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`qS -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 RS -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 RS -bimodules (with RS -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 MN : : 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 MN Hom MN Hom MN “ ψ p q “ p qψ : M N ψ . p q R R In case N M, we write End M : Hom MM 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 MN 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 MN 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 MN . 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 LM 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 CO [ 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.