A Buchsbaum theory for tight closure

TL;DR

This paper extends the concept of Buchsbaum rings to tight closure theory in prime characteristic, characterizing when a certain invariant remains constant and linking it to the parameter test ideal.

Contribution

It introduces a tight closure analog of Buchsbaum rings, providing a characterization via derived categories and relating it to the parameter test ideal in excellent local rings.

Findings

The difference $e(rak{q}, R) - ext{length}(R/rak{q}^*)$ is constant iff the parameter test ideal contains $rak{m}$.

Provides a derived category characterization similar to Schenzel's criterion.

Establishes conditions in unmixed excellent local rings of prime characteristic.

Abstract

A Noetherian local ring $(R,\mathfrak{m})$ is called Buchsbaum if the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q})$, where $\mathfrak{q}$ is an ideal generated by a system of parameters, is a constant independent of $\mathfrak{q}$. In this article, we study the tight closure analog of this condition. We prove that in an unmixed excellent local ring $(R,\mathfrak{m})$ of prime characteristic $p>0$ and dimension at least one, the difference $e(\mathfrak{q}, R)-\ell(R/\mathfrak{q}^*)$ is independent of $\mathfrak{q}$ if and only if the parameter test ideal $\tau_{\text{par}}(R)$ contains $\mathfrak{m}$. We also provide a characterization of this condition via derived category which is analogous to Schenzel's criterion for Buchsbaum rings.