SFU Number Theory and Algebraic Geometry Seminar: Ben Williams

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.