A note on injectivity of Frobenius on local cohomology of global complete intersections

TL;DR

This paper investigates the injectivity of Frobenius action on top local cohomology of graded complete intersections over fields of positive characteristic, providing conditions and bounds related to isolated singularities and non-F-pure points.

Contribution

It extends known results by establishing Frobenius injectivity in negative degrees for complete intersections with isolated singularities, including explicit bounds on the characteristic p.

Findings

Frobenius acts injectively on top local cohomology in certain degrees

Computed the minimal degree of kernel elements under Frobenius

Provided bounds on p for injectivity in all negative degrees

Abstract

Given a graded complete intersection ideal $J = (f_1, \dots, f_c) \subseteq k[x_0, \dots, x_n] = S$, where $k$ is a field of characteristic $p > 0$ such that $[k:k^p] < \infty$, we show that if $S/J$ has an isolated non-F-pure point then the Frobenius action on top local cohomology $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in sufficiently negative degrees, and we compute the least degree of any kernel element. If $S/J$ has an isolated singularity, we are also able to give an effective bound on $p$ ensuring the Frobenius action on $H^{n+1-c}_\mathfrak{m}(S/J)$ is injective in all negative degrees, extending a result of Bhatt and Singh in the hypersurface case.