Length and multiplicity of the local cohomology with support in a hyperplane arrangement

TL;DR

This paper investigates the structure of the first local cohomology module supported on a hyperplane arrangement, revealing that its length and multiplicity as a D-module are equal and can be explicitly computed using combinatorial invariants.

Contribution

It provides an explicit formula for the length and multiplicity of the first local cohomology module in terms of the arrangement's Poincaré polynomial or Möbius function.

Findings

Length and multiplicity of H^1_{(f)}(R) coincide.

Explicit expressions in terms of combinatorial invariants.

Results apply to hyperplane arrangements over characteristic zero fields.

Abstract

Let $R$ be the polynomial ring in $n$ variables with coefficients in a field $K$ of characteristic zero. Let $D_n$ be the $n$-th Weyl algebra over $K$. Suppose that $f \in R$ defines a hyperplane arrangement in the affine space $K^n$. Then the length and the multiplicity of the 1st local cohomology group $H^1_{(f)}(R)$ as left $D_n$-module coincide and are explicitly expressed in terms of the Poincar\'e polynomial or the M\"obius function of the arrangement.