#### Vol. 309, No. 2, 2020

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Ulrich elements in normal simplicial affine semigroups

### Jürgen Herzog, Raheleh Jafari and Dumitru I. Stamate

Vol. 309 (2020), No. 2, 353–380
##### Abstract

Let $H\subseteq {ℕ}^{d}$ 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