Abstract

A classical question in commutative algebra is the following: given a finitely generated projective module M over a ring R, what is the minimal number of generators of M as an R-module? A classical theorem of Forster and Swan implies that, if R is of dimension d over a field k and M is of rank r, then M can always be generated by r+d elements. Work of Murthy shows that, if k is algebraically closed, the only obstruction to r+d-1 generation of M is vanishing of the top Segre class of M. I will report on an approach to this problem using motivic obstruction theory. This approach recovers and improves these classical bounds: we prove results depending only on the homotopy dimension of R over k, we remove hypotheses on the base field, and we study r+d-2 generation in certain cases. We also prove a symplectic Forster–Swan theorem. This is joint work with Aravind Asok, Brian Shin, and Tariq Syed.