Extraordinary involutions on Azumaya algebras

Abstract: This is joint work with Uriya First. If $R$ is a ring and $n$ is a natural number, then an Azumaya algebra of degree $n$ on $R$ is an $R$-algebra such that $A$ becomes isomorphic to the $n \times n$ matrix algebra after some faithfully flat extension of scalars. An involution of an Azumaya algebra is an additive self map of order $2$ that reverses the multiplication. One obtains examples of Azumaya algebras with involution by starting with a projective $R$-module $P$ of rank $n$, equipped with a hermitian form. The endomorphism ring $\End_R(P)$ has the structure of an Azumaya algebra with involution. One may even allow the hermitian form to take values in a rank-$1$ projective $R$-module, rather than in $R$ itself.

We will say that an Azumaya algebra with involution $(A, \sigma)$ is semiordinary if there it becomes isomorphic to one constructed from a hermitian form after a faithfully flat extension of scalars. Although this is an extremely broad class of Azumaya algebras with involution, I will show that it is not all of them: there exist Azumaya algebras with truly extraordinary involutions. The method is to find an obstruction to being semiordinary in equivariant algebraic topology.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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