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Vanishing of local cohomology in unramified mixed characteristic

Manav Batavia

Abstract

Given an ideal $I$ in a regular local ring $A$, the cohomological dimension of $I$ in $A$ is the index of the highest non-vanishing local cohomology of $A$ supported at $I$. Determining effective upper bounds on the cohomological dimension in terms of topological invariants of $\text{Spec}(A/I)$ is a central problem in commutative algebra. In equal characteristic, Faltings proved in 1980 a general bound on the cohomological dimension of an ideal in terms of its big height. In this article, we extend Faltings' result to the unramified mixed characteristic setting and show that the resulting bound is sharp.

Vanishing of local cohomology in unramified mixed characteristic

Abstract

Given an ideal in a regular local ring , the cohomological dimension of in is the index of the highest non-vanishing local cohomology of supported at . Determining effective upper bounds on the cohomological dimension in terms of topological invariants of is a central problem in commutative algebra. In equal characteristic, Faltings proved in 1980 a general bound on the cohomological dimension of an ideal in terms of its big height. In this article, we extend Faltings' result to the unramified mixed characteristic setting and show that the resulting bound is sharp.
Paper Structure (4 sections, 18 theorems, 75 equations)

This paper contains 4 sections, 18 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. The Induction Theorem
  3. Bounds on cohomological dimension
  4. Questions and Further Directions

Key Result

Theorem A

(cf. Theorem thm:cdbound) Let $A$ be a $d$-dimensional unramified regular local ring of mixed characteristic $(0,p)$. Let $I$ be a nonzero proper ideal of $A$ with big height $c$. Then

Theorems & Definitions (39)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 29 more