A Buchsbaum theory for tight closure

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 $e(\mathfrak{q},R)-\ell (R/{\mathfrak{q}}^{\ast })$ 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.

arXiv comment20 pages, final version