UBC Algeb. Geom. Seminar:Zinovy Reichstein

Let k be a field, A be a finite-dimensional k-algebra (not necessarily commutative, associative or unital), and R be a commutative ring containing k. An R-algebra B is called an R-form of A if there exists a faithfully flat ring extension S/R such that B and A become isomorphic after tensoring with S. In this talk, based on joint work with Uriya First, I will be interested in the following question: if A can be generated by n elements as a k-algebra, how many elements are required to generate B as an R-algebra? For example, if A is an n-dimensional k-algebra with trivial (zero) multiplication, then an R-form of A is the same thing as a projective R-module. Otto Forster (1964) showed that every projective R-module B can be generated by n + d elements, where d is the Krull dimension of R. Richard Swan subsequently showed that this number is optimal. I will discuss generalizations of Forster's result to other types of algebras, in particular to Azumaya algebras.