Counterexamples in Involutions of Azumaya Algebras

TL;DR

This paper constructs the first known examples of Azumaya algebras with involutions that cannot be simplified to trivial cases, revealing new cohomological obstructions and properties of fixed loci under involutions.

Contribution

It provides the first explicit examples of extraordinary involutions on Azumaya algebras and analyzes fixed loci, advancing understanding of algebraic involutions and their obstructions.

Findings

Existence of non-trivial involutions on Azumaya algebras with cohomological obstructions.

Construction of a ring with a disconnected fixed locus where involutions are uniformly orthogonal or symplectic.

Identification of conditions under which involutions cannot be simplified to trivial cases.

Abstract

Suppose $A$ is an Azumaya algebra over a ring $R$ and $\sigma$ is an involution of $A$ extending an order-$2$ automorphism $\lambda:R\to R$. We say $\sigma$ is extraordinary if there does not exist a Brauer-trivial Azumaya algebra $\mathrm{End}_R(P)$ over $R$ carrying an involution $\tau$ so that $(A, \sigma)$ and $(\mathrm{End}_R(P), \tau)$ become isomorphic over some faithfully flat extension of the fixed ring of $\lambda:R\to R$. We give, for the first time, an example of such an algebra and involution. We do this by finding suitable cohomological obstructions and showing they do not always vanish. We also give an example of a commutative ring $R$ with involution $\lambda$ so that the scheme-theoretic fixed locus $Z$ of $\lambda:\mathrm{Spec} R\to \mathrm{Spec} R$ is disconnected, but such that every Azumaya algebra over $R$ with involution extending $\lambda$ is either orthogonal at every point of $Z$, or symplectic at every point of $Z$. No examples of this kind were previously known.