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Ulrich elements in normal simplicial affine semigroups. (English) Zbl 1458.05268

Summary: 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 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 \(H\).

MSC:

05E40 Combinatorial aspects of commutative algebra
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13H15 Multiplicity theory and related topics
20M25 Semigroup rings, multiplicative semigroups of rings
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References:

[1] 10.1007/978-1-4612-3660-3_4 · doi:10.1007/978-1-4612-3660-3_4
[2] 10.1006/jabr.1996.6837 · Zbl 0874.13018 · doi:10.1006/jabr.1996.6837
[3] 10.1007/978-1-4939-2969-6 · Zbl 1339.52002 · doi:10.1007/978-1-4939-2969-6
[4] 10.1007/b105283 · Zbl 1168.13001 · doi:10.1007/b105283
[5] 10.1017/CBO9780511608681 · doi:10.1017/CBO9780511608681
[6] ; Danilov, Uspekhi Mat. Nauk, 33, 85 (1978)
[7] 10.1006/jabr.1997.6990 · Zbl 0884.13012 · doi:10.1006/jabr.1997.6990
[8] ; Dinu, Osaka J. Math., 57, 935 (2020) · Zbl 1451.13069
[9] 10.1515/9781400882526 · Zbl 0813.14039 · doi:10.1515/9781400882526
[10] 10.4099/math1924.2.1 · doi:10.4099/math1924.2.1
[11] 10.1016/j.jalgebra.2013.01.025 · Zbl 1279.13035 · doi:10.1016/j.jalgebra.2013.01.025
[12] 10.1016/j.jpaa.2014.09.022 · Zbl 1319.13017 · doi:10.1016/j.jpaa.2014.09.022
[13] 10.1007/s00233-019-10007-2 · Zbl 1467.20072 · doi:10.1007/s00233-019-10007-2
[14] 10.1007/s11856-019-1898-y · Zbl 1428.13037 · doi:10.1007/s11856-019-1898-y
[15] 10.2969/aspm/01110093 · doi:10.2969/aspm/01110093
[16] 10.1007/BF01204726 · Zbl 0758.52009 · doi:10.1007/BF01204726
[17] 10.2307/1970791 · Zbl 0233.14010 · doi:10.2307/1970791
[18] 10.1016/j.jalgebra.2017.09.033 · Zbl 1387.13052 · doi:10.1016/j.jalgebra.2017.09.033
[19] 10.1007/s00233-012-9397-z · Zbl 1267.20086 · doi:10.1007/s00233-012-9397-z
[20] 10.1016/0001-8708(78)90045-2 · Zbl 0384.13012 · doi:10.1016/0001-8708(78)90045-2
[21] 10.1080/00927872.2017.1339066 · Zbl 1400.13025 · doi:10.1080/00927872.2017.1339066
[22] 10.1007/BF01163869 · Zbl 0573.13013 · doi:10.1007/BF01163869
[23] 10.1007/978-1-4613-8431-1 · Zbl 0823.52002 · doi:10.1007/978-1-4613-8431-1
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