Koszul homology of $F$-finite module and applications

TL;DR

This paper investigates the properties of Koszul homology modules of F-finite modules over rings of positive characteristic, and applies these results to study the graded components of local cohomology modules, revealing new vanishing and invariance properties.

Contribution

It establishes that Koszul homology modules of F-finite modules are F-finite, and extends results on local cohomology components in positive characteristic, including invariance and vanishing theorems.

Findings

Koszul homology modules are F-finite for F-finite modules.

Vanishing of local cohomology components when certain geometric conditions hold.

Invariance of Koszul cohomology modules under specific ring maps.

Abstract

Let $k$ be an infinite field of characteristic $p > 0$ and let $R = k[Y_1,\ldots, Y_d]$ (or $R = k[[Y_1,\ldots, Y_d]]$). Let $F \colon \text{Mod}(R) \rightarrow \text{Mod}(R)$ be the Frobenius functor and let $\mathcal{M}$ be a $F_R$-finite module (in the sense of Lyubeznik \cite{Lyu-2}). We show that if $r \geq 1$ then the Koszul homology modules $H_i(Y_1,\ldots, Y_r; \mathcal{M})$ are $F_{\overline{R}}$-finite modules where $\overline{R} = R/(Y_1,\ldots, Y_r)$ for $i = 0, \ldots, r$. As an application if $A$ is a regular ring containing a field of characteristic $p > 0$ and $S = A[X_1,\ldots, X_m]$ is standard graded and $I$ is an arbitrary graded ideal in $S$ then we give a comprehensive study of graded components of local cohomology modules $H^i_I(S)$. This extends in positive characteristic results we proved in \cite{P}. We study $H^i_I(S) $ when $A$ is local and prove that if $S/I$ is equidimensional and $Proj(S/I)$ is Cohen-Macaulay then $H^i_I(S)_n = 0$ for all $n \geq 0$ and for all $i > \ height \ I$. If $B$ is a equicharacteristic local Noetherian ring with infinite residue field and with a surjective map $\pi \colon T \rightarrow B$ where $(T,\mathfrak{n})$ is regular local then we show that the Koszul cohomology modules $H^j(\mathfrak{n}, H^{\dim T - i}_{\ker \pi }(T))$ depend only on $A, i, j$ and not on $T$ and $\pi$.