Local Cohomology of Certain Determinantal Thickenings

TL;DR

This paper investigates the local cohomology modules of rings defined by powers of maximal minors, providing explicit descriptions of their structure, annihilators, and connections to differential operators, especially for matrices of size n by n-1.

Contribution

It offers a complete characterization of the local cohomology modules of determinantal thickenings, including their cyclicity, annihilators, and explicit module structures, extending understanding of these algebraic objects.

Findings

Local cohomology modules are cyclic for large t.

Explicit computation of annihilators of local cohomology modules.

Description of Ext modules as submodules of top local cohomology.

Abstract

Let $R=\mathbb{C}[\{x_{ij}\}]$ be the ring of polynomial functions in $mn$ variables where $m> n$. Set $X$ to be the $m\times n$ matrix in these variables and $I:=I_n(X)$ the ideal of maximal minors of $X$. We consider the rings $R/I^t$; for $t\gg 0$ the depth of $R/I^t$ is equal to $n^2-1$, and we show that each local cohomology module $H^{n^2-1}_{\frak{m}}(R/I^t)$ is a cyclic $R$-module. We also compute the annihilator of $H^{n^2-1}_{\frak{m}}(R/I^t)$ thereby completely determining its $R$-module structure. In the case that $X$ is a $n\times (n-1)$ matrix we describe a map between the Koszul complex of the $t$-powers of the maximal minors and a free resolution of $R/I^t$. We use this map to explicitly describe the modules $\operatorname{Ext}_R ^n(R/I^t,R)$ as submodules of the top local cohomology module $H_I^n(R)$. Moreover, we can realize the filtration $\bigcup_i\operatorname{Ext}_R ^n(R/I^t,R)= H_I^n(R)$ in terms of differential operators. Utilizing this description, along with an explicit isomorphism $H_I^n(R) \cong H_{\frak{m}}^{n(n-1)}(R)$, we determine the annihilator of $\operatorname{Ext}_R ^n(R/I^t,R)$ and hence by graded local duality give another computation of the annihilator of $H^{(n-1)^2-1}_{\frak{m}}(R/I^t)$.