Exercise 18, § 6

Let G be a finite group and M a $\mathbf{Z}^{(G)}$-module. We say that M is cohomologically trivial if for every subgroup H of G and for every $q \in \mathbf{Z}$, the group $\hat{H}^q(H, M)$ is zero. a) Now prove that a $\mathbf{Z}^{(G)}$-module projective relatively to $\mathbf{Z}$ (in particular an induced module, cf. exercise 12) is cohomologically trivial. b) Let n, r, k be three integers such that r divides $(k^n - 1)$; let G be a cyclic group of order n, $\sigma$ a generator of G. We endow the $\mathbf{Z}$-module $M = \mathbf{Z}/r\mathbf{Z}$ with the structure of $\mathbf{Z}^{(G)}$-module defined by $\sigma m = km$ for every $m \in M$. Prove that M is projective relatively to $\mathbf{Z}$ if and only if n and r are relatively prime. c) With the notations of b), show that for $\hat{H}^q(G, M) = 0$ for every $q \in \mathbf{Z}$ to hold, it is necessary and sufficient that one has $(r, k - 1)(r, 1 + k + \cdots + k^{n-1}) = r$. In particular if $r = k^n - 1$, show that M is cohomologically trivial. d) Deduce from b) and c) an example of a $\mathbf{Z}^{(G)}$-module that is cohomologically trivial which is not projective relatively to $\mathbf{Z}$, and of a $\mathbf{Z}^{(G)}$-module M such that $\hat{H}^q(G, M) = 0$ for every $q \in \mathbf{Z}$ which is not cohomologically trivial. ¶19) Let p be a prime number, G a p-group, M a $\mathbf{Z}^{(G)}$-module. We put $A = F_p^{(G)}$. a) Suppose that $pM = 0$. Show that the following conditions are equivalent: α) There exists a rational integer q such that $\hat{H}^q(G, M) = 0$.

EXERCISES

β) M is cohomologically trivial (exercise 18). γ) M is an induced $\mathbf{Z}^{(G)}$-module. δ) The A-module M is free. (To prove that α) implies δ), reduce to the case where $q = -2$; show that one then has

