Lecture notes on homological algebra Hamburg SS 2019 T. Dyckerhoff June 17, 2019 Contents 1 Chain complexes 1 2 Abelian categories 6 3 Derived functors 15 4 Application: Syzygy Theorem 33 5 Ext and extensions 35 6 Quiver representations 41 7 Group homology 45 8 The bar construction and group cohomology in low degrees 47 9 Periodicity in group homology 54 1 Chain complexes Let R be a ring. A chain complex (C•; d) of (left) R-modules consists of a family fCnjn 2 Zg of R-modules equipped with R-linear maps dn : Cn ! Cn−1 satisfying, for every n 2 Z, the condition dn ◦ dn+1 = 0. The maps fdng are called differentials and, for n 2 Z, the module Cn is called the module of n-chains. Remark 1.1. To keep the notation light we usually abbreviate C• = (C•; d) and simply write d for dn. Example 1.2 (Syzygies). Let R = C[x1; : : : ; xn] denote the polynomial ring with complex coefficients, and let M be a finitely generated R-module. (i) Suppose the set fa1; a2; : : : ; akg ⊂ M generates M as an R-module, then we obtain a sequence of R-linear maps ' k ker(') ,! R M (1.3) 1 k 1 where ' is defined by sending the basis element ei of R to ai. We set Syz (M) := ker(') and, for now, ignore the fact that this R-module may depend on the chosen generators of M. Following D. Hilbert, we call Syz1(M) the first syzygy module of M. In light of (1.3), every element of Syz1(M), also called syzygy, can be interpreted as a relation among the chosen generators of M as follows: every element 1 of r 2 Syz (M) can be expressed as an R-linear combination λ1e1 +λ2e2 +···+λkek of the basis elements of Rk. The fact that, by definition, '(r) = 0, now reads as the equation λ1m1 + λ2m2 + ··· + λkmk = 0 which is a relation between the generators fmig in M. (ii) By Hilbert's basis theorem, the module Syz1(M) is again finitely generated, so we may choose a set fb1; b2; : : : ; blg of generators. Repeating (1), we obtain a sequence of R-linear maps l ker( ) ,! R M (1.4) l 2 where is defined by sending the basis element ei of R to bi. We set Syz (M) := ker( ), called the second syzygy module of M. In light of (1.3) and (1.4), every element of Syz2(M), also called second syzygy, can be interpreted as a relation among the relations among generators of M. (iii) We can reiterate this procedure ad infinitum. Piecing together the sequences (1.3), (1.4), . , we obtain a sequence of R-linear maps 2 Syz (M n ) (1.5) ? ? d d d ··· / Rm / Rl / Rm ? / Syz1(M) which, by construction, satisfies d2 = 0. The chain complex C = (C•; d) of free R-modules constructed in (1.5) is called a free resolution of M. Note that, by construction, C comes equipped with a surjective map C0 ! M called augmentation map. Hilbert's phisolophy behind the construction of the sequence of higher syzygies is that, while the original module M may be very complicated, the modules Syzk(M) become easier to understand as k increases. This statement is made precise in his famous syzygy theorem1: Every finitely generated R-module M admits a free resolution of length ≤ n. We will give a proof a local variant of this theorem in the course. Hilbert's original proof was very explicit, our proof will be a corollary of a fairly abstract modern theory. 1Hilbert stated and proved this theorem under the additional assumption that M is a graded R- module. The statement is true in general but relies on a deep theorem of Quillen and Suslin which says that every finitely generated projective module over a polynomial ring is free. 2 Example 1.6 (Simplicial complexes). Let N be a natural number. An (abstract) sim- plicial complex K on [N] := f0; 1;:::;Ng is a collection of subsets of [N] such that • if σ 2 K and τ ⊂ σ, then τ 2 K. One thinks of the collection of all subsets of [N] as corresponding to all subsimplices of a geometric N-dimensional simplex ∆N . A simplicial complex K on [N] then corresponds to a union of subsimplices in ∆N . For every n ≥ 0, we introduce the set Kn := fσ 2 K jσj = n + 1g of n-simplices in K. Any σ 2 Kn can be uniquely expressed as σ = fx0; x1; : : : ; xng with x0 < x1 < ··· < xn. We then define, for every 0 ≤ i ≤ n, the face map @i : Kn −! Kn−1; σ 7! fx0; : : : ; xi−1; xi+1; : : : ; xng: Let R be a ring and define M