MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

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1 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to do this first so that grad students will be more familiar with the ideas when they are applied to algebraic topology (in 121b). At the same time, it is not my intention to cover the same material twice. The topics are (1) abelian categories (2) projective and injective resolutions (3) derived functors (4) interpretation of Ext 1 1. Additive categories On the first day I talked about additive categories. Definition 1.1. An additive category is a category C for which every hom set Hom C (X, Y ) is an additive group and (1) composition is biadditive, i.e., (f 1 + f 2 ) g = f 1 g + f 2 g and f (g 1 + g 2 ) = f g 1 + f g 2. (2) The category has finite direct sums. I should have gone over the precise definition of direct sum: Definition 1.2. The direct sum n i=1a i is an object X together with morphisms j i : A i X, p i : X A i so that (1) p i j i = id : A i A i (2) p i j j = 0 if i j. (3) j i p i = id X : X X. Theorem 1.3. A i is both the product and coproduct of the A i Proof. Suppose that f i : Y A i are morphisms. Then there is a morphism f = j i f i : Y A i which has the property that p i f = p i j i f i = f i. Conversely, given any morphism g : Y A i satisfying p i g = f i for all i, then we have: f = j i f i = j i p i g = id X g = g Date: January 23,

2 2 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA So, f is unique and A i is the product of the A i. By an analogous argument, it is also the coproduct. The converse is also true: Proposition 1.4. Suppose that X = A i = A i and the composition of the inclusion j i : A i X with p j : X A j is p j j i = δ ij : A i A j I.e., it is the identity on A i for i = j and it is zero for i j. Then ji p i = id X Proof. Let f = j i p i : X X. Then p j f = i p j j i p i = i δ ij p i = p j So, f = id X by the universal property of a product. In class I pointed out the sum of no objects is the zero object 0 which is both initial and terminal. Also, I asked you to prove the following. Problem. Show that a morphism f : A B is zero if and only if it factors through the zero object. 2. Abelian categories 2.1. some definitions. First I explained the abstract definition of kernel, monomorphism, cokernel and epimorphism. Definition 2.1. A morphism f : A B in an additive category C is a monomorphism if, for any object X and any morphism g : X A, f g = 0 if and only if g = 0. (g acts like an element of A. It goes to zero in B iff it is zero.) Another way to say this is that f : A B is a monomorphism and write 0 A B if 0 Hom C (X, A) f Hom C (X, B) is exact, i.e., f is a monomorphism of abelian groups. The lower sharp means composition on the left or post-composition. The lower sharp is order preserving: (f g) = f g Epimorphisms are defined analogously: f : B C is an epimorphism if for any object Y we get a monomorphism of abelian groups: 0 Hom C (C, Y ) f Hom C (B, Y )

3 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 3 An abelian category is an additive category which has kernels and cokernels satisfying all the properties that one would expect which can be stated categorically. First, I explained the categorical definition of kernel and cokernel. Definition 2.2. The kernel of a morphism f : A B is an object K with a morphism j : K A so that (1) f j = 0 : K B (2) For any other object X and morphism g : X A so that f g = 0 there exists a unique h : X K so that g = j h. Since this is a universal property, the kernel is unique if it exists. Theorem 2.3. A is the kernel of f : B C if and only if 0 Hom C (X, A) Hom C (X, B) Hom C (X, C) is exact for any object X. In particular, j : ker f B is a monomorphism. If you replace A with 0 in this theorem you get the following statement. Corollary 2.4. A morphism is a monomorphism if and only if 0 is its kernel. Cokernel is defined analogously and satisfies the following theorem which can be used as the definition. Theorem 2.5. The cokernel of f : A B is an object C with a morphism B C so that 0 Hom C (C, Y ) Hom C (B, Y ) Hom C (A, Y ) is exact for any object Y. Again, letting C = 0 we get the statement: Corollary 2.6. A morphism is an epimorphism if and only if 0 is its cokernel. These two theorems can be summarized by the following statement. Corollary 2.7. For any additive category C, Hom C is left exact in each coordinate definition of abelian category. Definition 2.8. An abelian category is an additive category C so that (1) Every morphism has a kernel and a cokernel. (2) Every monomorphism is the kernel of its cokernel.

