Regularity and cohomology of determinantal thickenings

TL;DR

This paper studies the algebraic and geometric properties of determinantal ideals, providing formulas for regularity, resolutions, and cohomology, and characterizing invariant ideals with specific Ext and local cohomology behaviors.

Contribution

It offers explicit descriptions of Ext modules, formulas for regularity of powers, and characterizations of invariant ideals, advancing understanding of determinantal thickenings and their cohomological properties.

Findings

Derived formulas for regularity of powers and symbolic powers.

Characterized when Ext maps are injective for invariant ideals.

Verified Kodaira vanishing for determinantal thickenings.

Abstract

We consider the ring S=C[x_ij] of polynomial functions on the vector space C^(m x n) of complex m x n matrices. We let GL= GL_m x GL_n and consider its action via row and column operations on C^(m x n) (and the induced action on S). For every GL-invariant ideal I in S and every j>=0, we describe the decomposition of the modules Ext^j_S(S/I,S) into irreducible GL-representations. For any inclusion I into J of GL-invariant ideals we determine the kernels and cokernels of the induced maps Ext^j_S(S/I,S) -> Ext^j_S(S/J,S). As a consequence of our work, we give a formula for the regularity of the powers and symbolic powers of generic determinantal ideals, and in particular we determine which powers have a linear minimal free resolution. As another consequence, we characterize the GL-invariant ideals I in S for which the induced maps Ext^j_S(S/I,S) -> H_I^j(S) are injective. In a different direction we verify that Kodaira vanishing, as described in work of Bhatt-Blickle-Lyubeznik-Singh-Zhang, holds for determinantal thickenings.