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The arithmetic rank of determinantal nullcones

Jack Jeffries, Vaibhav Pandey, Anurag K. Singh, Uli Walther

Abstract

We compute the arithmetic rank as well as the local/étale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations to establish striking vanishing results for local cohomology modules supported at these nullcone ideals; this is achieved via a careful analysis of the critical local cohomology modules. The vanishing theorems that we prove are sharp in various respects.

The arithmetic rank of determinantal nullcones

Abstract

We compute the arithmetic rank as well as the local/étale cohomological dimension of nullcone ideals arising from the classical actions of the symplectic group, the general linear group, and the orthogonal group. We use these calculations to establish striking vanishing results for local cohomology modules supported at these nullcone ideals; this is achieved via a careful analysis of the critical local cohomology modules. The vanishing theorems that we prove are sharp in various respects.

Paper Structure

This paper contains 17 sections, 55 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Integer torsion in local cohomology
  3. Local cohomology of Pfaffian nullcones
  4. Local cohomology of generic determinantal nullcones
  5. Local cohomology of symmetric determinantal nullcones
  6. Preliminaries on cohomology
  7. Notation and terminology
  8. Singular cohomology
  9. Étale cohomology
  10. Additional lemmas
  11. Topology of Pfaffian nullcones
  12. Topology of generic determinantal nullcones
  13. Topology of symmetric determinantal nullcones
  14. Some locally trivial fiber bundles
  15. The Pfaffian case
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $S$ be a polynomial ring over a field $\mathbb{K}$, and let $G$ be a linearly reductive group acting on $S$ by degree-preserving $\mathbb{K}$-algebra automorphisms. Let $S^G$ denote the ring of invariants, and $\mathfrak{m}_{S^G}$ the homogeneous maximal ideal of $S^G$. Then the nullcone ideal $

Theorems & Definitions (110)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Theorem 2.1
  • proof
  • proof : Proof of Theorem \ref{['theorem:vector:space']}
  • Theorem 2.2
  • ...and 100 more