ, so that for each ￿ I there ￿ ￿ ￿ ￿ ›￿ÑI ￿ p￿ q“ @existsP ￿ M such that ψ ￿ α ￿ . Hence, there is a map β : ￿ M￿￿ ￿ . By the ˜ ˜ ￿ ￿ universal property of free modules this˝ inducesp q“ anpR-linearq“ p mapq β : P M such that β ￿ ￿ ￿ I. Thus ￿ ￿ ￿ ˝ ψ β˜ ￿ t ψu P￿ α ￿ ￿ “ ˝ “ ˚ ˜ ￿ ￿ I ˜ so that ψ `β˘and α coincide on the basis ￿ . By the uniqueness of β, we must have α ψ β˜ ψ β˜ . Skript zur Vorlesung: Cohomology of Groups SS 2018 27

Injective modules

Proposition-Definition 6.5 (Injective module) Let I be an R-module. Thenp´ q the following are equivalent: R (a) TheP functor pHom q ￿I is exact. ˚ p q›Ñ p q R ›Ñ R R ›Ñ (b) If ￿˝ “Hom L￿M is a injective morphism, then ￿ : Hom M￿I Hom L￿I is surjective (hence, any R-linear map α : L I can be lifted to an R-linear map β : M I, i.e., β ￿ ›αÑ). ›Ñ ˝ “ (c) If η : I M is an injective R-linear map, then η splits, i.e., there exists ρ : M I such I that ρ η Id .

If I satisfies these equivalent conditions, then I is called injective.

Proof : Exercise.

Remark 6.6 p q § Note that Condition b is particularly interesting when L M and ￿ is the inclusion.

Projective modules

Proposition-Definition 6.7 (Projective module) Let P be an R-module.p Then´q the following are equivalent: R (a) The functorP Homp Pq￿ is exact. ˚ R p q›Ñ p q (b) If ψ Hom M￿N is a surjective morphism of R-modules, then the morphism of abelian R R groups ψ›Ñ: Hom P￿M Hom P￿N is surjective. ›Ñ ˝ “ (c) If π : M P is a surjective R-linear map, then π splits, i.e., there exists σ : P M such P that π σ Id .

(d) P is isomorphic to a direct summand of a free R-module.

If P satisfies these equivalent conditions, then P is called projective. Z Z Z Example 7 “ (a) If R , then every submodule of a free “-module is again– free‘ (mainp ´ theoremq on -modules). ‰ 2 (b) Let ￿ be an idempotent in R, that is ￿ ￿. Then, R R￿ R 1 ￿ and R￿ is projective but not free if ￿ 0￿ 1t. u P ￿ ￿ J ￿ (c) A product of modules I t uisP injective if and only if each I is injective. ￿ ￿ I ￿ (d) A direct sum of modules P is projective if and only if each P is projective. Skript zur Vorlesung: Cohomology of Groups SS 2018 28

7 Tensor Products DefinitionZ 7.1 (Tensor product of R-modules) ˆ Let M be a right R-module and let N be a left R-module. Let F be the free abelian group (= free -module) with basisp M` N.q´p Let G beq´p the subgroupq of@ F generatedP @ byP all the elements 1 2 1 2 1 2 p￿ ￿` ￿￿q´p￿ ￿￿q´p￿ ￿￿q ￿ @ ￿P￿￿ @ M￿ ￿P N￿ 1 2 1 2 1 2 p￿￿ ￿ q´p￿ ￿￿q ￿@ P ￿￿ ￿@ ￿P ￿@ PM￿ ￿ ￿￿ N￿ and ￿￿￿ ￿ ￿￿ ￿￿ ￿ ￿ M￿ ￿ N￿ ￿ R￿ b “ { p qP b b R The tensor product of M and N (balanced over R), is the abelian group M N : F . The R G class of ￿￿ ￿ F in M N is denoted by ￿ ￿.

Remark 7.2 b “x b | P P yZ R (a) M bN ￿ ￿ ￿ M￿￿ N . R (b) In M N, wep have` theqb relations“ b ` b @ P @ P 1 2 1 2 1 2 ￿bp ￿` ￿q“ ￿b ￿` ￿b ￿￿ @ ￿P￿￿ @ M￿ ￿P N￿ 1 2 1 2 1 2 ￿ b ￿ “ ￿ b ￿ @￿ P ￿ @ ￿P￿ @￿P M￿ ￿ ￿￿ N￿ and ￿￿b ￿“ ￿“ b￿￿￿ @ ￿P M￿@ ￿ P N￿ ￿ p´R￿ qb “´p b q“ bp´ q @ P @ P In particular, ￿ 0 0 0 ￿ ￿ M, ￿ N and ￿ ￿ ￿ ￿ ￿ ￿ ￿ M, ￿ N.

