Test elements, excellent rings, and content functions

TL;DR

This paper extends the theory of test elements and F-jumping numbers to broader classes of rings, establishing new properties of excellent regular rings and developing parallel theories of algebraic flatness and content.

Contribution

It introduces the parallel theories of Ohm-Rush and intersection flat algebras, proving their local checkability and descent properties, and applies these to advance understanding of test elements and F-singularity invariants.

Findings

Reduced quotients of excellent regular rings of characteristic p admit big test elements.

The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is discrete in Q.

There is a uniform bound on Hartshorne-Speiser-Lyubeznik numbers for certain rings.

Abstract

Broadening existing results in the literature to much wider classes of rings, we prove among other things: 1. Reduced quotients of excellent regular rings of characteristic $p$ admit big test elements, 2. The set of F-jumping numbers of a principal ideal in a locally excellent regular ring is a discrete subset of $\mathbb Q$, and 3. If $R$ is a quotient of a locally excellent regular ring of prime characteristic, then there is a uniform upper bound on the Hartshorne-Speiser-Lyubeznik numbers of the injective hulls of the residue fields of $R$. To do so, we develop the parallel theories of Ohm-Rush and intersection flat algebras. We show that both properties can be checked locally in flat maps of Noetherian rings. We show that intersection-flatness admits a content theory parallel to that of Ohm-Rush content for Ohm-Rush algebras. We develop descent results for these properties. Using the descent result for intersection flatness, we obtain a local condition under which a faithfully flat map of Noetherian rings must be intersection-flat. The local condition for intersection-flatness allows us to conclude that finitely generated faithfully flat algebras over a Noetherian ring are intersection-flat. Combining the local condition for intersection flatness with results of Kunz and Radu yields the conclusion that the Frobenius endomorphism associated to a locally excellent regular ring of prime characteristic is intersection-flat, thus answering a question of Sharp. Applications of the latter result include the three enumerated results above. We also get applications to tight closure and parameter test ideals.