Big tight closure test elements for some non-reduced excellent rings

TL;DR

This paper establishes the existence of big tight closure test elements in certain non-reduced excellent rings of prime characteristic, under specific conditions related to Gorenstein and $F$-regular properties.

Contribution

It proves the existence of big test elements in non-reduced excellent rings without using the Gamma construction, under various conditions including $F$-purity and localness.

Findings

Big test elements exist in certain excellent rings under specified conditions.

The results apply to rings that are homomorphic images of regular rings with intersection-flat Frobenius.

The paper does not use the Gamma construction in its proofs.

Abstract

This paper is concerned with existence of big tight closure test elements for a commutative Noetherian ring $R$ of prime characteristic $p$. Let $R^{\circ}$ denote the complement in $R$ of the union of the minimal prime ideals of $R$. A big test element for $R$ is an element of $R^{\circ}$ which can be used in every tight closure membership test for every $R$-module, and not just the finitely generated ones. The main results of the paper are that, if $R$ is excellent and satisfies condition $(R_0)$, and $c \in R^{\circ}$ is such that $R_c$ is Gorenstein and weakly $F$-regular, then some power of $c$ is a big test element for $R$ if (i) $R$ is a homomorphic image of an excellent regular ring of characteristic $p$ for which the Frobenius homomorphism is intersection-flat, or (ii) $R$ is $F$-pure, or (iii) $R$ is local. The Gamma construction is not used.