Definition 7.3 (R-balanced map) ˆ ›Ñ Let M and N be as above and let A be an abelian group. A map ￿ : M N A is called R-balanced if p ` q“ p q` p q @ P @ P 1 2 1 2 1 2 ￿p￿ ￿` ￿￿q“ ￿p ￿ ￿￿q` ￿p ￿ ￿￿q ￿ @ ￿P￿￿ @ M￿ ￿P N￿ 1 2 1 2 1 2 ￿p￿￿ ￿ q“￿ p ￿ ￿￿q ￿@ P￿ ￿￿@ ￿ P￿ @￿ P M￿ ￿ ￿￿ N￿ ￿ ￿￿￿ ￿ ￿ ￿￿ ￿￿ ￿ ￿ M￿ ￿ N￿ ￿ R￿

Remark 7.4 ˆ ›Ñ b p qfiÑ b R The canonical map ￿ : M N M N￿ ￿￿ ￿ ￿ ￿ is R-balanced.

Proposition 7.5 (Universal property of the tensor product) Z ˆ ›Ñ b ›Ñ Let M be a right R-module and let N be a left R-module./ For every abelian group A and every ˆ ; R R-balanced map ￿ : M N A there exists a uniqueö ￿ -linear map ￿ : M N A such that the following diagram commutes: M ✏ N A ￿ b ￿ R M N Skript zur Vorlesung: Cohomology of Groups SS 2018 29 ˆ ›Ñ ›Ñ { Z Z ›Ñ Proof : Let ￿ : M N F denote the canonical inclusion, and let π : F F G denote the canonical ˝ “ Ñ p q ˜ projection. By the universal property of the free -module, there exists a unique -linear{ › mapÑ ￿ : F A such that ˜￿ ￿ ￿. Since ￿ is R-balanced, we have that G ker ˜￿ . Therefore, the universal property of ˝ “ the quotient yields the existence of a unique homomorphism of abelian/ groups ￿ : F G A such that ˆ = K ￿ π ˜￿: ￿ M N A

￿ ✏ ˜￿

￿ F ￿ π ✏ b – { R “ ˝ ˝ “ M˝ ˝N“ F˝ G“ Clearly ￿ π ￿, and hence ￿ ￿ ￿ π ￿ ˜￿ ￿ ￿.

Remark 7.6 t u P t ￿ u￿ PI (a) Let M be a collection of right R-modules, M be a right R-module, N be a left R-module ￿ ￿ J and N be a collection of left R-modules.b – Then,p web haveq P P à ￿ R à ￿ R M N M N ￿ Ib – ￿ I p b q P P R à ￿ à R ￿ M ￿ J N ￿ J M N ￿ b – b fiÑ R (b) For every R-module M, we have R M M viab￿ ￿– ￿￿P. R À (c) If P be a free left R-modulep with basisq X, then M P ￿p X M.q b p q R (d) Let Q be a ring. Let M be a Q￿R -bimodule and let N be an R￿S -module. Then M N can be endowed withp b theq structure“ b of a Q@￿S P-bimodule@ P via@ P @ P ￿ ￿ ￿ ￿ ￿￿ ￿￿￿ ￿ Q￿ ￿ S￿ ￿ M￿ ￿ N￿ p q b (e) If R is commutative, then any R-module can be viewed as an R￿R -bimodule. Then, in R particular, M N becomes an R-module. ›Ñ 1 ›Ñ 1 Z 1 1 (f) Tensor product of morphisms: Let ￿ : M M beb a morphismb › ofÑ rightbR-modules and ￿ : N N be a morphism of left R-modules. Then, by the universal property of the p b qp b q“ p qb p q R R tensor product, there exists a unique -linear map ￿ ￿ : M N M N such that ￿ ￿ ￿ ￿ ￿ ￿ ￿ ￿ .

Proposition 7.7 (Right exactness of the tensor´b product) ›Ñ R R (a) Let N be a left R-module. Then bN ´: Mod ›ÑAb is a right exact covariant functor. R R (b) Let M be a right R-module. Then M : Mod Ab is a right exact covariant functor. Skript zur Vorlesung: Cohomology of Groups SS 2018 30

Remark 7.8 ´b b ´ R R The functors N and M are not left exact in general.

Definition 7.9 (Flat module) ´b ›Ñ R R A left R-module N is called flat if the functor N : Mod Ab is a left exact functor.

Proposition 7.10 Any projective R-module is flat.

“ P Proof : To begin with, we note that a direct sum of modules is flat if and only if each module in the sum is À￿ X flat. Next, consider the free R-module P/ / R￿. If / / ￿ ψ 1 2 3 0 M M M 0

is a short exact sequence/ of right R-modules,b / then we obtain b / / b P P b P P b P ￿ Id ψ Id 1 R ` À ˘ 2 R ` À ˘ 3 R ` À ˘ 0 M – ￿ X R M – ￿ X R M – ￿ X R 0 ✏ ✏ ✏ / p q P / p q P / / P ￿ X P ￿ X P ￿ ψ À 1 À 2 À 3 0 ￿ X M ￿ X M ￿ X M 0￿ ´b Since the original sequence is exact, so is the bottom1 sequence,1 and therefore so is1 the top sequence. R ‘ “ Hence, 1 P is exact when P is free. Now, if N is a projective R-module, then N N P for some free R-module P and for some R- module N . It follows that N is flat, by the initial remark.