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. Throughout this chapter R denotes a ring, and unless otherwise specified, all rings are assumed to be unital and associative. Reference: [Rot09] J. J. R, An introduction to homological algebra. Second ed., Universitext, Springer, New York, 2009. [Wei94] C. A. W, An introduction to homological algebra, Cambridge Studies in Advanced Math- ematics, vol. 38, Cambridge University Press, Cambridge, 1994. 8 Chain and Cochain Complexes Definition 8.1 (Chain complex) (a) A chain complex (or simply a complex) of R-modules` is a sequence p ‚ ‚q“ ¨¨¨›Ñ ` ›Ñ1 ›Ñ ´ ›Ñ ¨ ¨ ¨ ˆ ˙ 1 1 ZC C C C P P p ´ q ˝ ` “ ‚ p ‚ ‚q R 1 1 where for each , C is an R-modules and Hom C C satisfies 0. We often write simply C instead of C . Z (b) The integer is called theP degree of the R-module C . Z (c) The R-linear‚ maps ( ) are called the differential maps“ . P † Z P § 0 (d) A complex C is called non-negative (resp. positive) if C 0, for all (resp. for all 0 ). Notice that sometimes we will omit the indices and write for all differential maps, and thus the 31 Skript zur Vorlesung: Cohomology of Groups SS 2018 32 ˝ ` “ “ “ 2 § 1 condition 0 can be written as 0. If there is an integer N such that C 0 for all N, then we omit to write the zero modules and` zero maps` on the right-hand side of the complex: ¨¨¨›Ñ ` ›Ñ ` ›Ñ N 2 N 1 N 2 N 1 N C “ C C• Similarly, if there is an integer N such that C 0 for all N, then we omit to write the zero modules and zero maps on the left-hand side of the´ complex: ›Ñ ´ ›NÑ1 ´ ›Ñ ¨ ¨ ¨ N N N 1 N 2 C C C 1 p ‚ ‚q p ‚ ‚q Definition 8.2 (Morphism of complexes) Z ‚ ‚ ›Ñ ‚ ›Ñ P ˝ ` “ A1 morphism of complexesZ (or a chain map) between two chain complexes C and D , ` ˝ ` P 1 written : C D , is a familiy of R-linear maps : C D ( ) such that 1 1 for each , that is` such/ that` the/ following/ diagram´ commutes:/ ¨¨¨ 2 ` 1 ´ 1 ¨¨¨ ` 1 ´ 1 1 C ✏ 1 C✏ 1 C ✏ 1 `/ 1 ` / / 1 ´ / ¨¨¨ ` ´ ¨¨¨ 2 1 1 1 1 D D D Notation. Chain complexes together with morphisms of chain complexes (and composition given by R degreewise composition of R-morphisms) form a category, which we will denote by Ch( Mod). Definition 8.3 (Subcomplex1 / quotient complex) 1 Z ‚ p ‚ ‚q § P 1 1 Z p qÄ ´ P (a) A subcomplex C1 of a chain complex C is a family of R-modules 1Cã C ( ), such p ‚ ‚q ‚ Ñ ‚ 1 1 Z that C C for every . P In this case, C becomes a chain complex and the inclusion C C given by the 1 1 1 canonical‚ inclusion of C ‚ into C for each is a chain‚{ ‚ map. { Z 1 1 P { ›Ñ ´ { ´ (b) If C is a subcomplex of C , then the quotient complex C C is the familiy of R-modules C C 1 ( ) together with the differential maps : C1 C C C 1 uniquelyZ determined ‚ ‚ ›Ñ ‚{ ‚ P by the universal property›Ñ { of1 the quotient. In this case, the quotient map π : C C C defined for each by the canonical projection π : C C C is a chain map. 1 Definition‚ 8.4‚ (›KernelÑ ‚ / image / cokernel) p ‚ ‚q p ‚ ‚q Z Let : C D be‚ a morphism of chain complexes‚ between ‚C“pt and uDP ‚q. Then, Z 1 (a) the kernel of ‚ is the subcomplex of C‚ defined by ker ‚ “pt: ker u P ‚q ; (b) the image of is‚ the subcomplex of D defined by‚ Im“ ‚{: Im‚ ; and (c) the cokernel of is the quotient complex coker : D Im . Skript zur Vorlesung: Cohomology of Groups SS 2018 33 R With these notions of kernel and cokernel, one can show that Ch( Mod) is in fact an abelian category. Definitionp ‚ 8.5‚q (Cycles, boundaries, homology) Let C be a chain complex of R-modules.“ p ‚q “ (a) An -cycle is an element of ker : `Z “C : p Z‚q. “ Z 1 (b) An -boundary is˝ an element` “ of Im Ñ : B ÑC :Ñ B .@ P 1 [Clearly, since 0, we have 0 B Z‚ C p ‚q “.]