A Buchsbaum theory for tight closure

Abstract:

A Noetherian local ring $(R,\frak {m})$ is called Buchsbaum if the difference $\ell (R/\mathfrak {q})-e(\mathfrak {q}, R)$, 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 $(R,\frak {m})$ 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 $\tau _{\mathrm {par}}(R)$ contains $\frak {m}$. We also provide a characterization of this condition via derived category which is analogous to Schenzel’s criterion for Buchsbaum rings.