"Infinite" properties of certain local cohomology modules of determinantal rings

TL;DR

This paper investigates the infinite-dimensional properties of certain local cohomology modules in determinantal rings, revealing new structural insights and generalizations of previous examples, especially in Gorenstein cases.

Contribution

It introduces examples of ideals in determinantal rings with infinite socle dimensions of local cohomology modules and analyzes their endomorphism rings, extending prior work.

Findings

Socle dimensions of specific local cohomology modules are not finite.

Endomorphism rings are finitely generated as algebras but not as modules.

Results generalize known examples and provide new structural understanding.

Abstract

For given integers $m,n \geq 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}), \dim R = m+n-1, \operatorname{grade} I = n-1,$ with the canonical module $\omega_R$ and the property that the socle dimensions of $H^{m+n-2}_I(\omega_R)$ and $H^m_{\mathfrak{m}}(H^{n-1}_I(\omega_R))$ are not finite. In the case of $m = n$, i.e. a Gorenstein ring, the socle dimensions provide further information about the $\tau$-numbers as studied in \cite{MS}. Moreover, the endomorphism ring of $H^{n-1}_I(\omega_R)$ is studied and shown to be an $R$-algebra of finite type but not finitely generated as $R$-module generalizing an example of \cite{Sp6}.