The Fedder action and a simplicial complex of local cohomologies

Abstract

Let $R$ be a regular ring of prime characteristic $p > 0$, and let $\underline{\mathbf{f}}=f_1,\ldots,f_c$ be a permutable regular sequence of codimension $c\geq 1$. We describe a complex of $R\langle F \rangle$-modules, denoted $\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R)$, whose terms include $\Delta\hspace{-2.65mm}\Delta^0_{\underline{\mathbf{f}}}(R)=R/\underline{\mathbf{f}}$ equipped with its natural Frobenius action, and $\Delta\hspace{-2.65mm}\Delta^c_{\underline{\mathbf{f}}}(R)=H^c_{\underline{\mathbf{f}}}(R)$ equipped with a Frobenius action we refer to as the Fedder action. We show that $H^i(\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R))=0$ for all $i<c$, and that $H^c(\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R))$ is a copy of $H^c_{\underline{\mathbf{f}}}(R)$ equipped with the usual Frobenius action. Using the $\Delta\hspace{-2.65mm}\Delta^\bullet_{\underline{\mathbf{f}}}(R)$ 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 $H^{\text{ht}(I/\underline{\mathbf{f}})+c}_{I/\underline{\mathbf{f}}}(R/\underline{\mathbf{f}})$ has Zariski closed support.