Counterexamples in Involutions of Azumaya Algebras

Abstract

Suppose $A$ is an Azumaya algebra over a ring $R$ and $\ddot{I}ƒ$ is an involution of $A$ extending an order-$2$ automorphism $\hat{I}»:R\to R$. We say $\ddot{I}ƒ$ is extraordinary if there does not exist a Brauer-trivial Azumaya algebra ${\mathrm{End}}_R(P)$ over $R$ carrying an involution $\ddot{I}„$ so that $(A,\ddot{I}ƒ)$ and $({\mathrm{End}}_R(P),\ddot{I}„)$ become isomorphic over some faithfully flat extension of the fixed ring of $\hat{I}»:R\to R$. 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 $\hat{I}»$ so that the scheme-theoretic fixed locus $Z$ of $\hat{I}»:\mathrm{Spec}R\to \mathrm{Spec}R$ is disconnected, but such that every Azumaya algebra over $R$ with involution extending $\hat{I}»$ is either orthogonal at every point of $Z$, or symplectic at every point of $Z$. No examples of this kind were previously known.

21 pages