On syzygies for rings of invariants of abelian groups. (English) Zbl 1440.13059
Facchini, Alberto (ed.) et al., Advances in rings, modules and factorizations. Selected papers based on the presentations at the international conference on rings and factorizations, Graz, Austria, February 19–23, 2018. Cham: Springer. Springer Proc. Math. Stat. 321, 105-124 (2020).
Summary: It is well known that results on zero-sum sequences over a finitely generated abelian group can be translated to statements on generators of rings of invariants of the dual group. Here the direction of the transfer of information between zero-sum theory and invariant theory is reversed. First it is shown how a presentation by generators and relations of the ring of invariants of an abelian group acting linearly on a finite-dimensional vector space can be obtained from a presentation of the ring of invariants for the corresponding multiplicity free representation. This combined with a known degree bound for syzygies of rings of invariants yields bounds on the presentation of a block monoid associated to a finite sequence of elements in an abelian group. The results have an equivalent formulation in terms of binomial ideals, but here the language of monoid congruences and the notion of catenary degree is used.
For the entire collection see [Zbl 1445.13002].
For the entire collection see [Zbl 1445.13002].
MSC:
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13A50 | Actions of groups on commutative rings; invariant theory |
| 13F65 | Commutative rings defined by binomial ideals, toric rings, etc. |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
References:
| [1] | Blanco, V., García-Sánchez, P.A., Geroldinger, A.: Semigroup-theoretical characterizations of arithmetical invariants with applications to numerical monoids and Krull monoids. Illinois J. Math. 55, 1385-1414 (2011) · Zbl 1279.20072 · doi:10.1215/ijm/1373636689 |
| [2] | Bruns, W.: The quest for counterexamples in Toric geometry, Proc. CAAG 2010, Ramanujan Math. Soc. Lect. Notes Series No. 17, 1-17 (2013) · Zbl 1317.14112 |
| [3] | Bruns, W., Gubeladze, J.: Polytopes, Rings, and K-theory. Springer, New York (2009) · Zbl 1168.13001 |
| [4] | Chapman, S.T., García-Sánchez, P.A., Llena, D., Ponomarenko, V., Rosales, J.C.: The catenary and tame degree in finitely generated commutative cancellative monoids. Manuscripta Math. 120, 253-264 (2006) · Zbl 1117.20045 · doi:10.1007/s00229-006-0008-8 |
| [5] | Conca, A., De Negri, E., Rossi, M.E.: Koszul algebras and regularity. In: Peeva, I. (ed.) Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday, pp. 285-315. Springer (2013) · Zbl 1262.13017 |
| [6] | Cox, D.: The homogeneous coordinate ring of a toric variety. J. Algebr. Geom. 4, 15-50 (1995) · Zbl 0846.14032 |
| [7] | Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York, Berlin, Heidelberg (1992) · Zbl 0756.13017 |
| [8] | Cox, D., Little, J., Schenck, H.: Toric Varieties. Amer. Math. Soc, Providence, RI (2010) · Zbl 1223.14001 |
| [9] | Cziszter, K., Domokos, M., Geroldinger, A.: The interplay of invariant theory with multiplicative ideal theory and with arithmetic combinatorics. In: Chapman, S.T., Fontana, M., Geroldinger, A., Olberding, B. (eds.), Multiplicative Ideal Theory and Factorization Theory, pp. 43-95, Springer (2016) · Zbl 1349.13014 |
| [10] | Derksen, H.: Degree bounds for syzygies of invariants. Adv. Math. 185, 207-214 (2004) · Zbl 1067.13002 · doi:10.1016/S0001-8708(03)00167-1 |
| [11] | Derksen, H., Kemper, G.: Computational Invariant Theory, Encyclopaedia of Mathematical Sciences 130. Springer, Berlin, Heidelberg, New York (2002) · Zbl 1011.13003 |
| [12] | Domokos, M., Joó, D.: On the equations and classification of toric quiver varieties. Proc. Royal Soc. Edinburgh: Sect. A Math. 146, 265-295 (2016) · Zbl 1386.14188 · doi:10.1017/S0308210515000529 |
| [13] | Domokos, M., Joó, D.: Toric quiver cells. arXiv:1609.03618 · Zbl 1386.14188 |
| [14] | Geroldinger, A., Halter-Koch, F.: Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, Pure and Applied Mathematics, Vol. 278, Chapman & Hall/CRC (2006) · Zbl 1113.11002 |
| [15] | Geroldinger, A., Zhong, Q.: The catenary degree of Krull monoids II. J. Australian Math. Soc. 98, 324-354 (2015) · Zbl 1373.20074 · doi:10.1017/S1446788714000585 |
| [16] | Gilmer, R.: Commutative Semigroup Rings. The University of Chicago Press (1984) · Zbl 0566.20050 |
| [17] | Haase, C., Paffenholz, A.: Quadratic Gröbner bases for smooth \(3 \times 3\) transportation polytopes. J. Algebr. Comb. 30, 477-489 (2009) · Zbl 1200.13046 · doi:10.1007/s10801-009-0173-4 |
| [18] | Herzog, J.: Generators and relations of abelian semigroups and semigroup rings. Manuscripta Math. 3, 175-193 (1970) · Zbl 0211.33801 · doi:10.1007/BF01273309 |
| [19] | Joó, D.: Complete intersection quiver settings with one dimensional vertices. Algebr. Represent. Theory 16, 1109-1133 (2013) · Zbl 1293.16010 · doi:10.1007/s10468-012-9348-0 |
| [20] | Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, Vol. 227. Springer (2005) · Zbl 1090.13001 |
| [21] | Ohsugi, H., Hibi, T.: Toric rings and ideals of nested configurations. J. Commut. Algebra 2, 187-208 (2010) · Zbl 1237.13053 · doi:10.1216/JCA-2010-2-2-187 |
| [22] | Philipp, A.: A characterization of arithmetical invariants by the monoid of relations. Semigroup Forum 81, 424-434 (2010) · Zbl 1213.20059 · doi:10.1007/s00233-010-9218-1 |
| [23] | Philipp, A.: A characterization of arithmetical invariants by the monoid of relations II: the monotone catenary degree and applications to semigroup rings. Semigroup Forum 90, 220-250 (2015) · Zbl 1323.20057 · doi:10.1007/s00233-014-9616-x |
| [24] | Polishchuk, A., Positelski, L.: Quadratic Algebras, University Lecture Series 37. AMS, Providence, Rhode Island (2005) · Zbl 1145.16009 |
| [25] | Rédei, L.: The theory of Finitely Generated Commutative Semigroups. Pergamon, Oxford-Edinburgh-New York (1965) · Zbl 0133.27904 |
| [26] | Shibuta, T.: Gröbner bases of contraction ideals. J. Algebr. Comb. 36, 1-19 (2012) · Zbl 1250.13028 · doi:10.1007/s10801-011-0320-6 |
| [27] | Sturmfels, B.: Gröbner Bases and Convex Polytopes, University Lecture Series 8. AMS, Providence, Rhode Island (1996) · Zbl 0856.13020 |
| [28] | Villareal, R.H.: Monomial Algebras, CRC Press, Chapman and Hall Book, Monographs and Research Notes in Mathematics, Second edition, Boca Raton, Florida (2015) · Zbl 1325.13004 |
| [29] | Wehlau, D.: Constructive invariant theory for tori. Ann. Inst. Fourier Grenoble 43, 1055-1066 (1993) · Zbl 0789.14009 |
| [30] | Wehlau, D. · Zbl 0809.14008 · doi:10.1007/BF02567695 |
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