“Infinite” Properties of Certain Local Cohomology Modules of Determinantal Rings

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}.