4 4 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA (3) Every epimorphism is the cokernel of its kernel. (4) Every morphism f : A B can be factored as the composition of an epimorphism A I and a monomorphism I B. (5) A morphism f : A B is an isomorphism if and only if it is both mono and epi. Proposition 2.9. The last condition follows from the first four conditions. Proof. First of all, isomorphisms are always both mono and epi. The definition of an isomorphism is that it has an inverse g : B A so that f g = id B and g f = id A. The second condition implies that f is mono since (g f) = g f = id = id which implies that f is mono and f is mono. Similarly, f g = id B implies that f is epi. Conversely, suppose that f : A B is both mono and epi. Then, by (2), it is the kernel of its cokernel which is B 0. So, by left exactness of Hom we get: 0 Hom C (B, A) Hom C (B, B) Hom C (B, 0) In other words, f : Hom C (B, A) = Hom C (B, B). So, there is a unique element g : B A so that f g = id B. Similarly, by (3), there is a unique h : B A so that h f = id A. If we can show that g = h then it will be the inverse of f making f invertible and thus an isomorphism. But this is easy: h = h id B = h f g = id A g = g 2.3. examples. The following are abelian categories: (1) The category of abelian groups and homomorphisms. (2) The category of finite abelian groups. This is an abelian category since any homomorphism of finite abelian groups has a finite kernel and cokernel and a finite direct sum of finite abelian groups is also finite. (3) R-mod= the category of all left R-modules and homomorphisms (4) R-Mod= the category of finitely generated (f.g.) left R-modules is an abelian category assuming that R is left Noetherian (all submodules of f.g. left R-modules are f.g.) (5) mod-r=the category of all right R-modules and homomorphisms (6) Mod-R=the category of f.g. right R-modules is abelian if R is right Noetherian.

5 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 5 The following are examples of additive categories which are not abelian. (1) Free abelian groups. (This category does not have cokernels) (2) Let R be a non-noetherian ring, for example a polynomial ring in infinitely many variables: R = k[x 1, X 2, ] Then R-Mod, the category of f.g. R-modules is not abelian since it does not have kernels. E.g., the kernel of the augmentation map R k is infinitely generated. 3. Projective and injective objects At the end of the second lecture we discussed the definition of injective and projective objects in any additive category. And it was easy to show that the category of R-modules has sufficiently many projectives. Definition 3.1. An object P of an additive category C is called projective if for any epimorphism f : A B and any morphism g : P B there exists a morphism g : P A so that f g = g. The map g is called a lifting of g to A. Theorem 3.2. P is projective if and only if Hom C (P, ) is an exact functor. Proof. If 0 A B C 0 is exact then, by left exactness of Hom we get an exact sequence: 0 Hom C (P, A) Hom C (P, B) Hom C (P, C) By definition, P is projective if and only if the last map is always an epimorphism, i.e., iff we get a short exact sequence 0 Hom C (P, A) Hom C (P, B) Hom C (P, C) 0 Theorem 3.3. Any free R-module is projective. Proof. Suppose that F is free with generators x α. Then every element of F can be written uniquely as r α x α where the coefficients r α R are almost all zero (only finitely many are nonzero). Suppose that g : F B is a homomorphism. Then, for every index α, the element

6 6 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA g(x α ) comes from some element y α A. I.e., g(x α ) = f(y α ). Then a lifting g of g is given by g( r α x a ) = r α y α The verification that this is a lifting is straightforward or I would say obvious but it would go like this: The claim is that, first, g : F A is a homomorphism of R-modules and, second, it is a lifting: f g = g. The second statement is easy: f g( r α x a ) = f( r α y α ) = r α f(y α ) = r α g(x α ) = g( r α x α ) The first claim says that g is additive: ( g rα x α + ) ( ) s α x α = g (rα + s α )x α = (r α + s α )y α = g( r α x α ) + g( s α x α ) and g commutes with the action of R: g (r ) ( ) r α x α = g rrα x α = rr α y α = r ( ) r α x α = r g rα x α For every R-module M there is a free R-module which maps onto M, namely the free module F generated by symbols [x] for all x M and with projection map p : F M given by ( ) p rα [x α ] = r α x α The notation [x] is used to distinguish between the element x M and the corresponding generator [x] F. The homomorphism p is actually just defined by the equation p[x] = x. Corollary 3.4. The category of R-modules has sufficiently many projectives, i.e., for every R-module M there is a projective R-module which maps onto M. This implies that every R-module M has a projective resolution 0 M P 0 P 1 P 2... This is an exact sequence in which every P i is projective. The projective modules are constructed inductively as follows. First, P 0 is any projective which maps onto M. This gives an exact sequence: P 0 M 0