and conclude with exercise 4, p. 185 and exercise 22 of VIII, § 8). b) Suppose that multiplication by p is injective in M. Show that the following conditions are equivalent: α) There exists an integer q such that the groups $\hat{H}^q(G, M)$ and $\hat{H}^{q+1}(G, M)$ are zero. β) M is cohomologically trivial. γ) The A-module $M/pM$ is free. ¶ 20) Let G be a finite group, M a $\mathbf{Z}^{(G)}$-module. a) Suppose that the Z-module M is free. Show that the $\mathbf{Z}^{(G)}$-module M is projective if and only if for every Sylow subgroup H $\subset G$, the $\mathbf{Z}^{(H)}$-module M is cohomologically trivial. (To prove that the condition is sufficient, consider a resolution $0 \to R \to L \to M \to 0$ of the $\mathbf{Z}^{(G)}$-module M, where L is free; show that the $\mathbf{Z}^{(H)}$-module Q = Hom$_\mathbf{Z}$(M, N) is cohomologically trivial by observing that Q/pQ is cohomologically trivial and by using exercise 19. Deduce from exercise 17, e) that one has $H^1(G, Q) = 0$, and conclude that M is projective.) b) Show that the following conditions are equivalent: α) The $\mathbf{Z}^{(G)}$-module M is cohomologically trivial. β) The $\mathbf{Z}^{(G)}$-module M admits a finite projective resolution. γ) The $\mathbf{Z}^{(G)}$-module M admits a projective resolution of length one. (To show that α) implies γ), consider an exact sequence $0 \to R \to L \to M \to 0$, where L is a free $\mathbf{Z}^{(G)}$-module, and apply a) to R.) c) Suppose that the Z-module M is divisible. Show that the $\mathbf{Z}^{(G)}$-module M is cohomologically trivial if and only if it is injective (consider an exact sequence $0 \to M \to I \to Q \to 0$, where I is an injective $\mathbf{Z}^{(G)}$-module, and show, using b), that the $\mathbf{Z}^{(G)}$-module Hom$_\mathbf{Z}$(Q, M) is cohomologically trivial). d) For a $\mathbf{Z}^{(G)}$-module to be cohomologically trivial, it is necessary and sufficient that it admit an injective resolution of length 1. ¶ 21) Let G be a cyclic group of finite order. a) If M is a $\mathbf{Z}^{(G)}$-module and q an even (resp. odd) rational integer, the group $\hat{H}^q(G, M)$ is isomorphic to $\hat{H}^0(G, M)$ (resp. $\hat{H}^1(G, M)$) (cf. exercise 7). b) Let $\mathscr{C}$ denote the set of classes of finitely generated $\mathbf{Z}^{(G)}$-modules T such that the groups $\hat{H}^q(G, T)$ are finite for every $q \in \mathbf{Z}$; if the $\mathbf{Z}^{(G)}$-module M is of type $\mathscr{C}$, put Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of $\mathbf{Z}^{(G)}$-modules. Show that if two of the modules M’, M, M” are of type $\mathscr{C}$, the third is also of type $\mathscr{C}$, and that one then has $h(M) = h(M')\ h(M'')$. In particular, the function h defines a homomorphism of the grothendieck group K($\mathscr{C}$) into the multiplicative group $\mathbf{Q}^*$. c) Let M be a finite $\mathbf{Z}^{(G)}$-module. Show that M is of type $\mathscr{C}$ and that one has $h(M) = 1$. d) Suppose that the order of G is a prime number p. Let $\mathscr{C}'$ denote the set of classes of finitely generated $\mathbf{Z}^{(G)}$-modules in which multiplication by p has a finite kernel and cokernel; if M is of type $\mathscr{C}'$, put $\varphi(M) = \operatorname{Card}(\operatorname{Coker} p_M)/\operatorname{Card}(\operatorname{Ker} p_M)$. Prove as in b) that $\varphi$ defines a homomorphism of K($\mathscr{C}'$) into $\mathbf{Q}^*$. e) Show that K($\mathscr{C}'$) is generated by the classes of the $\mathbf{Z}^{(G)}$-modules T possessing one of the following properties: (i) T is finite. (ii) Multiplication by p in T is bijective. (iii) $T = \mathbf{Z}$ endowed with the trivial action of G. (iv) $T = \mathbf{Z}^{(G)}$. (v) $T = \mathbf{Q}_p/\mathbf{Z}_p$ (cf. TG, III, p. 84, exercise 23), endowed with the trivial action of G. (vi) $T = \mathbf{Z}^{(G)} \otimes_\mathbf{Z} (\mathbf{Q}_p/\mathbf{Z}_p)$. (First show that every $\mathbf{Z}^{(G)}$-module of type $\mathscr{C}'$ can be written in K($\mathscr{C}'$) as the sum of a $\mathbf{Z}^{(G)}$-module of finite type over $\mathbf{Z}$ and of a $\mathbf{Z}^{(G)}$-module in which multiplication by p is surjective. Then determine the structure of the simple modules over the rings $\mathbf{Q}^{(G)}$ and $\mathbf{Q}_p^{(G)}$.)

g) Deduce from what precedes that one has $\mathscr{C}' \subset \mathscr{C}$ and that if M is a $\mathbf{Z}^{(G)}$-module of type $\mathscr{C}'$, one has

