A note on the growth of regularity with respect to Frobenius

TL;DR

This paper establishes a uniform linear bound on the growth of local cohomology regularity of Frobenius powers of ideals in standard graded algebras over fields of prime characteristic, independent of the power.

Contribution

It proves a new uniform linear bound on the regularity of local cohomology modules of Frobenius powers, extending understanding of Frobenius actions in algebraic geometry.

Findings

Regularity of local cohomology modules grows at most linearly with Frobenius powers.

The bound is independent of the Frobenius exponent $e$.

Applicable to ideals where the non-finite projective dimension locus has controlled dimension.

Abstract

Let $R=k[x_1,\dots,x_n]/I$ be a standard graded $k$-algebra where $k$ is a field of prime characteristic and let $J$ be a homogeneous ideal in $R$. Denote $(x_1,\dots,x_n)$ by $\mathfrak{m}$. We prove that there is a constant $C$ (independent of $e$) such that the regularity of $H^s_{\mathfrak{m}}(R/J^{[p^e]})$ is bounded above by $Cp^e$ for all $e\geq 1$ and all integers $s$ such that $s+1$ is at least the dimension of the locus where $R/J$ doesn't have finite projective dimension.