Ulrich elements in normal simplicial affine semigroups

Jürgen Herzog, Raheleh Jafari and Dumitru I. Stamate

Vol. 309 (2020), No. 2, 353–380

DOI: 10.2140/pjm.2020.309.353

Abstract

Let $H\subseteq {\mathbb{N}}^d$ be a normal affine semigroup, $R=K\left[H\right]$ 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 $x^h$ 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 property. 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 $\left(1,1\right)$ an Ulrich element in $H$.

Keywords

almost Gorenstein ring, Ulrich element, affine semigroup ring, lattice points

Mathematical Subject Classification

Primary: 05E40, 13H10

Secondary: 13H15, 20M25

Milestones

Received: 13 March 2020

Accepted: 23 October 2020

Published: 14 January 2021