(Verify the formula in the six cases considered in f).) 22) Let K be a commutative field, L a finite galois extension of K, G its galois group. a) Prove that the exact sequence of G-modules allows one to define an isomorphism $k_L : (L^n \cap K^*)/K^{*n} \to H^1(G, \mu_n(L))$. When $\mu_n(K)$ has n elements and when one identifies $H^1(G, \mu_n(L))$ with Hom $(G, \mu_n(K))$, prove that $k_L$ coincides with the homomorphism defined in V, § 11, No. 8. b) Let p be a prime number; suppose that K is of characteristic p. Prove that the exact sequence of G-modules (V, § 11, No. 9) allows one to define an isomorphism $a_L : (p(L) \cap K)/p(K) \to H^1(G, F_p)$ which coincides with the homomorphism defined in loc. cit. if one identifies $H^1(G, F_p)$ with Hom $(G, F_p)$. 23) Let G be a group, M a group with operators in G (I, p. 29, Def. 2). We shall denote by $^g m$ the composite of $g \in G$ and $m \in M$. We shall say that M is a G-group if one has $^h(^g m) = ^{hg} m$ for $m \in M, g, h \in G$, in other words if the mapping $\varphi$ from G into Aut (M) such that $\varphi(g).m = ^g m$ for $g \in G, m \in M$ is a homomorphism. If M is a G-group, we denote by $H^0(G, M)$ the subgroup of M formed by the elements $m \in M$ such that $^g m = m$ for all $g \in G$, and by $Z^1(G, M)$ the set of mappings $h$ from G into M such that a) We say that two elements $h_1, h_2$ of $Z^1(G, M)$ are cohomologous if there exists an element m of M such that $h_2(g) = m^{-1}\ h_1(g)\ ^g m$ for all $g \in G$. Prove that the relation thus defined is an equivalence relation on $Z^1(G, M)$; the quotient set is denoted by $H^1(G, M)$. The class in $H^1(G, M)$ of the mapping $i \in Z^1(G, M)$ such that $i(g) = 1_M$ for all $g \in G$ is denoted by $e_{G, M}$ or simply $e_M$. b) Let $f : M \to N$ be a homomorphism of G-groups (I, p. 30, Def. 3); prove that $f$ induces a group homomorphism $f^0 : H^0(G, M) \to H^0(G, N)$, and a mapping $f^1 : H^1(G, M) \to H^1(G, N)$ such that $f^1(e_M) = e_N$. c) Let $M \xrightarrow{i} N \xrightarrow{p} Q$ be an exact sequence of G-groups and homomorphisms, with i injective, p surjective and Prove that the group $N' = p^{-1}(H^0(G, Q))$ acts on $Z^1(G, M)$ in such a way that $(n.h)(g) = n^{-1}\ h(g)\ ^gn$ for $n \in N', h \in Z^1(G, M), g \in G$; prove that this action defines by passing to the quotient an action of $H^0(G, Q)$ on $H^1(G, M)$. The mapping $i^1 : H^1(G, M) \to H^1(G, N)$ passes to the quotient and defines a bijection of $H^1(G, M)/H^0(G, Q)$ onto $\operatorname{Im}(i^1)$. d) The action of $H^0(G, Q)$ on $H^1(G, M)$ allows one to define a mapping $\partial : H^0(G, Q) \to H^1(G, M)$ by $\partial(q) = q.e_M$ for $q \in H^0(G, Q)$. Prove that one has e) Suppose now that $i(M)$ is contained in the center of N. Prove that $\partial$ is a group homomorphism; construct an action of the group $H^1(G, M)$ on the set $H^1(G, N)$ such that the mapping $p^1$ passes to the quotient and defines a bijection of $H^1(G, N)/H^1(G, M)$ onto $\operatorname{Im}(p^1)$. f) Let E be the set of mappings $k$ from G into N such that $p \circ k \in Z^1(G, Q)$. For $k \in E$, prove that there exists an element $f_k$ of $Z^2(G, M)$ such that $i(f_k(x, y)) = k(x)\ ^xk(y)\ (k(xy))^{-1}$ for $x, y \in G$. Prove that the mapping $k \mapsto f_k$ defines by passing to the quotient a mapping $\partial^1 : H^1(G, Q) \to H^2(G, M)$. Prove that one has $\operatorname{Im}(p^1) = (\partial^1)^{-1}(0)$. 24) Let K be a commutative field, L a finite galois extension of K, with galois group G. Let V (resp. V’) be a K-vector space, t (resp. t’) a tensor of type $(p, q)$ on V (resp. V’) (III, p. 63). We call a K-isomorphism (or simply an isomorphism) of $(V, t)$ onto $(V', t')$ a K-linear isomorphism $u : V \to V'$ such that $(\mathbf{T}^p(u) \otimes \mathbf{T}^q(\tilde{u}))\ (t) = t'$. We shall denote by $V_L$ the L-vector space $L \otimes_K V$ and by $t_L$ the tensor of type $(p, q)$ on $V_L$ deduced from t by extension of scalars. We denote by $F_{L/K}(V, t)$ the set of isomorphism classes of pairs $(W, \tau)$ such that $(W_L, \tau_L)$ is L-isomorphic to $(V_L, t_L)$.

