Homological Algebra Andrew Kobin Fall 2014 / Spring 2017 Contents Contents Contents 0 Introduction 1 1 Preliminaries 4 1.1 Categories and Functors . .4 1.2 Exactness of Sequences and Functors . 10 1.3 Tensor Products . 17 2 Special Modules 24 2.1 Projective Modules . 24 2.2 Modules Over Noetherian Rings . 28 2.3 Injective Modules . 31 2.4 Flat Modules . 38 3 Categorical Constructions 47 3.1 Products and Coproducts . 47 3.2 Limits and Colimits . 49 3.3 Abelian Categories . 54 3.4 Projective and Injective Resolutions . 56 4 Homology 60 4.1 Chain Complexes and Homology . 60 4.2 Derived Functors . 65 4.3 Derived Categories . 70 4.4 Tor and Ext . 79 4.5 Universal Coefficient Theorems . 90 5 Ring Homology 93 5.1 Dimensions of Rings . 93 5.2 Hilbert's Syzygy Theorem . 97 5.3 Regular Local Rings . 101 5.4 Differential Graded Algebras . 103 6 Spectral Sequences 110 6.1 Bicomplexes and Exact Couples . 110 6.2 Spectral Sequences . 113 6.3 Applications of Spectral Sequences . 119 7 Group Cohomology 128 7.1 G-Modules . 128 7.2 Cohomology of Groups . 130 7.3 Some Results for the First Group Cohomology . 133 7.4 Group Extensions . 136 7.5 Central Simple Algebras . 139 7.6 Classifying Space . 142 i Contents Contents 8 Sheaf Theory 144 8.1 Sheaves and Sections . 144 8.2 The Category of Sheaves . 150 8.3 Sheaf Cohomology . 156 8.4 Cechˇ Cohomology . 161 8.5 Direct and Inverse Image . 170 ii 0 Introduction 0 Introduction These notes are taken from a reading course on homological algebra led by Dr. Frank Moore at Wake Forest University in the fall of 2014, with some additions made during a course taught by Dr. Benjamin Webster at the University of Virginia in Spring 2017. The companion text for both courses is Rotman's An Introduction to Homological Algebra, but many concepts were introduced to the discussion from outside and developed by students. For example, most of the material on spectral sequences (Chapter 6) and group cohomology (Chapter 7) follows Weibel's An Introduction to Homological Algebra. The main topics covered in this course are: Basic category theory Examples of functors including Hom and tensor products Free resolutions of modules, projectives, injectives and flat modules Homology of a complex Derived functors Ext and Tor Spectral sequences Group cohomology Other topics from commutative algebra Basic prerequisites for reading these notes are familiarity with groups, rings and modules. Consider the following motivating example. Let R be a ring and let R-Mod be the collec- tion of all R-modules. (This is an example of an abelian category, which will be defined in subsequent chapters.) Fix two modules M and N in R-Mod and define the set of extensions of M by N: Ext(M; N) := fshort exact sequences of R-modules0 ! N ! K ! M ! 0g= ∼ where ∼ is the equivalence relation of short exact sequences exhibited by the following diagram: 0 N K M 0 id id 0 N K0 M 0 Ext(M; N) has a natural \distinguished" element, which is an equivalence class represented by the canonial split exact sequence: 0 ! N ! N ⊕ M ! M ! 0: 1 0 Introduction p Definition. A splitting of a short exact sequence 0 ! N −!i K −! M ! 0 is a map q : M ! K (or equivalently, j : K ! N) such that p ◦ q = idM (or j ◦ i = idN ). For example, the canonical projections are splittings of 0 ! N ! N ⊕ M ! M ! 0. One can show that a short exact sequence has a splitting if and only if it is equivalent to the canonical split exact sequence. Example 0.0.1. Let M = Z=pZ (p prime) and N = Z, considered as modules over Z. An element of Ext(Z=pZ; Z) is represented by a short exact sequence of the form 0 ! Z ! K ! Z=pZ ! 0: Besides the split exact sequence (with K = Z ⊕ Z=pZ), there is a family of short exact sequences, p 0 ! Z −! Z ! Z=pZ ! 0 1 7! a−1 for each a 2 (Z=pZ)×. Suppose we have two such extensions that are equivalent: p a−1 0 Z Z Z=pZ 0 id ' id p b−1 0 Z Z Z=pZ 0 Since ' is an isomorphism of Z-modules that commutes with the identity on Z, it must in fact be the identity. But since the rest of the diagram commutes, this implies a = b. We have thus exhibited a bijection Ext(Z=pZ; Z) ! Z=pZ: It turns out that Ext(M; N) is always an abelian group. In the above example, the correspondence between a 2 (Z=pZ)× and 1 7! a−1 is actually an abelian group isomorphism. p To understand the structure of Ext(M; N), take two extensions 0 ! N −!i K −! M ! 