Chapter 29 Homological algebra 29.1 Homology of a chain complex A chain complex C∗ is a sequence of abelian groups and homomor- phisms / / / / C∗ : ··· Cn+1 Cn Cn−1 ··· ∂n+1 ∂n where (i) the elements of Cn are called n-chains, and / (ii) the homomorphisms ∂n : Cn Cn−1 are called boundary homo- morphisms. These are required to satisfy the condition ∂n+1 ◦ ∂n =0 for each integer n. We shall only consider those with Cn =0for n<0. Consequently, ∂n =0for n ≤ 0. The n-cycles are those elements in Cn whose boundaries are zero: / Zn(C):=ker(∂n : Cn Cn−1). In particular, Z0 = C0 On the other hand, the n-boundaries are those elements of Cn which are images of elements of Cn+1 under the boundary homomorphism ∂n+1: 1002 Homological algebra / Bn(C):=Im(∂n+1 : Cn+1 Cn). From the condition ∂n+1 ◦ ∂n =0,wehaveIm ∂n+1 ⊆ ker ∂n for each integer n. The homology of the chain complex C measures its deviation from exactness: for each integer n, Hn(C∗):=Zn(C)/Bn(C)=ker∂n/Im ∂n+1. In particular, Hn(C)=0for n<0. Theorem 29.1. Every short exact sequence of chain complexes gives rise to a long exact sequence in homology: if 0 /C /C /C /0 f g is an exact sequence of chain complexes, then the sequence / / / / / / ··· Hn+1(C ) Hn(C) Hn(C ) Hn(C ) Hn−1(C) ··· dn+1 f∗ g∗ dn is exact. Proposition 29.2. A commutative diagram 0 /C /C /C /0 0 /E /E /E /0 of short exact sequences of chain complexes induces a commutative diagram of exact homology sequences / / / / / / ··· Hn+1(C ) Hn(C) Hn(C ) Hn(C ) Hn−1(C) ··· / / / / / / ··· Hn+1(E ) Hn(E) Hn(E ) Hn(E ) Hn−1(E) ··· 29.2 Cohomology of a chain complex 1003 29.2 Cohomology of a chain complex A cochain complex C∗ is a sequence of abelian groups and homomor- phisms n−1 n C∗ : ··· /Cn−1 δ /Cn δ /Cn+1 /··· where (i) the elements of Cn are called n-cochains, and (ii) the homomorphisms δn : Cn /Cn+1 are called coboundary homo- morphisms. These are required to satisfied the condition δn ◦ δn−1 =0 for each integer n. The elements of ker δn are called the n-cocycles, and those of Im δn−1 the n-coboundaries. The n-th cohomology is the quotient ker δn : Cn /Cn+1 Hn(C∗):= . Im δn−1 : Cn−1 /Cn Example 29.1. Let C∗ be a chain complex. For each integer n, let Cn := Hom(Cn, Z) be the dual of Cn. These define a cochain complex C∗ with coboundary homomorphism δn : Cn /Cn+1 given by n = ∗ δ ∂n+1. 1004 Homological algebra Chapter 30 Tensor and torsion products 30.1 Tensor product Let A and B be abelian groups. The tensor product A ⊗ B is defined by the following universal property: there is a bilinear map τ : A × B /A ⊗ B such that for every bilinear map φ : A × B /C, there is a unique homomorphism A ⊗ B /C such that A × BCB φ /C ?? ?? ? ?? τ ?? f ? A ⊗ B is commutative. A ⊗ B is generated by elements of the form a ⊗ b, a ∈ A, b ∈ B, subject to the relations (i) a ⊗ 0=0⊗ b =0for a ∈ A, b ∈ B; (ii) n(a ⊗ b)=(na) ⊗ b = a ⊗ (nb) for a ∈ A, b ∈ B, and n ∈ Z; (iii) (a + a) ⊗ b = a ⊗ b + a ⊗ b and (iv) a ⊗ (b + b)=a ⊗ b + a ⊗ b. 30.1.1 Basic properties (1) A ⊗ B ≈ B ⊗ A. (2) Given f : A /B and g : A /B, there is an induced map f ⊗ g : A ⊗ B /A ⊗ B given by f ⊗ g(a ⊗ b)=f(a) ⊗ g(b). 1006 Tensor and torsion products (3) (f ◦ f) ⊗ (g ◦ g)=(f ⊗ g) ◦ (f ⊗ g). (4) If A ≈ A1 ⊕ A2 ⊕···⊕An, then A ⊗ B ≈ (A1 ⊗ B) ⊕ (A2 ⊗ B) ⊕···⊕(An ⊗ B). (6) If A is a free abelian group with basis {ai} and B is a free abelian group with basis {bj}, then A ⊗ B is a free abelian group with basis {ai ⊗ bj}. (7) If i : AtoA and j : B /B are inclusions, then A ⊗ B (A/A) ⊗ (B/B) ≈ . Im (i ⊗ ι)+Im(ι ⊗ j) Theorem 30.1. (1) Z ⊗ A ≈ A. (2) Zm ⊗ Zn ≈ Zgcd(m,n). Proposition 30.2. ⊗ is a right-exact functor: if f g A /B /C /0 is exact, then for any abelian group D, f⊗ι g⊗ι A ⊗ D /B ⊗ B /C ⊗ D /0 is also exact. 