EXERCISES

(a) Let M be the group of L-automorphisms of $(V_L, t_L)$. For $m \in M$ and $\sigma \in G$, put

where $\pi_V : G \to \mathrm{Aut}_K(V_L)$ denotes the action of G on $V_L = L \otimes_K V$ deduced from the action of G on L. Now prove that one thus defines a structure of G-group on M. (b) Let $(W, \tau)$ be an element of $F_{L/K}(V, t)$, and let $u$ be an L-isomorphism of $(V_L, t_L)$ onto $(W_L, \tau_L)$. For $\sigma \in G$, denote by $f_u(\sigma)$ the element $u^{-1} \circ \pi_W(\sigma) \circ u \circ \pi_V(\sigma)^{-1}$ of M. Now prove that $f_u \in Z^1(G, M)$ (exercise 23), and that its class in $H^1(G, M)$ is independent of the choice of $u$; denote it by $\theta(W, \tau)$. (c) Now prove that the mapping $\theta : F_{L/K}(V, t) \to H^1(G, M)$ is bijective, and that one has $\theta(V, t) = e_M$.

¶25) Let K be a commutative field, L a finite galois extension of K, G its galois group. We make G operate on the group $\mathbf{GL}(n, L)$ by putting $\sigma(a_{ij}) = (\sigma(a_{ij}))$ for $\sigma \in G$, $(a_{ij}) \in \mathbf{GL}(n, L)$. Every subgroup of $\mathbf{GL}(n, L)$ stable for this action is thus endowed with a structure of G-group (exercise 23). a) Now prove that the sets $H^1(G, \mathbf{GL}(n, L))$ and $H^1(G, \mathbf{SL}(n, L))$ are reduced to one element (cf. V, § 10, no 5, prop. 9). b) Now prove that the set $H^1(G, \mathbf{PGL}(n, L))$ identifies with the set of isomorphism classes of central simple algebras S such that the algebra $L \otimes_K S$ is isomorphic to $M_n(L)$ (use exercise 24). By associating with such an algebra its class in the group Br(K, L) (VIII, § 13) and by identifying this last group with $H^2(G, L^*)$ (loc. cit.), one obtains a mapping $H^1(G, \mathbf{PGL}(n, L)) \to H^2(G, L^*)$ which is the opposite of the mapping $\partial^1$ (exercise 23, f)) deduced from the extension $L^* \to \mathbf{GL}(n, L) \to \mathbf{PGL}(n, L)$. Now prove that the image of $\partial^1$ is formed by elements of order n in $H^2(G, L^*)$. c) Now prove that the set $H^1(G, \mathbf{Sp}(2n, L))$ is reduced to one element (cf. IX, § 5, no 1, th. 1). d) Let Q be a quadratic form on the K-vector space V, $Q_L$ the quadratic form which is deduced from it on $L \otimes_K V$. Now prove that $H^1(G, O(Q_L))$ identifies with the set of isomorphism classes of quadratic forms Q’ on V such that $Q'_L$ is isomorphic to $Q_L$. 26) We denote by $A^e$ the k-algebra $A \otimes_k A^0$; every $(A, A)$-bimodule is naturally endowed with a structure of left $A^e$-module and also with a structure of right $A^e$-module. Let M be an $(A, A)$-bimodule, $n$ an integer $\geqslant 0$; we put:

a) The k-module $H_0(A, M)$ is isomorphic to the quotient of M by the sub-k-module generated by the elements $(am - ma)$ for $a \in A, m \in M$; the k-module $H^0(A, M)$ is isomorphic to the sub-k-module of M formed by the elements m such that $am = ma$ for all $a \in A$. The k-module $H^1(A, M)$ identifies with the quotient of the k-module of derivations $D_k(A, M)$ by the sub-k-module of inner derivations, that is to say derivations $d_m$, for $m \in M$, defined by $d_m(a) = am - ma$ for all $a \in A$ (cf. III, p. 132). b) Under the assumption that A is a k-module projective. Now prove that the standard resolution B(A) (X, p. 58) defines a projective resolution of the left $A^e$-module A. Deduce that the k-modules $H^n(A, M)$ are isomorphic to the homology modules of the complex $C(A, M)$, where, for $p \geqslant 0$, $C^p(A, M)$ is the k-module of k-multilinear mappings from $A^p$ into M, $C^p(A, M) = 0$ for $p < 0$, the differential being given by the formula: for $f \in C^p(A, M)$, $x_0, ..., x_p \in A$. Give an analogous expression for the k-modules $H_n(A, M)$. c) Under the assumption of b), let $\rho : k \to k'$ be a homomorphism of commutative rings, $A'$ the $k'$-algebra $k' \otimes_k A$, $M'$ an $(A', A')$-bimodule. Now prove that the k-modules $H_n(A, M')$ and $H_n(A', M')$ (resp. $H^n(A, M')$ and $H^n(A', M')$) are isomorphic. d) We now suppose that k is a field. Let B be a second k-algebra (associative and unital), M an $(A, A)$-bimodule, N a $(B, B)$-bimodule. For all $n \geqslant 0$, define an isomorphism of k-modules If one supposes moreover that A and B are of finite dimension over k, now prove that there exists likewise a k-isomorphism (use X, p. 76, th. 3).

