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Characterization and finite descent of local cohomological invariants

Bradley Dirks, Sebastian Olano, Debaditya Raychaudhury

Abstract

We provide simple ``left-inverse characterizations'' of the recently introduced singularity invariants $c(Z)$, $w(Z)$, and ${\rm HRH}(Z)$ of an equidimensional variety $Z$. Combining this with a trace morphism, we establish descent results of these invariants for finite surjective morphisms.

Characterization and finite descent of local cohomological invariants

Abstract

We provide simple ``left-inverse characterizations'' of the recently introduced singularity invariants , , and of an equidimensional variety . Combining this with a trace morphism, we establish descent results of these invariants for finite surjective morphisms.
Paper Structure (5 sections, 14 theorems, 78 equations)

This paper contains 5 sections, 14 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Injectivity Theorems and Left-Inverse Characterizations
  4. Trace Morphism for Mixed Sheaves
  5. Finite Descent of Singularity Invariants

Key Result

Theorem 1

Let $X$ be a variety of pure dimension $n$. In particular, if ${\rm HRH}(X)\geq k-1$, then the natural morphism is an isomorphism for $i < k-n$, injective for $i=k-n$ and both the domain and codomain vanish for $i > k-n$.

Theorems & Definitions (40)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Remark 1.1
  • Corollary 5
  • Corollary 6
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • ...and 30 more