The Fedder action and a simplicial complex of local cohomologies

TL;DR

This paper constructs a complex of modules with Frobenius actions in prime characteristic rings, revealing new properties of local cohomology modules and their supports, especially in Cohen-Macaulay cases.

Contribution

It introduces a simplicial complex of local cohomologies with Frobenius actions, including the Fedder action, and demonstrates support properties of certain local cohomology modules.

Findings

The complex's cohomology vanishes below degree c.

The top cohomology is isomorphic to local cohomology with Frobenius action.

Supports of specific local cohomology modules are Zariski closed.

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.