Genus of numerical semigroups generated by three elements

Nari, Hirokatsu; Numata, Takahiro; Watanabe, Kei-ichi

doi:10.1016/j.jalgebra.2012.02.010

Cited by 16 publications

(9 citation statements)

References 7 publications

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“…In Section 4 we will explore 3-generated numerical semigroup rings over a field and their almost Gorenstein property. Corollary 4.2 has been reported by H. Nari [14] (see [15] also) at the 32-nd Symposium on Commutative Algebra in Japan (Hayama, 2010). Our research is independent of [14,15].…”

Section: Introductionsupporting

confidence: 53%

Almost Gorenstein rings

2013

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The notion of almost Gorenstein ring given by Barucci and Fröberg [2] in the case where the local rings are analytically unramified is generalized, so that it works well also in the case where the rings are analytically ramified. As a sequel, the problem of when the endomorphism algebra m : m of m is a Gorenstein ring is solved in full generality, where m denotes the maximal ideal in a given Cohen-Macaulay local ring of dimension one. Characterizations of almost Gorenstein rings are given in connection with the principle of idealization. Examples are explored.

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Section: Introductionsupporting

confidence: 53%

Almost Gorenstein rings

2013

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“…This can also be seen as follows. According to Nari et al [29], the structure matrix A j for an almost…”

Section: This Implies Thatmentioning

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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“…, (n − 1)α} ⊆ PF(H). If n = 3. then {α, 2α} = PF(H), so that by [4,13] the implication (3) ⇒ (2) follows (see Proposition 2.3 also). Therefore, in order to show the implication, we may assume that n ≥ 4 and that Theorem 1.2 holds true for n − 1.…”

Section: Proof Of Theorem 12mentioning

confidence: 91%