e) Let M, N be two left A-modules, P a right A-module, such that the k-modules Hom_k(M, N) and M $\otimes_k$ P have natural structures of (A, A)-bimodules. Define for every $n \geq 0$ isomorphisms of k-modules:

**T 27) An augmentation of the k-algebra A is called a unital homomorphism of k-algebras $\pi : A \to k$; the pair (A, $\pi$) is called an augmented k-algebra. The augmentation endows k with a structure of left A-module. Let M be a left A-module, N a right A-module; we set

a) We denote by $M_{(\pi)}$ (resp. $N_{(\pi)}$) the group M (resp. N) endowed with the structure of (A, A)-bimodule defined by:

Prove that the k-module $H_0(A, \pi ; N)$ (resp. $H^0(A, \pi ; M)$, resp. $H^1(A, \pi ; M)$) is isomorphic to the k-module $H_0(A, N_{(\pi)})$ (resp. $H^0(A, M_{(\pi)})$, resp. $H^1(A, M_{(\pi)})$) defined in exercise 26.

b) We now suppose that A is a projective k-module. Define, using the standard resolution (X, p. 58), k-isomorphisms $\varphi_n : H_n(A, N_{(\pi)}) \to H_n(A, \pi ; N)$ and $\varphi^n : H^n(A, M_{(\pi)}) \to H^n(A, \pi ; M)$ for all $n \geq 0$.

c) Suppose that there exists a homomorphism of k-algebras $\rho : A \to A^e$ which makes $A^e$ a projective right A-module and such that the left ideal of $A^e$ generated by $\rho(\mathrm{Ker}\ \pi)$ is the kernel of the homomorphism $A \otimes_k A^e \to A$ defined by multiplication in A. Prove that there exist, for every (A, A)-bimodule Q and every integer $n \geq 0$, k-isomorphisms $H_n(A, \pi ; Q) \to H_n(A, Q)$ (resp. $H^n(A, \pi ; Q) \to H^n(A, Q)$), Q being considered as a right A-module (resp. left A-module) via the homomorphism $\rho$. (Use the canonical homomorphisms of X, p. 109.)

d) Prove that the hypotheses of c) are verified in the following cases: α) A is the algebra (over k) of a group G, $\rho$ is defined by $\rho(e_g) = e_g \otimes e_{g^{-1}}$. β) A is the tensor algebra (resp. symmetric algebra) of a projective k-module V; $\rho$ is defined by

\* γ) A is the enveloping algebra of a Lie algebra g over k, free as a k-module; one has

(use cor. 5 to th. 1 of LIE I, § 2, no 7).

δ) A is the ring of functions of a smooth algebraic group G over a field k; one takes

\* 28) Let g be a Lie algebra over the commutative ring k, U its enveloping algebra. We endow k with the structure of left U-module corresponding to the trivial representation of g. Let N be a right g-module, M a left g-module, n an integer $\geq 0$; we set

a) The k-module $H_0(g, N)$ (resp. $H^0(g, M)$) is isomorphic to $N/Ng$ (resp. to the submodule of M formed by the elements annihilated by g). We denote by $Z^1(g, M)$ (resp. $B^1(g, M)$) the sub-k-module of $\mathrm{Hom}_k(g, M)$ formed by the elements f such that $f([x, y]) = xf(y) - yf(x)$ for $x, y \in g$ (resp. by the elements $f_m$, for $m \in M$, such that $f_m(x) = xm$ for $x \in g$). Prove that $H^1(g, M)$ is identified with the quotient $Z^1(g, M)/B^1(g, M)$.

b) We now suppose that g is a free k-module. Prove that the k-modules $H^n(g, M)$ are identified with the homology modules of the complex $C(g, M)$, where for $p \geq 0$, $C^p(g, M)$ is the k-module of alternating multilinear mappings from $g^p$ into M, $C^p(g, M) = 0$ for $p < 0$, the differential being given by the formula

for $f \in C^p(g, M)$, $x_1, ..., x_{p+1} \in g$ (where the sign $\hat{}$ on a letter means that it is to be omitted).

