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1. Chain Complexes Definition. a Sequence of Abelian Groups ...C−2 1. Chain complexes Definition. A sequence of abelian groups :::C−2;C−1;C0;C1;::: with homomorphisms @i : Ci+1 ! Ci is called a (homological) chain complex if @i−1 ◦ @i = 0 for all i 2 Z. A cohomological chain complex is almost the same thing, but with reversed grading: a sequence of abelian groups :::C−2;C−1;C0;C1;::: together with homomorphisms di : Ci−1 ! Ci such that di ◦ di−1 = 0 for all i 2 Z. We will concentrate on homological chain complexes; all results hold analogously for cohomological chain complexes. A chain complex (or just a sequence of abelian groups with homomorphisms) is called bounded below (bounded above) if Ci = 0 for i 0 (resp. i 0). It is called non-negatively graded (non-positively graded) if Ci = 0 for i < 0 (resp. i > 0). Definition. We call the subgroup Zi(C•) = ker(@i−1 : Ci ! Ci−1) < Ci the sub- group of i-cycles and the subgroup Bi(C•) = im(@i : Ci+1 ! Ci) < Ci the sub- group of i-boundaries. Lemma 1.1. For any chain complex C•, Bi(C•) is a subgroup of Zi(C•). Definition. The ith homology group of a chain complex C• is defined as the quotient group Hi(C•) = Zi(C•)=Bi(C•): If Zn = Bn for all n (and thus Hn = 0), we call C• exact or acyclic. An exact chain complex is more usually called exact sequence. An exact sequence of the form 0 ! A0 ! A ! A00 ! 0 is often called a short exact sequence. Lemma 1.2. Let A, B, C be abelian groups and f : A ! B and g : B ! C homomorphisms. f (1)0 ! A −! B is exact iff f is injective. f (2) A −! B ! 0 is exact iff f is surjective. f (3)0 ! A −! B ! 0 is exact iff f is an isomorphism. f g (4)0 ! A −! B −! C ! 0 is exact iff f is injective, g is surjective, and ker g = im f. Definition. Let A•, B• be sequences of abelian groups and homomorphisms (or chain complexes). A map of sequences (or map of chain complexes) is a commutative diagram A A @n @n−1 ··· An+1 An An−1 ··· fn+1 fn fn−1 B B @n @n−1 ··· Bn+1 Bn Bn−1 ··· 1 2 Lemma 1.3. A map of chain complexes f : C• ! D• induces maps Z(f): Zn(C•) ! Zn(D•) of n-cycles; B(f): Bn(C•) ! Bn(D•) of n-boundaries, and Hn(f) = f∗ : Hn(C•) ! Hn(D•) on homology. Lemma 1.4 (Five-lemma). Let A1 A2 A3 A4 A5 f1 f2 f3 f4 f5 B1 B2 B3 B4 B5 be a commutative diagram of abelian groups with exact rows. Then: (1) if f2, f4 are surjective and f5 is injective then f3 is surjective. (2) if f2, f4 are injective and f1 is surjective then f3 is injective. (3) in particular, if f1, f2, f4, f5 are isomorphisms then so is f3. Definition. A short exact sequence of the form 0 ! A0 ! A0 ⊕ A00 ! A00, where the first map is the inclusion into the first summand and the second map is the projection onto the second, is called split exact. See homework problem 1.2 for characterizations of split exact sequences. Definition. Let f : A ! B be a homomorphism between abelian groups. Define its cokernel coker(f) to be the quotient group B= im(f) and its coimage coim(f) to be A= ker(f). Lemma 1.5. For any homomorphism f : A ! B of abelian groups, we have: (1) f : coim(f) ! im(f) is an isomorphism; f (2)0 ! ker(f) ! A −! B ! coker(f) ! 0 is exact. Lemma 1.6 (Snake lemma). Given a diagram of abelian groups A1 A2 A3 0 0 B1 B2 B3 (1.7) with exact rows. Let Ki denote the kernel of Ai ! Bi and Ci its cokernel. Then there is a \snake homomorphism" K3 ! C1 such that the sequence K1 ! K2 ! K3 ! C1 ! C2 ! C3 3 is exact: 0 0 0 K1 K2 K3 A1 A2 A3 0 0 B1 B2 B3 C1 C2 C3: 0 0 0 If A1 ! A2 is injective then so is K1 ! K2, and if B2 ! B3 is injective then so is C2 ! C3. Furthermore, the snake map is natural, meaning that if we have a map (Ai;Bi) ! 0 0 (Ai;Bi) of diagrams of the type (1.7) then the following square commutes: K3 C1 0 0 K3 C1: i p Theorem 1.8. Let 0 ! A• −! B• −! C• ! 0 be a short exact sequence of chain complexes (meaning 0 ! An ! Bn ! Cn ! 