This Thursday at 12:00, U. Haifa Topology & Geometry seminar on May 16, 2019
Geometry & Topology Seminar
Speaker: Uriya First (University of Haifa)
Topic: Azumaya algebras without involution
Place: room 614 in the Science & Education Building
Date: Thursday, May 16, 2019
Let R/S be a quadratic Galois extension of commutative rings with nontrivial automorphism s (definitions will be recalled during the talk). A locally free R-algebra is called Azumaya if the specialization of A to every geometric point of Spec R is a matrix algebra. Brauer equivalence classes of Azumaya algebras form the Brauer group, an important cohomological invariant of R.
A theorem of Saltman characterizes the Brauer classes in Braur group of R admitting a representative with an s-involution, i.e., an involution whose restriction to the center R is s. Combined with a later proof of Knus, Parimala and Srinivas, it implies that if A is an Azumaya algebra that is Brauer equivalent to one admitting an s-involution, then it is Brauer equivalent to an algebra of degree 2*deg(A) admitting an s-involution.
I will discuss a joint work with Ben Williams where we use C_2-equivariant homomotopy theory to show that 2*deg(A) cannot be improved in general, at least for certain values of deg(A).
For comparison, when R is assumed to be semilocal, Saltman showed that A itself admits an involution, and so 2*deg(A) is not the lowest degree possible for a representative with an s-involution.