Cn(K; R) := Reσ σ2Kn the free R-module on the set Kn whose basis elements we label by feσg. We define differentials n X i dn : Cn(K; R) −! Cn−1(K; R); eσ 7! (−1) e@iσ i=0 where, as Cn(K; R) is a free R-module, it suffices to specify the map dn on a the basis 2 feσg. By explicit verification we have d = 0 and hence we obtain a chain complex (C•(K; R); d) called the simplicial chain complex of K with coefficients in R. As an explicit example, consider the simplicial complex on [2] given by the collection of subsets ff0; 1g; f1; 2g; f0; 2g; f0g; f1g; f2g; ;g: The corresponding simplicial chain complex with coefficients in the ring Z has two non- zero components given by C1(K; Z) = Zef0;1g ⊕ Zef1;2g ⊕ Zef0;2g and C0(K; Z) = Zef0g ⊕ Zef1g ⊕ Zef2g while the differential d : C1 ! C0 is given, with respect to the indicated basis, by the matrix 0−1 0 −11 @ 1 −1 0 A : 0 1 1 Note that, by convention, we omit the trivial modules of n-chains corresponding to the zero R-module. Example 1.7 (Cyclic bar complex). Let k be a field, and let A be an associative k- algebra. We define, for every n ≥ 0, the k-vector space Cn(A) = A ⊗k A ⊗k · · · ⊗k A | {z } n + 1 copies 3 given by iterating the above tensor product construction. This construction only depends on the k-vector space structure underlying A. Further, we define maps d : Cn(A) ! Cn−1(A) by k-linearly extending the formula d(a0 ⊗ a1 ⊗ · · · ⊗ an) = (a0a1) ⊗ a2 ⊗ · · · ⊗ an n−1 X i + (−1) a0 ⊗ · · · ⊗ (aiai+1) ⊗ · · · ⊗ an i=1 n + (−1) (ana0) ⊗ a1 ⊗ · · · ⊗ an−1: 2 An explicit computation shows d = 0. We obtain a chain complex (C•(A); d) of k-vector spaces called the cyclic bar complex of A. We introduce some terminology: Given a chain complex C• of R-modules, we call Zn := ker(dn) the module of n-cycles, and Bn := im(dn+1) the module of n-boundaries. Note that Bn ⊂ Zn ⊂ Cn so that we can further define the quotient module Hn(C•) := Zn=Bn; called the nth homology module of C•. The chain complex C• is called exact at n 2 Z, if ∼ Hn(C•) = 0. Example 1.8. (1) Let R = C[x1; : : : ; xn] be the polynomial ring, M a finitely gener- ated R-module and C• the free resolution of M as defined above. Then we have ( ∼ M for i = 0, Hi(C•) = 0 for i 6= 0. (2) Let K be a simplicial complex with corresponding simplicial chain complex C•(K; R) with coefficients in a ring R. The homology modules Hi(C•(K; R)) are called sim- plicial homology of K with coefficients in R. These modules, and variants, are of central importance in algebraic topology. In the explicit example above, given by the boundary of a 2-simplex, we have ( ∼ Z for i = 0; 1, Hi(C•(K; Z)) = 0 otherwise. (3) Let A be an associative k-algebra. The homology modules of the cyclic bar complex C•(A) are called Hochschild homology. Their calculation is in general very involved, but, for example if A is a finitely generated, commutative k-algebra satisfying a cer- tain smoothness condition, then the so-called Hochschild-Kostant-Rosenberg theo- rem states that, for all i ≥ 0, we have ∼ i Hi(C•(A)) = ΩA=k i where ΩA=k denotes the module of K¨ahlerdifferentials of A over k. 4 Given a chain complex C• of R-modules, we introduced, for every n, modules of cycles Zn = ker(dn), boundaries Bn = im(dn+1), and the homology module given by the quotient ∼ Hn(C•) = Zn=Bn. We say that C• is exact at n 2 Z, if Hn(C•) = 0. Convention 1.9. We say that a sequence d d d d Ck −! Ck−1 −! ::: −! Cl−1 −! Cl 2 of R-modules with d = 0 is exact if Hn(C•) = 0 for all l < n < k. Note the strict inequality so that no exactness condition is required at the end terms l and k. Example 1.10. An exact sequence of the form f g 0 −! A −! B −! C −! 0 is called short exact sequence. The exactness conditions translate into the requirements that f be injective, g be surjective, and ker(g) = im(f). The sequence 2 0 −! Z −! Z −! Z=2Z −! 0 of abelian groups is a short exact sequence. The sequence 2 Z=4Z −! Z=4Z −! Z=2Z −! 0 is exact but can not be extended to form a short exact sequence, since the map given by multiplication by 2 is not injective. A key insight of modern mathematics is that, to understand any class of objects of a certain kind, one ought to understand what kind of maps to allow between these objects.