Algebraic Geometry Seminar: Uriya First

Let K be a field, let  t : K à K be an automorphism of order 1 or 2. Let F denote the subfield of t-invariant elements in K. Then either K=F or K/F is a quadratic Galois extension. Given a central simple K-algebra A, a t-involution of A is an anti-automorphism s: A à A satisfying s2 = id_A and which restricting to t on the center K. The involution s is said to be of the first kind if K=F and of the second kind if K/F is quadratic Galois. A classical theorem of Albert gives a necessary and sufficient for A to have a t-involution.

Suppose now that R is a commutative ring, t: R à R is an automorphism of order 1 or 2 and S is the fixed subring of t. Over R, the role of central simple algebras is played by an Azumaya  R-algebra.  In this context, Albert's theorem fails, but Saltman showed that the condition given by Albert determines when an Azumaya algebra A is Brauer equivalent to another Azumaya algebra admitting a t-involution, provided S=R  (first kind) or R/S is quadratic etale (second kind). This was extended to Azumaya algebras over schemes by Knus, Parimala and Srinivias.

I will discuss recent work with Ben Williams in which we treat the case where R is neither S nor a quadratic etale extension of S. (Our results also apply in the even more general context of locally ringed spaces.) In this case, the t-involutions can be regarded as being "of the third kind". This setting features new phenomena and raises interesting open questions.

Relevant definitions will be recalled during the talk.