Give an analogous expression for the $k$-modules $H_p(\mathfrak{g},M)$. (Use X, p. 184, exercise 21.)

c) Suppose that $\dim_k(\mathfrak{g})=n<\infty$. Now prove that $H^p(\mathfrak{g},M)=0$ for all $p>n+1$ and all $\mathfrak{g}$-modules $M$, and that there exists a $\mathfrak{g}$-module $N$ such that $H^n(\mathfrak{g},N)\neq0$ (take $N=\Lambda^n\mathfrak{g}$, where $\mathfrak{g}$ operates by the formula

now prove that $df=0$ for all $f\in C^{n-1}(\mathfrak{g},N)$.

d) The adjoint representation of $\mathfrak{g}$ and the representation of $\mathfrak{g}$ on $M$ define a representation $\theta$ of $\mathfrak{g}$ in $C(\mathfrak{g},M)$, such that

for $f\in C^p(\mathfrak{g},M)$, $x,x_1,\ldots,x_p\in\mathfrak{g}$. Show that, for all $x\in\mathfrak{g}$, $\theta(x)$ is an endomorphism homotopic to zero of the complex $C(\mathfrak{g},M)$ (define a homotopy $i(x)$ by putting $(i(x)f)(x_1,\ldots,x_p)=f(x,x_1,\ldots,x_p)$). \* 29) We retain the notations of the preceding exercise; suppose in addition that $k$ is a field of characteristic zero, and that the Lie algebra $\mathfrak{g}$ is semisimple (LIE, I, § 6). Let $M$ be a $\mathfrak{g}$-module of finite dimension over $k$.

a) If $M$ is simple and nontrivial, one has $H^p(\mathfrak{g},M)=0$ for all $p>0$ (the bilinear form on $\mathfrak{g}$ associated with $M$ is non-degenerate (LIE, I, § 6, No. 1, prop. 1); observe that multiplication by the corresponding Casimir element (LIE, I, § 3, No. 7) is zero in $k$ and bijective in $M$).

b) Now prove that $H^1(\mathfrak{g},M)=0$ whatever $M$ may be (use a) and the fact that $\mathfrak{g}=\mathfrak{g}$). Conversely, a $k$-Lie algebra $\mathfrak{h}$ (of finite dimension) such that $H^1(\mathfrak{h},M)=0$ for every $\mathfrak{h}$-module $M$ of finite dimension is semisimple.

c) Equip $k$ with the structure of $\mathfrak{g}$-module corresponding to the trivial representation. Now prove that, for all $p>0$, the $k$-vector space $H^p(\mathfrak{g},k)$ is isomorphic to the vector subspace of $C^p(\mathfrak{g},k)$ formed by the alternating $p$-linear forms $f$ which are invariant, that is to say such that

(Deduce from the semisimplicity of the representations of $\mathfrak{g}$ and from exercise 28, d) that $H^p(\mathfrak{g},k)$ is isomorphic to $H^p(C_{\mathfrak{g}}(\mathfrak{g}),k)$, where $C_{\mathfrak{g}}(\mathfrak{g})$ is the subcomplex of $C(\mathfrak{g},k)$ annihilated by all the endomorphisms $\theta(x)$ for $x\in\mathfrak{g}$; then now prove that $C_{\mathfrak{g}}(\mathfrak{g})$ has zero differential.)

d) Deduce from c) that $H^2(\mathfrak{g},M)=0$ for every $\mathfrak{g}$-module $M$ of finite dimension over $k$. (Show that every invariant bilinear form on $\mathfrak{g}$ is symmetric, by reducing to the case where $\mathfrak{g}$ is simple and $k$ algebraically closed.) \*