On the defining axioms of Azumaya algebras (Q1090390)

An Azumaya algebra is classically defined as an algebra \(R\) with center \(Z\) such that (i) \(R\) is a finitely generated projective \(Z\)-module, and (ii) the map \(\nu : R\otimes_ZR^0 (=R^e)\to \Hom_Z(R,R)\) defined by \(\nu (a\otimes b^0)(x)=axb\) is a \(Z\)-algebra isomorphism. In this paper, the author shows that if \(R\) admits an anti-automorphism \(S\) which leaves \(Z\) elementwise fixed, then (i) follows from (ii). Also, since the algebra \(R^e\) has center \(Z\) and admits such a map \(S\), the following are equivalent: (1) \(\nu: R^e\otimes_Z(R^e)^0\to \Hom_Z(R^e,R^e)\) is a \(Z\)-algebra isomorphism, (2) \(R\) is an Azumaya algebra. The natural conjecture is then that (i) is superfluous for any \(R\).