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Chapter 3. Homological Algebra 8 Chain and Cochain Complexes

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Chapter 3. Homological Algebra 8 Chain and Cochain Complexes 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.
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