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

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, the Riemann-Hilbert correspondence for smooth complex varieties, N\'eron-Popescu desingularization, and a delicate Noetherian approximation argument.