Exercise 13, § 16

Let A be a commutative ring, B a commutative A-algebra, and G a finite group of automorphisms of the A-algebra B. Suppose that B is a finitely generated, projective, faithful A-module and that the A-linear homomorphism $\psi : B\otimes_AB\rightarrow B^G$ defined by $\psi (b\otimes b') = (\sigma (b)b')_{\sigma\in G}$ is bijective.

a) Let N be a B-module, endowed with a homomorphism $\sigma \mapsto u_{\sigma}$ from G to Aut$_A(N)$ satisfying $u_{\sigma}(bx) =\sigma (b)u_{\sigma}(x)$ for $\sigma \in G,b\in B,x\in N$. Let M be the A-submodule of N consisting of the elements $x$ such that $u_{\sigma}(x) =x$ for every $\sigma \in G$. Prove that the canonical B-homomorphism $B\otimes_AM\rightarrow N$ is bijective (use Exercise 7, d) of VIII, p. 270 to reduce to the case when B is the product algebra $A^G$ on which G acts by permuting the factors).

b) Let S be an A-algebra and M the automorphism group of the B-algebra $S_{(B)}$, on which G acts by $^{\sigma}\varphi =$ (Id$_S\otimes \sigma$ )$\circ \varphi \circ$ (Id$_S\otimes \sigma$ )$^{-1}$. Extend the construction of Exercise 6 to define a bijection from the set of isomorphism classes of pairs $(T, u)$, where T is an A-algebra and $u$ an isomorphism from $T_{(B)}$ to $S_{(B)}$, to the set $Z^1(G,M)$, as well as a bijection from the set of isomorphism classes of A-algebras T such that the B-algebras $T_{(B)}$ and $S_{(B)}$ are isomorphic to $H^1(G,M)$.

c) Assume from now on that every projective B-module is free. Adapt the reasoning of Exercise 8 to construct an isomorphism $\theta :$ Br(B$/A$)$\rightarrow H^2(G,B^*)$.