Ulrich elements in normal simplicial affine semigroups

TL;DR

This paper investigates Ulrich elements in normal simplicial affine semigroups, providing algebraic and combinatorial criteria for their existence, especially in the case of two-dimensional semigroups, and explores conditions for the ring to be almost Gorenstein.

Contribution

It introduces the concept of slim semigroups and offers new algebraic and combinatorial criteria for identifying Ulrich elements in normal affine semigroups.

Findings

All normal affine semigroups in dimension two are slim.

Provided a simple arithmetic criterion for the Ulrich property of (1,1) in H.

Established an algebraic criterion for testing the Ulrich property in slim semigroups.

Abstract

Let $H\subseteq \mathbb{N}^d$ be a normal affine semigroup, $R=K[H]$ its semigroup ring over the field $K$ and $\omega_R$ its canonical module. The Ulrich elements for $H$ are those $h$ in $H$ such that for the multiplication map by $\mathbf{x}^h$ from $R$ into $\omega_R$, the cokernel is an Ulrich module. We say that the ring $R$ is almost Gorenstein if Ulrich elements exist in $H$. For the class of slim semigroups that we introduce, we provide an algebraic criterion for testing the Ulrich propery. When $d=2$, all normal affine semigroups are slim. Here we have a simpler combinatorial description of the Ulrich property. We improve this result for testing the elements in $H$ which are closest to zero. In particular, we give a simple arithmetic criterion for when is $(1,1)$ an Ulrich element in $H$.