Algebra Seminar - Zinovy Reichstein (Math)

Speaker: Zinovy Reichstein (UBC)

Title: On the minimal number of generators of an algebra over a commutative ring.

Abstract

Let R be a commutative ring of Krull dimension d. A 1964 theorem of Forster asserts that every projective R-module of rank n can be generated by d+n elements. Chase and Swan subsequently showed that this bound is sharp: there exist examples that cannot be generated by fewer than d+n elements. We view projective R-modules as R-forms of the non-unital R-algebra where the product of any two elements is 0. A few years ago Uriya First and I generalized Forster's theorem to forms of other algebras (not necessarily commutative, associative or unital). For example, every etale algebra over R can be generated by d + 1 elements, every Azumaya algebra can be generated by d + 2 elements, every octonion algebra by d + 3 elements. Abhishek Shukla and Ben Williams then showed that this generalized Forster bound is optimal for etale algebras. In this talk, based on joint work with First and Williams, I will address the following question: Is the Forster bound optimal for other types of algebras?