Exercise 11, § 14

Let A be a commutative integral domain, K its field of fractions, and S an Azumaya A-algebra (Exercise 8).

a) Prove that the K-algebra $S_{(K)}$ is central simple. Identify S with a subalgebra of $S_{(K)}$. b) Assume from now on that A is the only A-subalgebra of K that is a finitely generated A-module, $*$that is, that the ring A is integrally closed, cf. Comm. Alg., V,

§1, No. $1, 2*$. Prove that S is maximal (for the inclusion) among the A-subalgebras of $S_{(K)}$ that are finitely generated as A-modules (apply Exercise 9, d)).

c) Suppose, moreover, that A is Noetherian and that the K-algebra $S_{(K)}$ is isomorphic to End$_K$(V), where V is a vector space over K. Prove that there exists a finitely generated A-module M without torsion such that the A-algebra S is isomorphic to End$_A(M)$ (let $x$ be a nonzero vector of V; take M to be the set of elements $s(x)$ as $s$ runs through S).

$*$d) Prove that in the previous construction, we may assume M reflexive (replace M

with its bidual). If, moreover, the ring A is regular (AC, X, §4, n$^o2$, p. 55, définition 1), then the A-module M is projective (apply AC, X, §4, p. 164, exercice $12$)$.*$