Involutions of Azumaya Algebras

Uriya A. First ; Ben Williams

Documenta mathematica, Tome 25 (2020), pp. 527-633

Résumé

We consider the general circumstance of an Azumaya algebra A of degree n over a locally ringed topos (X,OX​) where the latter carries a (possibly trivial) involution, denoted λ. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending λ are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra A is Brauer equivalent to an algebra carrying an involution extending λ, by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then A is Brauer equivalent to an Azumaya algebra of degree 2n carrying an involution. By comparison with the case of topological spaces, we show that the integer 2n is minimal, even in the case of a nonsingular affine variety X with a fixed-point free involution. As an incidental step, we show that if R is a commutative ring with involution for which the fixed ring S is local, then either R is local or R/S is a quadratic étale extension of rings.

Détail

DOI : 10.4171/dm/756

Classification : 11E39, 14F22, 16H05, 55P91
Mots-clés : Brauer group, involution, ramification, Azumaya algebra, sheaf cohomology, equivariant homotopy theory