A Buchsbaum theory for Frobenius closure

TL;DR

This paper explores conditions under which the difference between Hilbert--Samuel multiplicity and Frobenius closure length remains constant across parameter ideals in certain local rings, linking ideal theory, derived categories, and Buchsbaum properties.

Contribution

It provides a partial characterization involving derived categories for when the multiplicity difference is independent of parameter ideal choice in Frobenius-closed rings.

Findings

Characterizes when the multiplicity difference is constant across parameter ideals.

Connects Frobenius closure properties with Buchsbaum conditions.

Uses derived category techniques to establish ideal-theoretic equivalences.

Abstract

We give a partial characterization for when the difference $e(\mathfrak{q})-\ell_R(R/\mathfrak{q}^F)$ is independent of the choice of parameter ideal $\mathfrak{q}\subseteq R$ in an excellent equidimensional local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$. Here, $\mathfrak{q}^F$ is the Frobenius closure of $\mathfrak{q}$ and $e(\mathfrak{q})$ denotes the Hilbert--Samuel multiplicity of $\mathfrak{q}$. In addition to ideal-theoretic equivalences, our characterization involves the derived category and is motivated by Schenzel's criterion of the Buchsbaum property as well as similar results of Ma-Quy in the setting of tight closure.