Exercise 14, § 16

Let A be a Noetherian commutative local ring, $\mathfrak{m}$ its maximal ideal, $\kappa$ the field $A/\mathfrak{m}$, and S an Azumaya algebra over A.

a) Let $e$ be an idempotent in S whose image in $S/\mathfrak{m}S$ is an indecomposable idempotent. Deduce from Nakayama’s lemma (cf. VIII, p. 168, Exercise 16) that the canonical homomorphism $S\rightarrow$ End$_A(Se)$ is bijective.

b) Suppose that $\mathfrak{m}$ is nilpotent $*$(or, more generally, that A is complete for the $\mathfrak{m}$-adic topology)$*$. Prove that the homomorphism $r_{\kappa /A}:$ Br(A) $\rightarrow$ Br($\kappa$ ) is injective.

$*$c) Assume from now on that A is complete. Let L be a separable extension of $\kappa$ of

finite degree that is a splitting field for the $\kappa$-algebra $S/\mathfrak{m}S$. Let $x$ be a primitive element of $L,f$ its minimal polynomial, F a monic polynomial in A[X] whose image in $\kappa [X]$ is $f$, and $B = A[X]/(F)$. Prove that B is a local A-algebra, free and finitely generated as an A-module, with maximal ideal $\mathfrak{m}B$ and residue field isomorphic to L; the B-algebra $S_{(B)}$ is isomorphic to a matrix algebra.

d) Suppose, moreover, that L is a Galois extension of $\kappa$, with Galois group G. Prove that the action of G on L lifts to an action on the A-algebra B (let $G'$ be the automorphism group of this algebra; deduce from Hensel’s lemma (Comm. Alg., III, §4, No. 3, p. 215, Theorem 1) that the natural homomorphism $G'\rightarrow G$ is bijective). Deduce from Nakayama’s lemma that the homomorphism $\psi : B\otimes_AB\rightarrow B^G$ defined in Exercise 13 is bijective.

e) Prove that the homomorphism $H^q(\pi ) : H^q(G,B^*)\rightarrow H^q(G,L^*)$ is bijective for every $q\geqslant 1$. (For $i\geqslant 1$, let $U^i$ be the subgroup of elements $b$ of $B^*$ such that $b\equiv 1$ (mod $\mathfrak{m}^iB$). Observe that the $\mathbf{Z}$[G]-module $U^i/U^{i+1}$ is isomorphic to $\mathfrak{m}^iB/\mathfrak{m}^{i+1}B$, hence to a power of L, and deduce from Exercise 5, d) that $H^q(G,U^i/U^{i+1})$ is zero for $q\geqslant 1$. Let $c\in Z^q(G,U^1)$. Construct, by induction, a sequence $(b_n)$ with $b_n\in Z^{q-1}(G,U^n)$ such that $c-d^{q-1}(\sum^n_{i=1}b_i)$ belongs to $Z^q(G,U^{n+1})$. Conclude that $H^q(G,U^1)$ is zero for $q\geqslant 1.)$

f ) Conclude that the canonical homomorphisms Br(B$/A$)$\rightarrow$ Br(L$/\kappa$ ) and Br(A) $\rightarrow$ Br($\kappa$ ) are bijective.$*$