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Notes on Homological Algebra

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Notes on Homological Algebra Mariusz Wodzicki December 1, 2016 1 Foundations 1.1 Preliminaries 1.1.1 A tacit assumption is that A, B, ... , are abelian categories, i.e., additive categories with kernels, cokernels, every epimorphism being a kernel, and every monomorphism being a cokernel. 1.1.2 In an abelian category every morphism has an image-coimage factorization • fl [^ i ^ q [ (1) fl ••u a in which i is an image of a, i.e., a kernel of a cokernel of a, and q is a coimage of a, i.e., a cokernel of a kernel of a, cf. Notes on Category Theory. 1.1.3 Exact composable pairs A composable pair of arrows a b •••u u (2) is said to be exact if b factorizes k ◦ b0 with k being a kernel of a and b0 being an epimorphism. 1.1.4 If l is a cokernel of b, then l is a cokernel of k , i.e., a cokernel of b and a kernel of a form an extension l k . •••u u u (3) x Unlike the original condition this last condition is self-dual. 3 1.1.5 Note that a cokernel of b and a kernel of a form an extension precisely when k is an image of b and l is a coimage of a • fl [^^ a [l fl b •••u a u [^ [ k ^ flfl b¯ • in which case coimage factorization factorizes a through a cokernel of b with a being a monomorphism while image factorization factorizes b through a kernel of a with b¯ being an epimorphism. Let us record this fact for future use. Lemma 1.1 For any composable pair of arrows, the following three conditions are equivalent (a) b factorizes through a kernel of a b ••u [^ [ ff k ^ ffflfl b0 • (b) a factorizes being a cokernel of b • 0 fl [^^ a ff [l fffl ••u a (c) a cokernel of b and a kernel of a form an extension. 1.1.6 Condition a ◦ b = 0 A weaker condition, a ◦ b = 0, is equivalent to existence of a factorization of b b ••u u k l (4) v u ••u m x 4 with m being a monomorphism. Note that m satisfying identity a = k ◦ m ◦ l is unique in view of k being a monomorphism and l being an epimor- phism. 1.1.7 Homology A cokernel c of m is called a homology of composable pair (2) and it forms an extension c m •••u u u .(5) x 1.1.8 The target of c is usually referred to as the homology (group) of the com- posable pair. This terminology originally was applied to homomorphisms of abelian groups but the tradition is so well established that it continues to be used also in the context of general abelian categories. 1.1.9 Note that a homology morphism c is zero or, equivalently, in view of the fact that c is an epimorphism, its target is zero, precisely when m is an isomorphism, i.e., when (2) is exact. 1.1.10 Composable pairs (2) satisfying the condition a ◦ b = 0 form a full subcat- egory of the category of composable pairs A2,0 . Exercise 1 Show that any assignment of a morphism c on the class of such composable pairs admits a unique extension to a functor c : A2,0 −! Arr A. 1.1.11 The compositions Z ˜ s ◦ c and H ˜ t ◦ c 5 with the source and, respectively, the target functors s : Arr A −! A and t : Arr A −! A are referred to as a cycles and, respectively, homology functors. Exercise 2 Express the condition a ◦ b = 0 in terms of a certain factorization of a. 1.1.12 A dual approach to homology Based on the corresponding factorization of a instead of b one can develop a dual approach to homology. 1.1.13 Chain complices We shall use the term chain complex in two ways: either in a loose sense, to denote any sequence of composable morphisms of any length such that the composition of any two composable arrows is zero or, in a strict sense, as a Z-graded object C• = (Cl)l2Z equipped with an endomorphism ¶ of degree −1 such that ¶ ◦ ¶ = 0. The graded endomorphism ¶ is referred to as the boundary morphism or operator. 1.1.14 Notation We may use notation (C•, ¶•), (C, ¶), C• or, simply, C, to denote a chain complex (Cl, ¶l)l2Z . 1.2 Projective and injective objects 1.2.1 Projective objects in an abelian category An object P is said to be projective if it possesses the Lifting Property for epimorphisms, more precisely, if for any diagram P u ••u u e 6 with e being an epimorphism, there exists an arrow P \ \] • such that the diagram P \ \ \\] u ••u u e commutes. 