{ (c) The -th homologyZ module (or simply group) of C is H C : Z B . P p´q ›Ñ R R In fact, for each , H : Ch( Mod) Mod is a covariant additive functor (see Exercise 1, Exercise Sheet 6), which we define on morphisms as follows: 1 Lemma 8.6‚ ‚ ›Ñ ‚ p ‚ ‚q p ‚ ‚q ‚ Let : C D be a morphism of chain complexes between C and D . Then induces an R-linear map p ‚q p ‚q›Ñ p ‚q ` p ‚qfiÑ p q` p ‚q H : H C H D Z P B C ˚ B D p ‚q Z 1 for each P . To simplify, thisp ‚q map›Ñ isp often‚q{ denotedp ‚q by insteadp ‚q of›ÑH p ‚.q{ p ‚q 1 Proof : Fix , and let πp ‚:qZ ÄC p ‚q Z C B CP , resp. π : ˝Z Dp q“ Z´ ˝D pBq“D , be the canonical projections. p qP p` q ˘ 1 First, notice that Z C Z D because if Z , then 0. Hence, p ‚q Ä p ‚q P p ‚q “ ` p q P ` we have Z D . 1 ‚ ` ˘ p q“ ˝ ` p q“ ` ˝ ` p qP1 p ‚q 1 Similarly, we have B C B D . Indeed, if B C , then for some1 C , 1 1 ˝ and because is a chain map we have 1 B D . Therefore, by the universal property of the quotient, there exists a unique R-linear map π such 1 that the following diagram commutes: / / p ‚q p ‚q p ‚q{ π 3 p ‚q Z C Z 1D Z D B D ˝ π ✏ π p ‚q{ p ‚q 1 Z C B C p ‚q “ ˝ Set H : π . The claim follows. p ‚q It should be thought that the homology module` H/ C measures/ the "non-exactness" of the sequence ` 1 ´ Z 1 1 p´q P C C C Moroever, the functors H ( ) 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 R-modules. Skript zur Vorlesung: Cohomology of Groups SS 2018 34 / ‚ / ‚ / / Theorem‚ 8.7 (Long‚ exact‚ sequence‚ in homology‚ ) ψ Let 0 C D E 0 be a s.e.s. of chain complexes. Then there is a long exact sequence ` / ˚ / ˚ / / ˚ / ˚ / ¨¨¨ 1 p ‚q p ‚q p ‚q ´ p ‚q ´ p ‚q ¨¨¨ δ ψ δ ψ Z 1 1 HP C Hp ‚qD›Ñ ´Hp E‚q H C H D 1 where for each , δ : H E H C is an R-linear map, called connecting homomor- phism. ‚ Note: Here 0 simply denotes the zero¨¨¨› complexÑ ›Ñ, that›Ñ is the›Ñ complex ¨ ¨ ¨ 0 0 0 0 0 ‚ consisting of zero modules and zero morphisms. We often write simply 0 instead of 0 . ‚ ‚ ‚ Z P Proof : To simplify, we denote all differential maps of the three complexes C , D , E with the same letter , and we fix . First, we apply the/ “non-snake”/ part of the/ Snake Lemma/ to the commutative diagram ψ 0 C D E 0 ✏ ✏ ✏ / ´ / ´ / / ´ 1 ´ 1 ´ ψ 1 1 1 0 C D E 0 and we obtain two exact sequences / / / p ‚q p ‚q p ‚q ψ 0 Z C Z D Z E ´ / ´ / / ´ { 1 ´ { 1 ´ { and ψ 1 1 1 C Im D Im E Im 0´ Shifting indices in both sequences we obtain similar sequences in degrees 1, and respectively. Therefore, we have a commutative diagram with/ exact rows of the/ form: / { { { ` ` ψ ` 1 1 1 C Im D Im E Im 0 ✏ ✏ ✏ / ´ / ´ / ´ p ‚q 1 ´ p ‚q 1 ´ p ‚q ψ 1 1 1 { 0` ›Ñ Z ´ p C‚q Z D Z E 1 ›Ñ ´1 ` Ñ where : C Im Z C is the unique R-linear map induced by the universal property of ‚ ‚ 1 1 the quotient by : C C (as Im ker by definition of a chain complex), and similarly for D and E . Therefore, the Snake Lemma yieldsp ‚q › theÑ existencep of‚q the connecting homomorphisms “ p ‚q “ ´ p ‚q δ : ker E coker C loooomoooon loooooomoooooon 1 Z H E H C P for each ` as/ well as the˚ required/ long˚ exact/ sequence: / ˚ / ˚ / ¨¨¨ 1 p ‚q p ‚q p ‚q ´ p ‚q ´ p ‚q ¨¨¨ δ ψ δ ψ 1 1 H“ C H“ D H“ E H“ C H“ D loomoon loomoon loomoon loooomoooon loooomoooon ker ker ker coker coker Skript zur Vorlesung: Cohomology of Groups SS 2018 35 We now describe some important properties of chain maps and how they relate with the induced mor- phisms in homology. ‚ ‚ ›Ñ ‚ p ‚q DefinitionZ 8.8 (Quasi-isomorphism) P A chain map : C D is called a quasi-isomorphism if H is an isomorphism for all . ‚ ‚ Warning: A quasi-isomorphism does not imply that the complexes C and D are isomorphic as chain complexes. See Exercise 2, Sheet 5 for a counter-example.