0 and 0 p0 0 ! N −!i K0 −! M ! 0. A natural composition would be to take the direct sum of the two sequences, but this would yield an extension of M ⊕ M by N ⊕ N. In fact, there is a universal object (called pullback) making the following diagram commute: K Ke M K0 2 0 Introduction Explicitly Ke = f(k; k0) 2 K × K0 : p(k) = p0(k0)g. Then we define the space K00 := K=e f(n; −n): n 2 Ng gives an extension of R-modules which is the natural composition of the two initial extensions: 0 ! N ! K ! M ! 0 + 0 ! N ! K0 ! M ! 0 = 0 ! N ! K00 ! M ! 0: It's obvious that this construction is commutative, but associativity and inverses require some proof. Example 0.0.2. Given two extensions p a−1 p b−1 0 ! Z −! Z −−! Z=pZ ! 0 and 0 ! Z −! Z −−! Z=pZ ! 0; we have the composite space K00 = f(x; y) 2 Z ⊕ Z j bx = ay (mod p)g=h(p; −p)i. One can show that this extension represents the class in Ext(Z=pZ; Z) corresponding to a + b in Z=pZ. One question going forward will be: how do we compute these Ext groups in a reasonable fashion? 3 1 Preliminaries 1 Preliminaries 1.1 Categories and Functors Category theory is a way of describing `classes' of things in mathematics and the relations between them in a general way. Definition. A category C is a class of objects obj(C) and for every pair of objects A; B in obj(C) a set of morphisms HomC(A; B) with a composition law: HomC(B; C) × HomC(A; B) −! HomC(A; C) (g; f) 7−! g ◦ f = gf: For every A 2 obj(C) there is an identity morphism 1A 2 HomC(A; A) such that for all f 2 HomC(A; B), f ◦ 1A = f and likewise for HomC(B; A). Examples. 1 Veck is a category consisting of vector spaces over a field k as the objects and linear transformations as the morphisms. This is the same category as k-Mod (see below). 2 For a ring R with identity, R-Mod is the category of (left) R-modules with R-linear maps as the morphisms. In this case we will denote HomR-Mod(A; B) as HomR(A; B). 3 Groups (or Grps) is the category consisting of groups together with group homomor- phisms. Even more generally, we can define the category Sets: objects are sets and morphisms are just regular functions. Other algebraic categories arise in a similar fash- ion: AbGrps or Ab is the category of abelian groups with the usual homomorphisms; Rings is the category of rings with ring homomorphisms, where we assume all rings have unity; likewise, ComRings are commutative rings. 4 For a topological space X, we can form a category by taking the open subsets of X as objects and for any open U; V ⊂ X, ( ? if U 6⊆ V HomX (U; V ) = fiU;V g if U ⊆ V where iU;V is the inclusion U,! V . This category can be generalized to any space X which has a partial ordering . 5 As another topological example, Top is the category of topological spaces together with continuous functions between them. How do we compare two categories? Definition. A functor F from C to D is an assignment F : obj(C) −! obj(D) HomC(A; B) −! HomD(F (A);F (B)) A 7−! F (A) f : A ! B F (f): F (A) ! F (B) such that F (gf) = F (g)F (f). 4 1.1 Categories and Functors 1 Preliminaries Remark. The functor defined above is sometimes called a covariant functor: the image of a morphism f : A ! B is a morphism F (A) ! F (B). On the other hand, a contravariant functor takes f : A ! B to F (f): F (B) ! F (A), and F (gf) = F (f)F (g). Examples. 1 The forgetful functor is useful in many contexts. For example, Forget : Veck −! Ab V 7−! (V; +) f 7−! f takes a vector space V and \forgets" its vector space structure. We can go further and forget a group structure, giving a functor Ab ! Sets. 2 For any category C, the identity functor 1C : C!C takes A 7! A and f 7! f. 3 In topology, the fundamental group is a functor: π1 : Top −! Grps X 7−! π1(X) ∗ f : X ! Y f : π1(X) ! π1(Y ): 3 The nth homology group of a topological space is a functor Hn(·): Top ! Ab. 4 Let C be the category of field extensions K=k with morphisms ' : K ! L such that 'jk = idk, i.e.