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 d be a normal affine semigroup, R = K[H] its semigroup ring over the field K and ωR its canonical module. The Ulrich elements for H are those h in H such that for the multiplication map by xh from R into ωR, 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 (1,1) 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
Authors
Jürgen Herzog
Fakultät für Mathematik
Universität Duisburg-Essen
Essen
Germany
Raheleh Jafari
Mosaheb Institute of Mathematics
Kharazmi University
Tehran
Iran
School of Mathematics
Institute for Research in Fundamental Sciences (IPM)
Tehran
Iran
Dumitru I. Stamate
Faculty of Mathematics and Computer Science
University of Bucharest
Bucharest
Romania