Regularity and cohomology of Pfaffian thickenings

TL;DR

This paper studies the algebraic and geometric properties of Pfaffian ideals in skew-symmetric matrix coordinate rings, providing explicit Ext module descriptions, regularity formulas, and insights into their resolutions and cohomological behavior.

Contribution

It offers explicit Ext module descriptions for invariant ideals, formulas for regularity and linear resolutions, and answers to longstanding questions about cohomology maps in Pfaffian thickenings.

Findings

Explicit Ext module descriptions for invariant ideals

Formulas for regularity and linear resolutions

Verification of a case of Kodaira vanishing

Abstract

Let $S$ be the coordinate ring of the space of $n\times n$ complex skew-symmetric matrices. This ring has an action of the group $\textrm{GL}_n(\mathbb{C})$ induced by the action on the space of matrices. For every invariant ideal $I\subseteq S$, we provide an explicit description of the modules $\textrm{Ext}^{\bullet}_S(S/I,S)$ in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusion of invariant ideals $I\supseteq J$, we compute the (co)kernel of the induced map $\textrm{Ext}^j_S(S/I,S)\to \textrm{Ext}^j_S(S/J,S)$ for all $j\geq 0$. As a consequence, we show that if an invariant ideal $I$ is unmixed, then the induced maps $\textrm{Ext}_S^j(S/I,S)\to H_I^j(S)$ are injective, answering a question of Eisenbud-Musta\c{t}\u{a}-Stillman in the case of Pfaffian thickenings. Finally, using our Ext computations and local duality, we verify an instance of Kodaira vanishing in the sense described in the recent work of Bhatt-Blickle-Lyubeznik-Singh-Zhang.