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

Abstract

For given integers $m,n\ge 2$ there are examples of ideals $I$ of complete determinantal local rings $(R,\mathfrak{m}),\dim R=m+n-1,\mathrm{grade}I=n-1,$ with the canonical module ${\omega }_R$ and the property that the socle dimensions of $H_I^{m+n-2}({\omega }_R)$ and $H_{\mathfrak{m}}^m(H_I^{n-1}({\omega }_R))$ 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 $H_I^{n-1}({\omega }_R)$ 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}.

arXiv commentto appear in Acta Mathematica Vietnamica