7 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 7 By induction, we get an exact sequence P n P n 1 P n 2 P 0 M 0 Let K n be the kernel of d n : P n P n 1. Since there are enough projectives, there is a projective module P n+1 which maps onto K n. The composition P n+1 K n P n is the map d n+1 : P n+1 P n which extends the exact sequence one more step. Definition 3.5. An object Q of C is injective if, for any monomorphism A B, any morphism A Q extends to B. I.e., iff As before this is equivalent to: Hom C (B, Q) Hom C (A, Q) 0 Theorem 3.6. Q is injective if and only if Hom C (, Q) is an exact functor. The difficult theorem we need to prove is the following: Theorem 3.7. The category of R-modules has sufficiently many injectives. I.e., every R-module embeds in an injective R-module. As in the case of projective modules this theorem will tell us that every R-module M has an injective co-resolution which is an exact sequence: where each Q i is injective. 0 M Q 0 Q 1 Q 2 4. Injective modules I will go over Lang s proof that every R-module M embeds in an injective module Q. Lang uses the dual of the module dual module. Definition 4.1. The dual of a left R-module M is defined to be the right R-module with right R-action given by for all φ M, r R. M := Hom Z (M, Q/Z) φr(x) = φ(rx)

8 8 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA Proposition 4.2. Duality is a left exact functor ( ) : R-mod mod-r which is additive and takes sums to products: ( M α ) = M α Proof. We already saw that the hom functor Hom Z (, X) is left exact for any abelian group X. It is also obviously additive which means that (f + g) = f + g for all f, g : N M. I.e., the duality functor induces a homomorphism (of abelian groups): Hom R (N, M) Hom Z (M, N ) Duality also takes sums to products since a homomorphism f : M α X is given uniquely by its restriction to each summand: f α : M α X and the f α can all be nonzero. (So, it is the product not the sum.) 4.2. constructing injective modules. In order to get an injective left R-module we need to start with a right R-module. Theorem 4.3. Suppose F is a free right R-module. (I.e., F = R R is a direct sum of copies of R considered as a right R-module). Then F is an injective left R-module. This theorem follows from the following lemma. Lemma 4.4. (1) A product of injective modules is injective. (2) Hom R (M, RR ) = Hom Z (M, Q/Z) (3) Q/Z is an injective Z-module. Proof of the theorem. Lemma (3) implies that Hom Z (, Q/Z) is an exact functor. (2) implies that Hom R (, RR ) is an exact functor. Therefore, RR is an injective R-module. Since duality takes sums to products, (1) implies that F is injective for any F with is a sum of R R s, i.e. F is a free right R-module. We need one more lemma to prove the main theorem. Then we have to prove the lemmas. Lemma 4.5. Any left R-module is naturally embedded in its double dual: M M Assume this 4th fact for a moment.

9 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 9 Theorem 4.6. Every left R-module M can be embedded in an injective left R-module. Proof. Let F be a free right R-module which maps onto M : Since duality is left exact we get: F M 0 0 M F By the last lemma we have M M F. So, M embeds in the injective module F proof of lemmas. There are four lemmas to prove. Suppose for a moment that T = Q/Z is injective then the other three lemmas are easy: Proof of Lemma 4.5. A natural embedding M M is given by the evaluation map ev which sends x M to ev x : M T which is evaluation at x: ev x (φ) = φ(x) Evaluation is additive: ev x+y (φ) = φ(x + y) = φ(x) + φ(y) = ev x (φ) + ev y (φ) = (ev x + ev y ) (φ) Evaluation is an R-module homomorphism: ev rx (φ) = φ(rx) = (φr)(x) = ev x (φr) = (rev x ) (φ) Finally, we need to show that ev is a monomorphism. In other words, for every nonzero element x M we need to find some additive map φ : M T so that ev x (φ) = φ(x) 0. To do this take the cyclic group C generated by x C = {kx k Z} This is either Z or Z/n. In the second case let f : C T be given by f(kx) = k n + Z Q/Z This is nonzero on x since 1/n is not an integer. If C = Z then let f : C T be given by f(kn) = k 2 + Z Then again, f(x) is nonzero. Since T is Z-injective, f extends to an additive map φ : M T. So, ev x is nonzero and ev : M M is a monomorphism.

10 10 MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA Proof that products of injectives are injective. Suppose that J α are injective. Then we want to show that Q = J α is injective. Let p α : Q J α be the projection map. Suppose that f : A B is a monomorphism and g : A Q is any morphism. Then we want to extend g to B. Since each J α is injective each composition p α g : A J α extends to a morphism g α : B J α. I.e., g α f = p α g for all α. By definition of the product there exists a unique morphism g : B Q = J α so that p α g = g α for each α. So, p α g f = g α f = p α g : A J α The uniquely induced map A J α is g f = g. Therefore, g is an extension of g to B as required. Finally, we need to prove that Hom R (M, R R) = Hom Z (M, Q/Z)

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