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COHOMOLOGY of GROUPS 1. Some Homological Algebra 1.1

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COHOMOLOGY of GROUPS 1. Some Homological Algebra 1.1 COHOMOLOGY OF GROUPS J. WARNER Abstract. Notes on Kenneth Brown's book Cohomology of Groups. 1. Some Homological Algebra 1.1. Review of Chain Complexes. Let R be a ring, and let (C; d) and (C0; d0) be two chain complexes 0 of left R-modules. Define a complex of abelian groups HR(C; C ) as follows. Let 0 Y 0 HR(C; C )n = HomR(Cq;Cq+n) q2Z 0 n and define the boundary map Dn by Dn(f) = d f − (−1) fd. 1.5. The Standard Resolution. For any group G, we can always form the following free resolution of Z over ZG. Let Fn be the free ZG module with basis given by the (n+1)-tuples of elements of G whose first component is 1: (1; g1; g2; : : : ; gn). The G-action on Fn is defined on basis elements component-wise. We introduce the shorthand bar notation: [g1jg2j ::: jgn] = (1; g1; g1g2; : : : ; g1g2 : : : gn) If n = 0, there is only one basis element which we denote by [ ]. Define the boundary morphisms by n X i @ = (−1) di i=0 where 8 g [g j ::: jg ] i = 0 <> 1 2 n di([g1jg2j ::: jgn]) = [g1jg2j ::: jgigi + 1j ::: jgn] 0 < i < n > :[g1jg2j ::: jgn−1] i = n For F0 = ZG, the boundary morphism is the standard augmentation. We will refer to this resolution as the standard resolution or bar resolution 2. The Homology of a Group 2.1. Generalities. In homological algebra, we can construct invariants of algebraic objects using the fol- lowing procedure. Let R be a ring, and let T be a covariant additive functor from the category of R-modules to the category of abelian groups. Recall, this means that the map induced by T on Hom sets HomR(M; N) ! HomZ(T M; T N) is a homomorphism of abelian groups. Now, let " : F ! M be a free (or projective) resolution of M. Applying T to F we obtain a chain complex of abelian groups. Since T is additive, it preserves homotopy equivalences, so that the homology groups Hn(TF ) depend only on T and M. Notice that if T is exact, then Hn(TF ) = 0 for n > 0 and H0(TF ) = TM. So, in some sense, the higher homology groups measure the failure of T to be exact. Date: Fall 2013. 1 2.2. Co-invariants. Here we introduce the co-invariants functor, which we will use in the context of the previous section to define the homology groups of a group G. Definition 2.1. The group of co-invariants of M, denoted MG, is the quotient of M by the additive subgroup of elements of the form gm − m. MG is the largest quotient of M on which G acts trivially. Exercise 2.2. If f : M ! N is a map of ZG-modules, show that f : MG ! NG which maps m to f(m) is a well-defined map of abelian groups. We thus obtain a covariant, additive (check!) functor (·)G from ZG-modules to abelian groups, called the co-invariants functor. The following proposition will reveal some information (·)G. ∼ Proposition 2.3. MG = Z ⊗ZG M as abelian groups, where Z is a trivial right ZG-module. Proof. The map M ! Z ⊗ZG M defined by m 7! 1 ⊗ m vanishes on elements of the form gm − m, so it factors through MG to a map of the form m 7! 1 ⊗ m. The map Z × M ! MG defined by (a; m) 7! am is ZG biadditive, so by the universal property of tensor products, we obtain a well-defined map Z ⊗ZG M ! M given by a ⊗ m 7! am. These two maps are inverses of each other. By properties of the tensor product, we see that (·)G is right exact, and that it take a free ZG-module with basis (ei) to a free Z-module with basis (ei). Also, here the map f : MG ! NG sends a⊗m to a⊗f(m). 2.3. The Definition of H∗G. We now use the co-variants functor to define the homology of a group G. The following definition is independent of choice of projective resolution, as described in x2:1. Definition 2.4. Let " : F ! Z be a projective resolution of Z by ZG-modules, and define the homology groups of G by HiG = Hi(FG) 2 n−1 Example 2.5. Let G = Zn = hti, let N = 1 + t + t + ::: + t and consider the projective resolution: t−1 N t−1 " ··· −−! ZG −! ZG −−! ZG −! Z ! 