×
Ojeda, I.; Vigneron-Tenorio, A.

The short resolution of a semigroup algebra. (English) Zbl 1373.13015

Bull. Aust. Math. Soc. 96, No. 3, 400-411 (2017).
Summary: This work generalises the short resolution given by P. Pisón Casares [Proc. Am. Math. Soc. 131, No. 4, 1081–1091 (2003; Zbl 1038.13009)] to any affine semigroup. We give a characterisation of Apéry sets which provides a simple way to compute Apéry sets of affine semigroups and Frobenius numbers of numerical semigroups. We also exhibit a new characterisation of the Cohen-Macaulay property for simplicial affine semigroups.

Cited in 9 Documents

MSC:

13D02 Syzygies, resolutions, complexes and commutative rings
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
68W30 Symbolic computation and algebraic computation

Keywords:

free resolution; Betti number; semigroup algebra; affine semigroup; Apéry set; Frobenius problem

Citations:

Zbl 1038.13009

Software:

Frobby
Cite Review PDF
Full Text: DOI arXiv

References:

[1] E.Briales, A.Campillo, C.Marijuán and P.Pisón, ‘Minimal systems of generators for ideals of semigroups’, J. Pure Appl. Algebra124 (1998), 7-30.10.1016/S0022-4049(96)00106-5 · Zbl 0913.20036 · doi:10.1016/S0022-4049(96)00106-5
[2] E.Briales, A.Campillo, P.Pisón and A.Vigneron, ‘Simplicial complexes and syzygies of lattice ideals’, in: Symbolic Computation: Solving Equations in Algebra, Geometry, and Engineering (South Hadley, MA, 2000), Contemporary Mathematics, 286 (American Mathematical Society, Providence, RI, 2001), 169-183.10.1090/conm/286/04762 · Zbl 1093.13507 · doi:10.1090/conm/286/04762
[3] W.Bruns and J.Herzog, Cohen-Macaulay Rings, Cambridge Studies in Advanced Mathematics, 39 (Cambridge University Press, New York, 1993). · Zbl 0788.13005
[4] A.Campillo and P.Gimenez, ‘Syzygies of affine toric varieties’, J. Algebra225 (2000), 142-161.10.1006/jabr.1999.8102 · Zbl 0973.14027 · doi:10.1006/jabr.1999.8102
[5] T.Cortadellas Benítez, R.Jafari and S.Zarzuela Armengou, ‘On the Apéry sets of monomial curves’, Semigroup Forum86(2) (2013), 289-320.10.1007/s00233-012-9445-8 · Zbl 1282.13043 · doi:10.1007/s00233-012-9445-8
[6] W.Decker, G.-M.Greuel, G.Pfister and H.Schönemann, ‘Singular 4-0-2—A computer algebra system for polynomial computations’, 2015, http://www.singular.uni-kl.de.
[7] D.Einstein, D.Lichtblau, A.Strzebonski and S.Wagon, ‘Frobenius numbers by lattice point enumeration’, Integers7 (2007), A15, 63 pages. · Zbl 1229.11049
[8] P. A.García-Sánchez and J. C.Rosales, Finitely Generated Commutative Monoids (Nova Science, Commack, NY, 1999). · Zbl 0966.20028
[9] G.Marquez-Campos, I.Ojeda and J. M.Tornero, ‘On the computation of the Apéry set of numerical monoids and affine semigroups’, Semigroup Forum91(1) (2015), 139-158.10.1007/s00233-014-9631-y · Zbl 1347.13010 · doi:10.1007/s00233-014-9631-y
[10] E.Miller and B.Sturmfels, Combinatorial Commutative Algebra, Graduate Texts in Mathematics, 227 (Springer, New York, 2005). · Zbl 1090.13001
[11] M.Morales and N. T.Dung, ‘Gröbner basis, a “pseudo-polynomial” algorithm for computing the Frobenius number’, Preprint, 2015, arXiv:1510.01973 [math.AC].
[12] I.Ojeda and A.Vigneron-Tenorio, ‘Simplicial complexes and minimal free resolution of monomial algebras’, J. Pure Appl. Algebra214 (2010), 850-861.10.1016/j.jpaa.2009.08.009 · Zbl 1195.13015 · doi:10.1016/j.jpaa.2009.08.009
[13] P.Pisón Casares, ‘The short resolution of a lattice ideal’, Proc. Amer. Math. Soc.131(4) (2003), 1081-1091.10.1090/S0002-9939-02-06767-9 · Zbl 1038.13009 · doi:10.1090/S0002-9939-02-06767-9
[14] B. H.Roune, ‘Solving thousand-digit Frobenius problems using Gröbner bases’, J. Symbolic Comput.43(1) (2008), 1-7.10.1016/j.jsc.2007.06.002 · Zbl 1132.13311 · doi:10.1016/j.jsc.2007.06.002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.