Local cohomology modules of a smooth Z-algebra have finitely many associated primes

TL;DR

This paper proves that local cohomology modules of smooth Z-algebras have finitely many associated primes, confirming a key case of Lyubeznik's conjecture for regular rings.

Contribution

It establishes the finiteness of associated primes for local cohomology modules of smooth Z-algebras, advancing the understanding of Lyubeznik's conjecture.

Findings

Finiteness of associated primes for local cohomology modules of smooth Z-algebras.

Resolution of a crucial case of Lyubeznik's conjecture.

Supports the broader conjecture for regular rings.

Abstract

Let $R$ be a commutative Noetherian ring that is a smooth $\mathbb Z$-algebra. For each ideal $I$ of $R$ and integer $k$, we prove that the local cohomology module $H^k_I(R)$ has finitely many associated prime ideals. This settles a crucial outstanding case of a conjecture of Lyubeznik asserting this finiteness for local cohomology modules of all regular rings.