Finiteness of associated primes for local cohomology modules of excellent locally unramified regular rings of finite Krull dimension

TL;DR

This paper proves that local cohomology modules over certain excellent regular rings have finitely many associated primes, advancing the longstanding conjecture in algebraic geometry and commutative algebra.

Contribution

It establishes finiteness of associated primes for local cohomology modules over a broad class of excellent regular rings, including those with regular and F-finite reductions modulo primes.

Findings

Finiteness of associated primes proven for rings with specified regularity and F-finiteness conditions.

Results apply to excellent regular Q-algebras of finite Krull dimension.

Uses advanced techniques like perverse sheaves, D-modules, and the Riemann-Hilbert correspondence.

Abstract

Thirty years ago, Huneke (for local rings) and Lyubeznik (in general) conjectured that for all regular rings $R$, the local cohomology modules $H^i_I(R)$ have finitely many associated prime ideals. We prove substantial new cases of their conjecture by proving that the local cohomology modules $H^i_I(R)$ have finitely many associated prime ideals whenever $R$ is an excellent regular ring of finite Krull dimension such that $R/pR$ is regular and $F$-finite for every prime number $p$. Our result is new even for excellent regular $\mathbf{Q}$-algebras of finite Krull dimension, for example for finitely generated rings over formal power series rings over fields of characteristic zero. Our proof uses perverse sheaves, $\mathscr{D}$-modules, the Riemann-Hilbert correspondence for smooth complex varieties, N\'eron-Popescu desingularization, and a delicate Noetherian approximation argument.