Test elements, excellent rings, and content functions

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.