30.2 Torsion product However, ⊗ is in general not left-exact. This means that if 0 A/ B/ C/ is exact, the tensored sequence 0 /A ⊗ D /B ⊗ D /C ⊗ D may not be exact. This leads to the notion of torsion product. Given an abelian group A and a free resolution / f1 / f0 / / ··· F1 F0 A 0, for any abelian group B, the kernel of f1 ⊗ ι in the sequence f1⊗ι / / / F1 ⊗ B F0 ⊗ B A ⊗ B 0 is independent of the choose of the free abelian groups F0 and F1.Itis denoted by Tor(A, B) and is called the torsion product of A and B. 30.3 Homology with coefficients 1007 30.2.1 Basic properties of torsion product (1) If 0 /A /B /C /0 is exact, then for arbitrary abelian group D, 0 Tor/ (A, D) Tor/ (B,D) Tor/ (C, D) A/⊗D B/ ⊗D C/⊗D 0 is exact. (2) The torsion product is an additive functor. (3) If A is torsion-free, then Tor(A, B)=0for any B. (4) Tor(A, B) ≈ Tor(B,A). (5) Tor(Zn,B)={b ∈ B : nb =0}. 30.3 Homology with coefficients Let C∗ be a chain complex, and G an abelian group. The homology of C∗ with coefficients G is the homology of the chain complex / / / / C∗ ⊗ G : ··· Cn+1 ⊗ G Cn ⊗ G Cn−1 ⊗ G ··· ∂n+1⊗ι ∂n⊗ι Thus, ker ∂n+1 ⊗ ι Hn(C∗; G):= . Im ∂n ⊗ ι THEOREM (Universal coefficient theorem). Let C∗ be a free chain complex of abelian groups and G an abelian group, then Hn(C; G) ≈ Hn(C) ⊗ G ⊕ Tor(Hn−1(C),G). In fact, this follows from an exact sequence / α / / / 0 Hn(C) ⊗ G Hn(C ⊗ G) Tor(Hn−1(C,G) 0, where α([c] ⊗ g)=[c ⊗ g] In particular, if G is torsion free, then α is an isomorphism between Hn(C∗; G)=Hn(C ⊗ G) and Hn(C) ⊗ G. 1008 Tensor and torsion products Chapter 31 The Hom and Ext functors 31.1 The Ext functor Let G be an abelian group. The functor Hom(−,G) is a contravariant f g functor, and is left exact. This means that if 0 /A /B /C /0 is exact, then g∗ f ∗ 0 /Hom(C, G) /Hom(B,G) /Hom(A, G) is also exact. However, it may not be right exact. If / f1 / f0 / / F1 F0 A 0 is a free resolution of A, then for arbitrary abelian group B, the cokernel of ∗ f1 / Hom(F0,B) Hom(F1,B) is independent of F0 and F1. We denote this by Ext(A, B) and call it the extension of A by B. 31.1.1 Basic properties of the ext functor (1) If 0 /A /B /C /0 is exact, then for arbitrary abelian group G, 0Hom(/Hom(C, G)Hom() /Hom(B,G))Hom(/Hom(A, G) ggg ggggg ggggg sggggg Ext(C, G))Ext(/Ext(B,G))Ext(/Ext(A, G))0/ 1010 The Hom and Ext functors is exact. (2) The ext functor is additive. (3) If A is free (abelian), then Ext(A, B)=0for any B. (4) If B is divisible in the sense that for every b ∈ B and every integer n>0 there exists b ∈ B such that b = nb, then Ext(A, B)=0for every abelian group A (5) Ext(Zn,G)=G/nG. In particular, Ext(Zn, Z)=Zn. (6) Ext(Zm, Zn)=Zgcd(m,n). In particular if p is a prime, then Ext(A, Zp)=0for every abelian group A. 31.2 Cohomology with coefficients Z2 Let C∗ be a free chain complex. The cohomology of the cochain complex C∗ with n C := Hom(Cn, Z2) satisfies n ∗ H (C )=Hom(Hn(C∗), Z2) for each integer n ≥ 0. Chapter 32 Cellular structures of projective spaces 32.1 Relative n-cells Let f : Sn−1 /Y be a given map. The mapping cone of f is the space n n Z := Y ∪f e := Y ∪ D /(f(x)=x). We say that Z is obtained from Y by adjoining an n-cell via the map f. From the exact homotopy sequence of the pair (Z, Y ),wehave n πk(Y ∪f e ,Y) ≈ πk(Y ) for k<n− 1. / n Proposition 32.1. A map g : Y Z can be extended to Y ∪f e if and only if the composite g ◦ f is nullhomotopic. 32.2 Cellular structure of the real projective spaces Let p : Sn /RPn be the double covering map of the real projec- (En Sn−1) ≈ tive space. This restricts to a relative homeomorphism +, (RPn, RPn−1). It follows that n n−1 n RP = RP ∪p e . Since RP1 = e0 ∪ e1, it follows by induction that RPn has a cellular structure consisting of one cell in each dimension through n: RPn = e0 ∪ e1 ∪···∪en. 1102 Cellular structures of projective spaces 32.3 Cellular structure of the complex projective spaces Let q : S2n−1 /CPn−1 be the quotient map.