On ring homomorphisms of Azumaya algebras

Kossivi Adjamagbo (1) , Jean-Yves Charbonnel (1) , Arno van den Essen (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 Department of Mathematics

The main theorem (Theorem 4.1) of this paper claims that any ring morphism from an Azumaya algebra of constant rank over a commutative ring to another one of the same constant rank and over a reduced commutative ring induces a ring morphism between the centers of these algebras. The second main theorem (Theorem 5.3) implies (through its Corollary 5.4) that a ring morphism between two Azumaya algebras of the same constant rank is an isomorphism if and only if it induces an isomorphism between their centers. As preliminary to these theorems we also prove a theorem (Theorem 2.8) describing the Brauer group of a commutative Artin ring and give an explicit proof of the converse of the Artin-Procesi theorem on Azumaya algebras (Theorem 2.6).

Mots clés: Azumaya algebras; ring homomorphisms; ring of matrices; center

Domaines: Anneaux et algèbres [math.RA]

Kossivi Adjamagbo, Jean-Yves Charbonnel, Arno van den Essen. On ring homomorphisms of Azumaya algebras. 2005. ⟨hal-00008556⟩