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GROUP COHOMOLOGY Let G Be a Group. We Can Form the Group Ring Z

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GROUP COHOMOLOGY Let G Be a Group. We Can Form the Group Ring Z GROUP COHOMOLOGY AKHIL MATHEW 1. COHOMOLOGY AND HOMOLOGY Let G be a group. We can form the group ring Z[G] over G; by definition it is the set P of formal finite sums aigi, where ai 2 Z; gi 2 G, and multiplication is defined in the obvious manner. We shall call an abelian group A a G-module if it is a left Z[G]-module. This means, of course, that there exists a homomorphism G ! AutZ(A). We can also make A into a right Z[G]-module simply by writing ag := g−1a for a 2 A; g 2 G. This is important for tensor products. An example of a G-module is any abelian group with trivial action by G. For instance, we shall in the future denote by Z the integers with trivial G-action. Finally, if A and B are G-modules, then a G-homomorphism between them is a map φ : A ! B which is a Z[G] homomorphism. The set of G-homomorphisms between A and B is denoted by HomG(A; B). It is a left exact functor of A and B, covariant in B and contravariant in A. As usual its derived functors are denoted by Exti. Let A be a G-module. Then we define the cohomology groups as Hi(G; A) := Exti ( ;A): Z[G] Z Then the Hi(G; ·) are covariant functors from the category of G-modules to the category 0 of abelian groups. Now we have clearly H (G; A) = HomG(Z;A), by basic properties of Ext over any ring. Also, a Z-homomorphism Z ! A is determined by its value at 1; it is a G-homomorphism if and only if its image a 2 A is fixed by G, i.e ga = a for all G G g 2 G. Denote the set of such a by A . So we see that HomG(Z;A) = A ; in particular A ! AG is a left exact functor. Then another way of stating our definitions is that Hi(G; ·) are the derived functors of the functor A ! AG. i Go back to the initial definition of H (G; ·) in terms of Ext. Let fPig be a projective resolution of Z, i.e. an exact sequence ···! Pn ! Pn−1 !···! P1 ! P0 ! Z; where all the Pi are projective (e.g. free) G-modules. Then we have a chain complex 0 ! HomG(P0;A) ! HomG(P1;A) :::: The homology groups of this complex are the derived functors Hi(G; A). The above characterization yields the following: Proposition 1. Let A be a co-induced G-module, i.e. A = HomZ(Z[G];B) for a Z- module B. (This can be made into a G-module by multiplication on the right.) Then we have Hi(G; A) = 0 if i > 0. Proof. Consider the chain complex HomG(Pi;A) = HomG(Pi; HomZ(Z[G];B)) = HomZ(Pi;B), as is easily seen. Since the Pi are projective, this sequence is exact except possibly at i = 0, and the proposition follows. Date: January 25, 2009. 1 2 AKHIL MATHEW Moreover, by basic properties of derived functors, we have the following result: If 0 ! A ! B ! C ! 0 is a “short” exact sequence of G-modules, then there exists a “long” exact sequence 0 ! AG ! BG ! CG ! H1(G; A) ! H1(G; B) ! H1(G; C) ! H2(G; A) ! H2(G; B) :::: The definition of homology is similar. Let A be a G-module. We define Hi(G; A) := TorZ[G](Z;A): Here we have used the making of A into a right Z[G]-module, as above. Then by defini- tion, the Hi(G; ·) are the derived functors of the functor A ! Z ⊗Z[G] A, which is also H0(G; A), by elementary properties of Tor. In the future, we suppress the Z[G] in the tensor product ⊗Z[G], for convenience. Let us compute the functor Z ⊗Z[G] A. We have an exact sequence of G-modules 0 ! IG ! Z[G] ! Z ! 0; where IG is the augmentation ideal, i.e. the G-submodule of Z[G] generated by the g − 1, g 2 G; and the homomorphism Z[G] ! Z is the augmentation homomorphism that maps each g to 1. Since the tensor product is right exact, we have the following exact sequence: IG ⊗Z[G] A ! Z[G] ⊗Z[G] A ! Z ⊗Z[G] A ! 0: The second term is just A. Thus we see that Z⊗A = A=IGA. In other words, the Hi(G; ·) are the derived functors of A ! A=IGA. There are derived functor properties of the homology groups. As usual, if fPig is a projective resolution of Z as before, then the Hi(G; A) are the homology groups of the chain complex (tensor products over Z[G]) ···! P2 ⊗ A ! P1 ⊗ A ! P0 ⊗ A ! 