Local Cohomology of Certain Determinantal Thickenings

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 $H_{\mathfrak{m}}^{n^2-1}(R/I^t)$ is a cyclic $R$-module. We also compute the annihilator of $H_{\mathfrak{m}}^{n^2-1}(R/I^t)$ 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 ${\mathrm{Ext}}_R^n(R/I^t,R)$ as submodules of the top local cohomology module $H_I^n(R)$. Moreover, we can realize the filtration ${\bigcup }_i{\mathrm{Ext}}_R^n(R/I^t,R)=H_I^n(R)$ in terms of differential operators. Utilizing this description, along with an explicit isomorphism $H_I^n(R)\cong H_{\mathfrak{m}}^{n(n-1)}(R)$, we determine the annihilator of ${\mathrm{Ext}}_R^n(R/I^t,R)$ and hence by graded local duality give another computation of the annihilator of $H_{\mathfrak{m}}^{(n-1{)}^2-1}(R/I^t)$.

23 pages