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let be $R$ a commutative ring

Auslander and Goldman define a central separabl $R$-algebra $A$ as a faithful $R$-module with $Z(A) = R$ , and such that $A$ is a projective $A\otimes_R A^{\mathrm{op}}$-module

Following Milne; an Azumaya algebra $A$ over a local ring $R $ is an algebra (with $Z(A) = R$) which is a free module of finite rank over $R$ such that the natural map $A\otimes_R A^{\mathrm{op}}\to \mathrm{End}_R(A)$ is an isomorphism

How to prove that a central separable algebra over a local ring is just an Azumaya algebra.

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    $\begingroup$ How about Theorem 2.1 of Auslander, Goldman, "The Brauer group of a commutative ring" (link), the equivalence of (a) and (d) ? $\endgroup$
    – Minseon Shin
    Sep 12, 2019 at 16:33
  • $\begingroup$ thank you dear @MinseonShin,even the theorem give the inverse implication ; but for local rings, i guess we have an equivalence, since over such a ring (projective implies free); $\endgroup$
    – Moutand Mohammed
    Sep 12, 2019 at 23:05

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