Schubert complexes and degeneracy loci

TL;DR

This paper introduces Schubert complexes, a new tool for constructing Cohen-Macaulay modules with linear resolutions on degeneracy loci, extending Schubert functors and providing formulas in K-theory.

Contribution

The paper develops Schubert complexes that extend Schubert functors, offering a new method to resolve Cohen-Macaulay modules on degeneracy loci with linear resolutions.

Findings

Constructed maximal Cohen-Macaulay modules supported on degeneracy loci.

Provided a K-theoretic formula as a linear approximation of the structure sheaf.

Extended Schubert functors to new complexes with applications in algebraic geometry.

Abstract

Given a generic map between flagged vector bundles on a Cohen-Macaulay variety, we construct maximal Cohen-Macaulay modules with linear resolutions supported on the Schubert-type degeneracy loci. The linear resolution is provided by the Schubert complex, which is the main tool introduced and studied in this paper. These complexes extend the Schubert functors of Kra\'skiewicz and Pragacz, and were motivated by the fact that Schur complexes resolve maximal Cohen-Macaulay modules supported on determinantal varieties. The resulting formula in K-theory provides a "linear approximation" of the structure sheaf of the degeneracy locus, which can be used to recover a formula due to Fulton.