Symmetries on almost symmetric numerical semigroups

Nari, Hirokatsu

doi:10.1007/s00233-012-9397-z

Cited by 55 publications

(51 citation statements)

References 9 publications

(2 reference statements)

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“…In other words, one has the following result: The type of a numerical semigroup C is defined in [11] as the number of elements in the set C ′ = {x ∈ Z/ x / ∈ C and x + s ∈ C , ∀s ∈ C \ {0}}. In [15,Prop. 6.6] it is shown that the type of a numerical semigroup obtained by gluing two semigroups is the product of the types of the two semigroups.…”

Section: Main Results and Consequencesmentioning

confidence: 99%

The structure of the minimal free resolution of semigroup rings obtained by gluing

Gimenez

Srinivasan

2019

Journal of Pure and Applied Algebra

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We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when C is a numerical semigroup obtained by gluing two numerical semigroups A and B . Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in N n .The first author was partially supported by Ministerio de Ciencia e Innovación (Spain), MTM2016-78881-P.

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Section: Main Results and Consequencesmentioning

confidence: 99%

The structure of the minimal free resolution of semigroup rings obtained by gluing

Gimenez

Srinivasan

2019

Journal of Pure and Applied Algebra

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“…It is known by Nari [27] that K [H] is almost Gorenstein, if and only if (10) f The following result shows that a numerical semigroup generated by an arithmetic sequence is nearly Gorenstein. We also characterize when such semigroups are almost symmetric, taking into account that the symmetric case was known from work of Gimenez, Sengupta and Srinivasan in [13].…”

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confidence: 99%

The trace of the canonical module

2019

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The trace of the canonical module (the canonical trace) determines the non-Gorenstein locus of a local Cohen-Macaulay ring. We call a local Cohen-Macaulay ring nearly Gorenstein, if its canonical trace contains the maximal ideal. Similar definitions can be made for positively graded Cohen-Macaulay K-algebras. We study the canonical trace for tensor products and Segre products of algebras, as well as of (squarefree) Veronese subalgebras. The results are used to classify the nearly Gorenstein Hibi rings. We also consider the canonical trace of onedimensional rings with a focus on numerical semigroup rings.By using a result of Goto and Watanabe [15] one deduces then from Theorem 4.6 that all Veronese subalgebras of a standard graded K-algebra R over an infinite field K are nearly Gorenstein, if R is Gorenstein.The situation for squarefree Veronese subalgebras is more complicated. Given integers n ≥ d > 0, the d-th squarefree Veronese subalgebra R n,d of the polynomial ring S in n variables over a field K is the K-algebra generated by the squarefree monomials in S of degree d. Based on a theorem of Bruns, Vasconcelos and Villarreal [5] we give in Theorem 4.12 an explicit description of the anti-canonical ideal of R n,d , and we use this result to show in Theorem 4.14 that the following conditions are equivalent: (i) R n,d is nearly Gorenstein, (ii) R n,d is Gorenstein, (iii) d = 1 or d = n − 1 or n = 2d.By another result of Goto and Watanabe [15], the canonical module for the Segre product T = R#S of positively graded Cohen-Macaulay K-algebras R and S of Krull dimension at least 2 is just the Segre product of the respective canonical modules, assuming that T is Cohen-Macaulay. We use this result in Theorem 4.15 to compute tr(ω T ) in the case that R and S are standard graded Gorenstein Kalgebras. It is shown in Theorem 4.15 that m |r−s| T ⊆ tr(ω T ), where m T is the graded maximal ideal of T and where r and s are the respective a-invariants of R and S. Equality holds, if T is a domain. If follows that under these conditions T is nearly Gorenstein if and only if |r − s| ≤ 1, see Corollary 4.16. The section ends with Proposition 4.18 in which the anti-canonical ideal of the Segre product is computed in the case that R and S are polynomial rings.These results on Segre products are used in Section 5 to give a complete classification of all nearly Gorenstein Hibi rings. Given a finite distributive lattice L and a field K, the Hibi ring of L defined over K is the toric ring R K [L] whose relations are the meet-join relations of L. By a fundamental theorem of Birkhoff, L is the ideal lattice J (P ) of its poset of join irreducible elements. It is shown in Theorem 5.4 3

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“…Unfortunately there exist almost symmetric numerical semigroups with odd type that cannot be constructed in this way. For example consider T = 9, 10, 14, 15 = {0, 9,10,14,15,18,19,20 in any case E is not contained in S and then E is not a proper ideal of S.…”

Section: One Half Of Almost Symmetric Numerical Semigroups With Odd Typementioning

confidence: 99%

One half of almost symmetric numerical semigroups

Strazzanti

2014

Semigroup Forum

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Let S,T be two numerical semigroups. We study when S is one half of T, with T almost symmetric. If we assume that the type of T, t(T), is odd, then for any S there exist infinitely many such T and we prove that 1≤t(T)≤2t(S)+1. On the other hand, if t(T) is even, there exists such T if and only if S is almost symmetric and different from N; in this case the type of S is the number of even pseudo-Frobenius numbers of T. Moreover, we construct these families of semigroups using the numerical duplication with respect to a relative ideal

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Symmetries on almost symmetric numerical semigroups

Cited by 55 publications

References 9 publications

The structure of the minimal free resolution of semigroup rings obtained by gluing

The structure of the minimal free resolution of semigroup rings obtained by gluing

The trace of the canonical module

One half of almost symmetric numerical semigroups

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