Non-commutative desingularization of determinantal varieties, I

TL;DR

This paper constructs a non-commutative desingularization for determinantal varieties, providing a new approach to resolving singularities using Cohen-Macaulay modules and endomorphism rings, including a non-commutative crepant resolution for square matrices.

Contribution

It introduces a novel non-commutative desingularization method for determinantal varieties via Cohen-Macaulay modules with finite global dimension.

Findings

Existence of a non-commutative desingularization for maximal minors

Construction of a Cohen-Macaulay module with finite global dimension

Non-commutative crepant resolution for square matrices

Abstract

We show that determinantal varieties defined by maximal minors of a generic matrix have a non-commutative desingularization, in that we construct a maximal Cohen-Macaulay module over such a variety whose endomorphism ring is Cohen-Macaulay and has finite global dimension. In the case of the determinant of a square matrix, this gives a non-commutative crepant resolution.