Non-commutative Desingularization of Determinantal Varieties, II: Arbitrary Minors

Abstract

In our paper "Non-commutative desingularization of determinantal varieties I", we constructed and studied non-commutative resolutions of determinantal varieties defined by maximal minors. At the end of the introduction, we asserted that the results could be generalized to determinantal varieties defined by non-maximal minors, at least in characteristic zero. In this paper, we prove theexistence of non-commutative resolutions in the general case in a manner which is still characteristic free, and carry out the explicit description by generators and relations in characteristic zero. As an application of our results, we prove that there is a fully faithful embedding between the bounded derived categories of the two canonical (commutative) resolutions of a determinantal variety, confirming a well-known conjecture of Bondal and Orlov in this special case.