A Note on Injectivity of Frobenius on Local Cohomology of Hypersurfaces

TL;DR

This paper establishes a precise bound on the degrees where Frobenius acts injectively on local cohomology of hypersurfaces, linking F-purity properties to Frobenius injectivity in algebraic geometry.

Contribution

It provides a sharp degree bound for Frobenius injectivity on local cohomology of hypersurfaces with isolated non-F-pure points, clarifying the relationship between F-purity and Frobenius action.

Findings

Frobenius action is injective in degrees ≤ -n(d-1) if and only if the hypersurface has an isolated non-F-pure point.

A sharp bound on degrees for Frobenius injectivity is established.

Characterization of non-F-purity points via Frobenius action in local cohomology.

Abstract

Let $k$ be a field of characteristic $p > 0$ such that $[k:k^p] < \infty$ and let $f \in R = k[x_0, ..., x_n]$ be homogeneous of degree $d$. We obtain a sharp bound on the degrees in which the Frobenius action on $H^n_\mathfrak{m}(R/fR)$ can be injective when $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$. As a corollary, we show that if $(R/fR)_\mathfrak{m}$ is not F-pure then $R/fR$ has an isolated non-F-pure point at $\mathfrak{m}$ if and only if the Frobenius action is injective in degrees $\le -n(d-1)$.