The Second Vanishing Theorem for Local Cohomology Modules

TL;DR

This work establishes the Second Vanishing Theorem for local cohomology in unramified regular local rings of mixed characteristic, and provides a new Frobenius-based proof in prime characteristic $p$. It reduces the problem to dimension two and extends the Lyubeznik-number framework to mixed characteristic by showing the highest Lyubeznik number $ u_{d,d}(A)$ matches the number of connected components of a Hochster–Huneke graph associated with the completed strict henselization of the completion of $A$. A dimension-bound in mixed characteristic is obtained via $F$-depth, and the results yield a unified approach tying vanishing, Frobenius actions, and topological invariants across equal- and mixed-characteristic settings. The paper also outlines numerous open questions and potential extensions, including induction theorems in mixed characteristic and invariants under group actions.

Abstract

We prove the Second Vanishing Theorem for local cohomology modules of an unramified regular local ring in its full generality and provide a new proof of the Second Vanishing Theorem in prime characteristic $p$. As an application of our vanishing theorem for unramified regular local rings, we extend our topological characterization of the highest Lyubeznik number of an equi-characteristic local ring to the setting of mixed characteristic. An upper bound of local cohomological dimension in mixed characteristic is also obtained by partially extending Lyubeznik's vanishing theorem in prime characteristic $p$ to mixed characteristic.

Theorems & Definitions (41)

• Definition 1.2
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Théorème III.5.1 in PeskineSzpiroDimensionProjective
• Remark 2.2
• proof : Proof of Theorem \ref{['vanishing unramified']}
• Remark 2.3
• Proposition 3.1
• proof
• Theorem 3.2