# Algebra and Algebraic geometry

February 6, 2023

An involution on a ring A is a self-map of order 2 that is additive and reverses the order of multiplication. I will motivate the work by explaining the classification of involutions of central simple algebras over fields. In this context, the involution acts either trivially or as a Galois action on the centre.

I will explain what an Azumaya algebra A on a scheme X is, where the action on X can be considerbly more complicated, and give a definition of "involution" that fits this context. I will then give a classification of involutions into types, based on the local classification of the involution at the fixed points of the action of the involution on X. I will then present work in progress dedicated to determining which types actually arise. If time permits, I will make some remarks about extending the definition of "Brauer group" to this setting. The entire talk is based on joint work with Uriya First.

**Time: **4:10-5:10pm

**Location: **MATH 126

**Seminar Website: **https://yifeng-huang-math.github.io/seminar_ubc_ag_23w.html