0 is exact for each n 2 Z). Then there is a connecting homomorphism δn : Hn+1(C•) ! Hn(A•) such that the following long sequence is exact: p∗ δn i∗ p∗ δn−1 i∗ ··· −! Hn+1(C•) −! Hn(A•) −! Hn(B•) −! Hn(C•) −−−! Hn−1(A•) −!· · · : The homomorphism δ is natural: given a map of short exact sequences of chain 0 0 0 complexes (A•;B•;C•) ! (A•;B•;C•), the following square commutes: δ Hn+1(C•) Hn(A•) 0 δ 0 Hn+1(C•) Hn(A•): 2. Categories and functors Definition. A category C consists of: • a class ob(C) of objects; • for each pair of objects X, Y 2 ob(C), a set of morphisms HomC(X; Y ); • for each object X 2 ob(C), an element idX 2 HomC(X; X) called identity morphism; 4 • for each three objects X, Y , Z 2 ob(C), a map ◦: HomC(Y; Z) × HomC(X; Y ) ! Hom(X; Z); (g; f) 7! g ◦ f called composition. These have to satisfy the following axioms: (1) The composition ◦ is associative; (2) For f 2 HomC(X; Y ), idY ◦f = f and f ◦ idX = f. A morphism f 2 HomC(X; Y ) is called an isomorphism (and the objects X, Y isomorphic) if there is another morphism g 2 HomC(Y; X) such that g ◦ f = idX −1 and g ◦ g = idY . If such a g exists, it is unique and is denoted by f . We will often abuse notation and write X 2 C for X 2 ob(C), f 2 Hom(X; Y ) or even just f : X ! Y for f 2 HomC(X; Y ), and id for idX . We will also use commutative diagrams to denote equalities between compositions of morphisms. Definition. We use the following standard notations for familiar categories: Set: The category of sets and functions; Ab: The category of abelian groups and homomorphisms; Top: The category of topological spaces and continuous maps. Definition. Let C, D be categories. A (covariant) functor F : C!D consists of: • a function ob(C) ! ob(D), also called F ; and • for every X, Y 2 ob(C), a function HomC(X; Y ) ! HomD(F (X);F (Y )) denoted by f 7! F (f) or f 7! f∗ satisfying (idX )∗ = idF (X) and (g ◦ f)∗ = g∗ ◦ f∗. A contravariant functor F : C!D consists of: • a function ob(C) ! ob(D), also called F ; and • for every X, Y 2 ob(C), a function HomC(X; Y ) ! HomD(F (Y );F (X)) denoted by f 7! F (f) or f 7! f ∗ ∗ ∗ ∗ ∗ satisfying (idX ) = idF (X) and (g ◦ f) = f ◦ g . (\It turns arrows around.") Definition. A natural transformation η : F ! G between two functors F , G: C!D consists of a morphism ηX 2 HomD(F (X);G(X)) for each object X 2 C such that for each morphism f : HomC(X; Y ), the following diagram commutes: ηX F (X) G(X) F (f) G(f) ηY F (Y ) G(Y ): Natural transformations between contravariant functors are defined analogously. A natural transformation η : F ! G is called natural isomorphism (and F and G isomorphic, F ' G) if ηX is an isomorphism for all X 2 C. Definition. A covariant functor F : C!D is called an equivalence of categories if there is another functor G: D!C such that G◦F ' IdC and F ◦G ' IdD, where IdC, IdD denote the identity functors on C and D, respectively. 5 Definition. Let C be a category and (Xi)i2I a family of objects in C, for some index set I. An object X together with morphisms ιi : Xi ! X is called coproduct of ` the Xi, and is denoted by i2I Xi, if for each test object Y 2 C, the map HomC (ιi;−) Y HomC(X; Y ) −−−−−−−! HomC(Xi;Y ) i2I is a bijection. The coproduct of only two objects is denoted by X1 t X2. Similarly, an object X with morphism πi : X ! Xi is called product of the Xi, Q and is denoted by i2I Xi, if for each test object Y 2 C, the map HomC (−,πi) Y HomC(Y; X) −−−−−−−−! HomC(Y; Xi) i2I is a bijection. The product of only two objects is denoted by X1 × X2. Lemma 2.1. In an arbitrary category C, (co-)products need not exist, but if they do, they are unique up to isomorphism. 3. Rings and modules Definition. A ring R is an abelian group together with a unity 1 2 R and an associative bilinear map R × R ! R,(x; y) 7! xy, such that 1x = x1 = x for all x 2 R. A ring is called commutative if xy = yx for all x, y 2 R. A map f : R ! S between rings is called a ring homomorphism or map of rings if it is linear, f(1R) = 1S, and f(xy) = f(x)f(y) for all x, y 2 R.
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