On Cohomology of Complexes Associated with a Generic Matrix

TL;DR

This paper studies a family of complexes related to a generic matrix, analyzing their cohomology through geometric and derived category methods, revealing connections to vector bundles and tilting theory.

Contribution

It refines the understanding of the cohomology of complexes including Eagon-Northcott and Buchsbaum-Rim, using geometric interpretations and tilting theory.

Findings

Cohomology computed via vector bundle methods

Non-exactness linked to failure of lifting exceptional sequences

Contrasts with known lifting results for differential forms

Abstract

In the appendix of the famous book "Commutative Algebra with a View Towards Algebraic Geometry" one can find an infinite family of complexes indexed by integers. This family includes Eagon-Northcott and Buschsbaum-Rim complexes. The objective of this paper is to study this family, and, in particular, refine the knowledge of its cohomology. First, we obtain these complexes from the derived images of twists of the Koszul complex on the projective space. This idea apparently goes back to Kempf [1970]. Taking this "geometric" point of view, we interpret the cohomology of these complexes as the cohomology of certain vector bundles on the projective space, and proceed with calculations. Finally, we put the above results in the realm of tilting theory: non-exactness of this family in certain regions can be seen as a failure of the exceptional sequence of line bundles on the projective space to lift to an exceptional sequence on a certain vector bundle. This observation creates a curious contrast with the results of Buchweitz-Leushke-Van den Bergh, stating that the exceptional sequence of twisted differential forms does lift to an exceptional sequence on the same vector bundle.