Local Cohomology and Degree Complexes of Monomial Ideals

TL;DR

This paper studies the local cohomology of monomial ideals using degree complexes, providing explicit computations for powers and sums of ideals, and linking these to regularity measures.

Contribution

It introduces explicit formulas for degree complexes of powers and sums of monomial ideals, enhancing understanding of their local cohomology and regularity.

Findings

Explicit computation of degree complexes for powers and sums of ideals

Connection between local cohomology and reduced homology of degree complexes

Formulas for regularity of symbolic powers of combined ideals

Abstract

This paper examines the dimension of the graded local cohomology $H_\mathfrak{m}^p(S/K^s)_\gamma$ and $H_\mathfrak{m}^p(S/K^{(s)})$ for a monomial ideal $K$. This information is encoded in the reduced homology of a simplicial complex called the degree complex. We explicitly compute the degree complexes of ordinary and symbolic powers of sums and fiber products of ideals, as well as the degree complex of the mixed product, in terms of the degree complexes of their components. We then use homological techniques to discuss the cohomology of their quotient rings. In particular, this technique allows for the explicit computation of $\text{reg} ((I + J + \mathfrak{m}\mathfrak{n})^{(s)})$ in terms of the regularities of $I^{(i)}$ and $J^{(j)}$.