Involutions of Azumaya Algebras

  • Uriya A. First

    Department of Mathematics, University of Haifa,199 Abba Khoushy Avenue, Haifa 3498838, Israel
  • Ben Williams

    Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
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We consider the general circumstance of an Azumaya algebra AA of degree nn over a locally ringed topos (X,OX)(\mathbf{X}, \mathcal O_\mathbf{X}) where the latter carries a (possibly trivial) involution, denoted λ\lambda. 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\lambda are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra AA is Brauer equivalent to an algebra carrying an involution extending λ\lambda, 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 AA is Brauer equivalent to an Azumaya algebra of degree 2n2n carrying an involution. By comparison with the case of topological spaces, we show that the integer 2n2n is minimal, even in the case of a nonsingular affine variety XX with a fixed-point free involution. As an incidental step, we show that if RR is a commutative ring with involution for which the fixed ring SS is local, then either RR is local or R/SR/S is a quadratic étale extension of rings.

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Uriya A. First, Ben Williams, Involutions of Azumaya Algebras. Doc. Math. 25 (2020), pp. 527–633

DOI 10.4171/DM/756