Non-normal affine monoids

TL;DR

This paper provides a geometric framework for understanding the holes in non-normal affine monoids and links these to algebraic properties of their monoid algebras, offering bounds on depth and applications to seminormal cases.

Contribution

It introduces a geometric description of holes in non-normal affine monoids and connects these to local cohomology, revealing how algebraic properties are encoded in geometry.

Findings

Holes are related to graded components of local cohomology.

A combinatorial upper bound for the depth of the monoid algebra is established.

Conditions for equality in the depth bound are identified.

Abstract

We give a geometric description of the set of holes in a non-normal affine monoid $Q$. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of $k[Q]$. From this, we see how various properties of $k[Q]$ like local normality and Serre's conditions $(R_1)$ and $(S_2)$ are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra $k[Q]$ is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.