The Fedder action and a simplicial complex of local cohomologies

Abstract

Let $R$ be a regular ring of prime characteristic $p>0$, and let $\underline{\mathbf{f}}=f_1,\dots ,f_c$ be a permutable regular sequence of codimension $c\ge 1$. We describe a complex of $R⟨F⟩$-modules, denoted $\Delta {\Delta }_{\underline{\mathbf{f}}}^{\bullet }(R)$, whose terms include $\Delta {\Delta }_{\underline{\mathbf{f}}}^0(R)=R/\underline{\mathbf{f}}$ equipped with its natural Frobenius action, and $\Delta {\Delta }_{\underline{\mathbf{f}}}^c(R)=H_{\underline{\mathbf{f}}}^c(R)$ equipped with a Frobenius action we refer to as the Fedder action. We show that $H^i(\Delta {\Delta }_{\underline{\mathbf{f}}}^{\bullet }(R))=0$ for all $i<c$, and that $H^c(\Delta {\Delta }_{\underline{\mathbf{f}}}^{\bullet }(R))$ is a copy of $H_{\underline{\mathbf{f}}}^c(R)$ equipped with the usual Frobenius action. Using the $\Delta {\Delta }_{\underline{\mathbf{f}}}^{\bullet }(R)$ complex, we show that if $I\supseteq \underline{\mathbf{f}}$ is an ideal such that $H_I^i(R)=0$ for $\text{ht}(I)<i<\text{ht}(I)+c$ (which is automatic if $R/I$ is Cohen-Macaulay), then the module $H_{I/\underline{\mathbf{f}}}^{\text{ht}(I/\underline{\mathbf{f}})+c}(R/\underline{\mathbf{f}})$ has Zariski closed support.

19 pages