An excellent F-pure ring of prime characteristic has a big tight closure test element

TL;DR

This paper proves that every excellent F-pure ring of prime characteristic possesses a big tight closure test element, establishing a deep connection between F-purity and the existence of test elements in tight closure theory.

Contribution

It demonstrates that F-purity implies the existence of a big test element in excellent rings, extending previous results to non-local and non-excellent cases.

Findings

F-pure local rings have a tight closure test element.

Every excellent F-pure ring has a big test element.

The converse relation between F-purity and module structure is established.

Abstract

In two recent papers, the author has developed a theory of graded annihilators of left modules over the Frobenius skew polynomial ring over a commutative Noetherian ring $R$ of prime characteristic $p$, and has shown that this theory is relevant to the theory of test elements in tight closure theory. One result of that work was that, if $R$ is local and the $R$-module structure on the injective envelope $E$ of the simple $R$-module can be extended to a structure as a torsion-free left module over the Frobenius skew polynomial ring, then $R$ is $F$-pure and has a tight closure test element. One of the central results of this paper is the converse, namely that, if $R$ is $F$-pure, then $E$ has a structure as a torsion-free left module over the Frobenius skew polynomial ring; a corollary is that every $F$-pure local ring of prime characteristic, even if it is not excellent, has a tight closure test element. These results are then used, along with embedding theorems for modules over the Frobenius skew polynomial ring, to show that every excellent (not necessarily local) $F$-pure ring of characteristic $p$ must have a so-called `big' test element.