Exercise 9, § 14

Let A be a commutative ring and S an Azumaya A-algebra (Exercise 8).

a) Prove that the center of S (which we identify with A) is a direct factor of the A-module S (use Exercise 7, d)).

b) Let M be an $(S$, S)-bimodule; denote by $\mathscr{Z}(M)$ the A-submodule of M consisting of the elements $m$ such that $sm=ms$ for every $s\in S$. Deduce from Exercise 8, a) that the canonical homomorphism $S\otimes_A\mathscr{Z}(M)\rightarrow M$ is bijective.

c) The mapping $N\mapsto \mathscr{Z}(N)$ is an increasing bijection from the set of $(S$, S)-sub-bimodules of M to the set of A-submodules of $\mathscr{Z}(M)$; the inverse bijection sends an A-submodule N of $\mathscr{Z}(M)$ to the $(S$, S)-sub-bimodule SN of M. In particular, the mapping $\mathfrak{s}\mapsto \mathfrak{s}\cap A$ is an increasing bijection from the set of two-sided ideals of S to the set of ideals of A.

d) Let $\rho : S\rightarrow T$ be an A-algebra homomorphism, and let $S'$ be the commutant of $\rho (S)$ in T. Prove that the canonical homomorphism $S\otimes_AS'\rightarrow T$ is bijective (apply a) to the $(S$, S)-bimodule T).

e) Suppose that every projective A-module L such that dim$_{A/\mathfrak{m}}(L/\mathfrak{m}L) = 1$ for every maximal ideal $\mathfrak{m}$ of A is free of rank 1 (this is, for example, the case when the ring A is local or a principal ideal domain). Prove that every endomorphism $f$ of the A-algebra S is an inner automorphism (apply a) to the $(S$, S)-bimodule $S^f$, using Exercise 8, a) to show that the A-module $\mathscr{Z}(S^f)$ is projective).