0 Applying (·)G we obtain the complex 0 n 0 ··· −! Z −! Z −! Z ! 0 from which it follows that 8 i = 0 <>Z HiG = Zn i odd :>0 i even, i > 0. Let Fn be the standard resolution of x1:5, and let Cn(G) = (Fn)G. By our observation of the functor (·)G, Cn(G) is a free Z-module with basis [g1jg2j ::: jgn]. Notice here by abuse of notation we have [g1jg2j ::: jgn] = (1; g1; g1g2; : : : ; g1g2 : : : gn) Since G acts trivially on (Fn)G, the boundary morphisms for C∗(G) only differ for i = 0. They are given by: n X i @ = (−1) di i=0 where 8 [g j ::: jg ] i = 0 <> 2 n di([g1jg2j ::: jgn]) = [g1jg2j ::: jgigi + 1j ::: jgn] 0 < i < n > :[g1jg2j ::: jgn−1] i = n The beginning of the resolution C∗(G) looks like @ 0 C2(G) −! C1(G) −! Z ! 0 where @[gjh] = [h] − [gh] + [g]. It follows that H0G = Z for any group G. 2 ∼ 0 Proposition 2.6. H1G is isomorphic to the abelianization of G: H1G = Gab = G=G . Proof. Let g denote the homology class of the cycle [g] 2 C1(G), and define a map ' : H1G ! Gab by g 7! gG0 and extend additively to make a homomorphism of abelian groups (that is g + h 7! ghG0). The map is well-defined because if g = h, then [g]−[h] = [a]+[b]−[ba], so that g is mapped to haba−1b−1G0 = hG0. The map is surjective, and injectivity follows from the identity gh = g + h. 0 0 2.6. Functoriality. Suppose α : G ! G is a map of groups, and define Hn(α): Hn(G) ! Hn(G ) via [g1j ::: jgn] 7−! [α(g1)j ::: jα(gn)] 3. Homology and Cohomology with Coefficients 3.0. Preliminaries on HomG. Let G be a group, and let M and N be ZG-modules. Exercise 3.1. Show that the formula (gu)(m) = g · u(g−1m) defines a left action of G on HomZ(M; N), called the diagonal action. Notice that gu = u if and only if the action of G commutes with u, so it follows that G HomZG(M; N) = HomZ(M; N) ⊂ HomZ(M; N) Exercise 3.2. If F is a projective ZG-module, and M is a ZG-module which is free as a Z-module, then F ⊗ M with diagonal action g · f ⊗ m = (g · f) ⊗ (g · m) is a projective ZG-module. 3.1. Definition of H∗(G; M). Let G be a group, and let F ! Z be a projective resolution of Z over ZG. For a G-module M, consider M as a chain complex concentrated in degree 0. Then HZG(F; M)n = HomZG(F−n;M), so we can reindex to consider the cochain complex defined by n HZG(F; M) = HZG(F; M)−n = HomZG(Fn;M) Since all boundary maps in the complex defined by M are 0, the boundary map of HZG(F; M) is n+1 (δnu)(x) = (−1) u(@x) for u 2 HomZG(Fn;M) and x 2 Fn+1. Remark 3.3. The sign convention here is used to follow the book. Changing the signs of coboundary maps does not affect cohomology. Definition 3.4. The cohomology groups of G with coefficients in M are the cohomology groups of the cochain complex HZG(F; M): n n H (G; M) = H (HZG(F; M)) Exercise 3.5. ∼ G G (1) Show that HomZG(Z;M) = M as abelian groups, where M is the subgroup of invariants, ie, of fixed points of M under the action of G. 0 G (2) Use the left exactness of HomZG(·;M) to show that H (G; M) = M . Example 3.6. Let G be an infinite cyclic group with generator t, and consider the projective resolution t−1 0 ! ZG −−! ZG ! Z ! 0 The cochain complex HZG(F; M) has the form: 0 ! M −−!1−t M ! 0 It follows that H0(G; M) consists of all m such that tm = m, ie, H0(G; M) = M G, and H1(G; M) is the quotient of M by elements of the form gm − m. This quotient, denoted MG, is called the group of co-invariants. 3 Example 3.7. Let G be a finite cyclic group of order n with generator t, let N = 1 + t + t2 + ::: + tn−1, and consider the projective resolution N t−1 N t−1 ··· −! ZG −−! ZG −! ZG −−! ZG ! Z ! 0 For a ZG-module M, the associated chain complex is then 0 ! M −−!1−t M −!N M −−!1−t M −!·N · · G Since Ng = N = gN for any g 2 G, we have a well-defined map, called the Norm map, N : MG ! M sending [m] to Nm. Exercise 3.8. Check that Hn(G; M) = ker N for odd n and Hn(G; M) = coker N for even n ≥ 2. (We already know that H0(G; M) = M G) 3.2. Ext. We obtain an immediate generalization of H∗(G; ·) by taking projective resolutions of ZG-modules other than Z Definition 3.9.
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