0: This immediately yields the following analog of Proposition 1: Proposition 2. Let A be an induced G-module, i.e. A = Z[G] ⊗Z B for a Z-module B. (This can be made into a G-module by multiplication on the left.) Then we have Hi(G; A) = 0 if i > 0. Proof. Indeed: Pn ⊗Z[G] A = Pn ⊗Z[G] (Z[G] ⊗Z B) = (Pn ⊗Z[G] Z[G]) ⊗Z B = Pn ⊗Z B; and the Pn are projective. We have a similar long exact sequence, which we leave the reader to write out. 2. EXAMPLES We construct an explicit resolution of Z, which will give us a practical criterion for computing the cohomology and homology groups. Define Pi to be the free abelian group generated by the symbols (g0; : : : ; gi) for gj 2 G; 0 ≤ j ≤ i. Let G act on Pi in the obvious manner, i.e. g(g0; : : : ; gi) := (gg0; : : : ; ggi). Then the Pi are G-modules. We give the boundary homomorphisms. The homomorphism P0 ! Z maps (g0) ! 1 for all g0 2 G, and is extended by linearity. The homomorphism di : Pi ! Pi−1 for i > 0 is given by i X j di(g0; : : : ; gi) := (−1) (g0;:::; g^j; : : : ; gi); j=0 GROUP COHOMOLOGY 3 where as usual the hat means a term is omitted. These are G-homomorphisms and form a chain complex (compute di ◦ di−1 directly). We must now check exactness. First, if d0(a) = 0, then we can write a as a sum of P (gj) − (hj), which is the boundary of (gj; hj). Next, fix g 2 G. We define a chain homotopy Di : Pi ! Pi+1 by Di(g0; : : : ; gi) := (g; g0; : : : ; gi). We see easily that Dd + dD = 1, by abuse of notation, whence the sequence is exact. In the future we will drop indexes like that. Hence, for instance, a n-cocycle in some G-module A is a G-homomorphism f : Pn ! A such that fd = 0. f is a n-coboundary if f = gd for some G-homomorphism Pn−1 ! A. There is, however, a more convenient notation. Pi is a free G-module on the set [g1; : : : ; gi] := (1; g1; g1g2; g1g2g3; : : : ; g1 : : : gi). So a G-homomorphism f : Pi ! A is equivalent to a map assigning an element of A to each [g1; : : : ; gi], which extends to the free module in the usual manner. Note that i−1 X j i d[g1; : : : ; gi] = g1[g2; : : : ; gi] + (−1) [g1; : : : ; gjgj+1; : : : ; gi] + (−1) [g1; : : : ; gi−1]: j=1 Now denote the set of [g1; : : : ; gi] by Si. Hence a n-cocycle becomes a map f : Si ! A such that i X j i+1 g1f([g2; : : : ; gi+1])+ (−1) f([g1; : : : ; gjgj+1; : : : ; gi])+(−1) f([g1; : : : ; gi]) = 0; j=1 where a n-coboundary becomes a map f such that there exists a map h : Si−1 ! A with i−1 X j i f([g1; : : : ; gi]) = g1h([g2; : : : ; gi])+ (−1) h([g1; : : : ; gjgj+1; : : : ; gi])+(−1) h([g1; : : : ; gi−1]: j=1 Take the case i = 1. Then a 1-cocyle is a map f : G ! A such that f(st) = sf(t) + f(s), or a crossed homomorphism. A 1-coboundary is a map f of the form f(s) = sa − a. This allows us to compute the cohomology groups in special cases. We will use one more example, this time for homology: Proposition 3. H1(G; Z) = Gab, the abelianization of G. Proof. We have an exact sequence 0 ! IG ! Z[G] ! Z ! 0; with the usual augmentation ideal and augmentation homomorphism. Taking the long ex- act sequence and noting that Z[G] is Z[G]-free (hence projective), we see that H1(G; Z) = 2 2 H0(G; IG) = IG=IG. So we have to define a map f : Gab ! IG=IG which is an isomor- phism. Let f(s) := s − 1; then f(st) − f(s) − f(t) = st − 1 − (s − 1) − (t − 1) = 0 in 2 2 IG, so f is a homomorphism. For the inverse, define g : IG=IG ! Gab by g(σ − 1) := σ. 2 Note that all elements in IG=IG are of that form. As above this is a homomorphism (since Gab is abelian), and it is clearly the inverse of f. 3. INFLATION,RESTRICTION,CORESTRICTION Let A be a G-module, and let f : G0 ! G be a homomorphism of groups. We can consider A as a G0 module by setting g0a := f(g0)a for g0 2 G0. We have obviously 0 AG ⊂ AG . Hence there is an inclusion homomorphism H0(G; A) ! H0(G0;A). Now the families of functors fHi(G; ·)g and fHi(G0; ·)g are universal δ-functors in the sense 4 AKHIL MATHEW of Grothendieck, (see [?]) since they are derived functors.
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