1.2.2 The above definition makes sense in any category, in general categories, however, there may be “too many” epimorphisms making the Lifting Property with respect to the class of all epimorphisms very restrictive. 1.2.3 An example: the category of topological spaces In the category of topological spaces and continuous mappings, epimor- phisms are mappings with dense image. For any nonsurjective epimor- phism f : Y −! Z and any nonempty space X the maping that sends all points in X to a point in Z n f (Y) cannot be lifted to Y. Thus, the only topological space that has Lifting Property for epimorphisms is the empty space. 1.2.4 Projective objects in a general category For the above reasons, projective objects in general categories are defined by the Lifting Property with respect to a specific class of epimorphisms, e.g., the class of all strong epimorphisms, see Notes on Category Theory. 7 1.2.5 Injective objects in an abelian category An object I is said to be injective if it possesses the Extension Property for monomorphisms, more precisely, if for any diagram I u ••v m w with m being a monomorphism, there exists an arrow I \^ \ • such that the diagram I u \^\ \ \ ••v m w commutes. Exercise 3 Show that P is projective if and only if for any commutative diagram with an exact row P 0 f fl u •••u u a b there exists an arrow P \ ˜ \]f • such that the diagram P \ \f˜ f \\] u •••u u a b commutes. 8 1.2.6 Dually, I is injective if and only if for any commutative diagram with an exact row I u [^[ [0 y [ •••u u a b there exists an arrow I y˜ fffi ff • such that the diagram I u y˜ fffffi ff y ff •••u u a b commutes. 1.2.7 Characterization of projectivity in terms of extensions Lemma 1.2 An object P is projective if and only if every extension of P, p i P u u ••u , x splits. Proof. Consider the diagram P idP . u p P u u • If P is projective, then there exists iP such that p ◦ iP = idP . 9 Vice-versa, suppose that a diagram P a u e A00 u u A is given. Consider a pullback by a of the extension e k A00 u u A u A0 x where k is a kernel of e, e¯ k¯ P u u A¯ u A0 x a a¯ u u e k A00 u u A u A0 x i If P A¯ splits the pullback extension, then a¯ ◦ i lifts a. Indeed, w e¯ ◦ (idA¯ −i ◦ e¯) = 0 implies that 0 idA¯ −i ◦ e¯ = k¯ ◦ p for a unique p0 and 0 a ◦ e¯ = e ◦ a¯ = e ◦ a¯ ◦ idA¯ = e ◦ a¯ ◦ i ◦ e¯ + e ◦ a¯ ◦ k¯ ◦ p = e ◦ a¯ ◦ i ◦ e¯ which in turn implies that a = e ◦ (a¯ ◦ i) since e¯ is an epimorphism. 1.3 Categories of chain complices 1.3.1 The (strict) category of graded objects The strict category of Z-graded objects has Z-sequences (Aq)q2Z as its objects and Z -sequences of morphisms (fq)q2Z , 0 fq Aq u Aq ,(6) as morphisms. In order to avoid confusion with morphisms in the category of Z-graded chain complices Ch(A), we shall denote such sequences graded maps. 10 1.3.2 Ch(A) Morphisms in the category of chain complices are graded maps that com- mute with the boundary operators, 0 ¶q ◦ fq = fq−1 ◦ ¶q (q 2 Z). 1.3.3 The graded category of graded objects We shall also consider graded maps of degree d, i.e., sequences of morphisms (fq)q2Z in A, 0 fq Aq+d u Aq (q 2 Z),(7) and often will refer to them simply as maps of degree d. The resulting graded category will be referred to as the graded category of graded objects of A. 1.3.4 Subcategories of Ch(A) The category of chain complices Ch(A) of an abelian category A has several naturally defined subcategories, for example, for any pair of subsets J ⊆ I ⊆ Z, let ChI,J(A) denote the full subcategory of chain complices with chain objects being zero in degrees outside I , Cl = 0 for l < I, and homology objects being zero in degrees outside J , Hl = 0 for l < J. Thus, ChZ,Æ(A) denotes the subcategory of acyclic chain complices. When I = J we shall use notation ChI (A). 1.3.5 Ch+(A) The subcategory Ch+(A) of complices vanishing at −¥, i.e., satisfying Cl = 0 for l 0, is the union of the subcatories Ch≥n(A) of complices satisfying Cl = 0 for l < n.
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