Lyubeznik numbers and injective dimension in mixed characteristic

TL;DR

This paper explores Lyubeznik numbers and injective dimensions of local cohomology modules over $Z$-algebras, showing their agreement in most cases and providing a counterexample to a longstanding conjecture.

Contribution

It proves the local agreement of mixed characteristic and standard Lyubeznik numbers for almost all reductions and provides a counterexample to Lyubeznik's bound conjecture on injective dimension.

Findings

Lyubeznik numbers agree locally in mixed and positive characteristic cases

Counterexample shows the injective dimension bound does not always hold

Illustrates differences between regular rings in equal and mixed characteristic

Abstract

We investigate the Lyubeznik numbers, and the injective dimension of local cohomology modules, of finitely generated $\mathbb{Z}$-algebras. We prove that the mixed characteristic Lyubeznik numbers and the standard ones agree locally for almost all reductions to positive characteristic. Additionally, we address an open question of Lyubeznik that asks whether the injective dimension of a local cohomology module over a regular ring is bounded above by the dimension of its support. Although we show that the answer is affirmative for several families of $\mathbb{Z}$-algebras, we also exhibit an example where this bound fails to hold. This example settles Lyubeznik's question, and illustrates one way that the behavior of local cohomology modules of regular rings of equal characteristic and of mixed characteristic can differ.