Generating algebras via versality

Abstract: Let R be a noetherian (commutative) ring of Krull dimension d. A classical theorem of Forster states that a rank-n locally free R-module can be generated by n+d elements. Swan and Chase observed that this upper bound cannot be improved in general. I will discuss joint works with Zinovy Reichstein and Ben Williams where similar upper and lower bounds are obtained for R-algebras, provided that R is of finite type over an infinite field k. For example, every Azumaya R-algebra of degree n (i.e. an n-by-n matrix algebra bundle over Spec R) can be generated by floor(d/(n-1))+2 elements, and there exist degree-n Azumaya algebras over d-dimensional rings which cannot be generated by fewer than floor(d/(2n-2))+2 elements. The case d=0 recovers the folklore fact that every central simple algebra is generated by 2 elements over its center. The proof reinterprets the problem as a question on "how much versal" are certain algebraic spaces approximating the classifying stack of the automorphism scheme of the algebra in question.

algebraic geometrygroup theorynumber theoryrepresentation theory

Audience: researchers in the topic

Algebraic groups and algebraic geometry: in honor of Zinovy Reichstein's 60th birthday