Content of Local Cohomology, Parameter Ideals, and Robust Algebras

TL;DR

This paper explores the properties of local cohomology, introduces a new closure operation called robust closure, and characterizes tight closure in complete local domains of positive characteristic, providing new insights and counterexamples.

Contribution

It introduces robust forcing algebras and a new closure operation that coincides with tight closure in certain domains, answering longstanding questions.

Findings

Existence of nonzero local cohomology modules with content zero.

Robust closure coincides with tight closure in complete local domains of positive characteristic.

Counterexample to a question of Lyubeznik.

Abstract

This paper continues the investigation of quasilength, of content of local cohomology with respect to generators of the support ideal, and of robust algebras begun in joint work of Hochster and Huneke. We settle several questions raised by Hochster and Huneke. In particular, we give a family of examples of top local cohomology modules both in equal characteristic 0 and in positive prime characteristic that are nonzero but have content 0. We use the notion of a robust forcing algebra (the condition turns out to be strictly stronger than the notion of a solid forcing algebra in, for example, equal characteristic 0) to define a new closure operation on ideals. We prove that this new notion of closure coincides with tight closure for ideals in complete local domains of positive characteristic, which requires proving that forcing algebras for instances of tight closure are robust, and study several related problems. This gives, in effect, a new characterization of tight closure in complete local domains of positive characteristic. As a byproduct, we also answer a question of Lyubeznik in the negative.