Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
-
Alberto Corso
,
Sonja Petrović
Abstract
We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen–Macaulay domains that are Koszul, that the initial ideals correspond to vertex decomposable simplicial complexes, and we determine their Hilbert functions and Castelnuovo–Mumford regularities. As a consequence, we find explicit minimal reductions for all Ferrers and many specialized Ferrers ideals, as well as their reduction numbers. These results can be viewed as extensions of the classical Dedekind–Mertens formula for the content of the product of two polynomials.
Funding source: National Security Agency
Award Identifier / Grant number: H98230-09-1-0032
Award Identifier / Grant number: H98230-12-1-0247
Funding source: Simons Foundation
Award Identifier / Grant number: #208869
Award Identifier / Grant number: #317096
Funding source: Air Force Office of Scientific Research
Award Identifier / Grant number: #FA9550-14-1-0141
Funding source: Defense Advanced Research Projects Agency
Award Identifier / Grant number: #FA9550-14-1-0141
Funding statement: The work for this paper was done while the second author was sponsored by the National Security Agency under grant numbers H98230-09-1-0032 and H98230-12-1-0247, and by the Simons Foundation under grants #208869 and #317096. The third author gratefully acknowledges partial support by AFOSR/DARPA grant #FA9550-14-1-0141 during the final phase of this project.
Acknowledgements
The third author thanks the University of Kentucky Math Department for its hospitality. The authors are also grateful to the referee for helpful suggestions that improved the exposition.
References
[1] S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux, Marcel Dekker, New York, 1988. Suche in Google Scholar
[2] J. L. Andersen, Determinantal rings associated with symmetric matrices: A counterexample, Ph.D. thesis, University of Minnesota, Minneapolis, 1992. Suche in Google Scholar
[3] A. Boocher, Free resolutions and sparse determinantal ideals, Math. Res. Lett. 19 (2012), 805–821. 10.4310/MRL.2012.v19.n4.a6Suche in Google Scholar
[4] F. Brenti, G. Royle and D. Wagner, Location of zeros of chromatic and related polynomials of graphs, Canad. J. Math. 46 (1994), 55–80. 10.4153/CJM-1994-002-3Suche in Google Scholar
[5] W. Bruns and A. Guerrieri, The Dedekind–Mertens formula and determinantal rings, Proc. Amer. Math. Soc. 127 (1999), 657–663. 10.1090/S0002-9939-99-04535-9Suche in Google Scholar
[6] W. Bruns and J. Herzog, Cohen–Macaulay Rings, Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993. Suche in Google Scholar
[7] W. Bruns, J. Herzog and U. Vetter, Syzygies and walks, Commutative Algebra (Trieste 1992), World Science Publisher, River Edge (1994), 36–57. Suche in Google Scholar
[8] W. Bruns and U. Vetter, Determinantal Rings, Lecture Notes in Math. 1327, Springer, Berlin, 1988. 10.1007/BFb0080378Suche in Google Scholar
[9] F. Butler, Rook theory and cycle-counting permutation statistics, Adv. in Appl. Math. 33 (2004), 655–675. 10.1016/j.aam.2004.03.004Suche in Google Scholar
[10] A. Conca, Gröbner Bases and Determinantal Rings, Ph.D. thesis, University of Essen, 1993. Suche in Google Scholar
[11] A. Conca and J. Herzog, On the Hilbert function of determinantal rings and their canonical module, Proc. Amer. Math. Soc. 112 (1994), 677–681. 10.1090/S0002-9939-1994-1213858-0Suche in Google Scholar
[12] A. Corso and U. Nagel, Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math. Soc. 361 (2009), 1371–1395. 10.1090/S0002-9947-08-04636-9Suche in Google Scholar
[13] A. Corso and U. Nagel, Specializations of Ferrers ideals, J. Algebraic Combin. 28 (2008), 425–437. 10.1007/s10801-007-0111-2Suche in Google Scholar
[14] A. Corso, W. V. Vasconcelos and R. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra 125 (1998), 117–127. 10.1016/S0022-4049(97)80001-1Suche in Google Scholar
[15] D. Cox, J. Little and D. O’Shea, Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3rd ed., Undergrad. Texts Math., Springer, New York, 2007. Suche in Google Scholar
[16] D. Cox, J. Little and H. Schenck, Toric Varieties, Grad. Stud. Math. 124, American Mathematical Society, Providence, 2011. 10.1090/gsm/124Suche in Google Scholar
[17] A. M. del Campo, S. Cepeda and C. Uhler, Exact goodness-of-fit testing for the Ising model, preprint (2014), https://arxiv.org/abs/1410.1242. Suche in Google Scholar
[18] E. De Negri and E. Gorla, Invariants of ideals generated by Pfaffians, Commutative Algebra and Its Connections to Geometry, Contemp. Math. 555, American Mathematical Society, Providence (2011), 47–62. 10.1090/conm/555/10988Suche in Google Scholar
[19] M. Develin, Rook poset equivalence of Ferrers boards, Order 23 (2006), 179–195. 10.1007/s11083-006-9039-8Suche in Google Scholar
[20] P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distribution, Ann. Statist. 26 (1998), 363–397. 10.1214/aos/1030563990Suche in Google Scholar
[21] K. Ding, Rook placements and cellular decomposition of partition varieties, Discrete Math. 170 (1997), 107–151. 10.1016/S0012-365X(96)00002-7Suche in Google Scholar
[22] A. Dochtermann and A. Engström, Algebraic properties of edge ideals via combinatorial topology, Electron. J. Combin. 16 (2009), no. 2, Research Paper R2. 10.37236/68Suche in Google Scholar
[23] R. Ehrenborg and S. van Willigenburg, Enumerative properties of Ferrers graphs, Discrete Comput. Geom. 32 (2004), 481–492. 10.1007/s00454-004-1135-1Suche in Google Scholar
[24] D. Eisenbud, C. Hunke and B. Ulrich, Heights of ideals of minors, Amer. J. Math. 126 (2004), 417–438. 10.1353/ajm.2004.0011Suche in Google Scholar
[25] R. Fröberg, On Stanley–Reisner rings, Topics in Algebra. Part 2: Commutative Rings and Algebraic Groups (Warsaw 1988), Banach Center Publ. 26, PWN, Warszawa (1990), 57–70. 10.1007/978-3-030-89694-2_10Suche in Google Scholar
[26] S. R. Ghorpade, Hilbert functions of ladder determinantal varieties, Discrete Math. 246 (2002), 131–175. 10.1016/S0012-365X(01)00256-4Suche in Google Scholar
[27] J. Goldman, J. T. Joichi and D. White, Rook Theory I. Rook equivalence of Ferrers boards, Proc. Amer. Math. Soc. 52 (1975), 485–492. 10.1090/S0002-9939-1975-0429578-4Suche in Google Scholar
[28] E. Gorla, Symmetric ladders and G-biliaison, Liaison, Schottky Problem and Invariant Theory, Progr. Math. 280, Birkhäuser, Basel (2010), 49–62. 10.1007/978-3-0346-0201-3_5Suche in Google Scholar
[29] E. Gorla, J. Migliore and U. Nagel, Gröbner bases via linkage, J. Algebra 384 (2013), 110–134. 10.1016/j.jalgebra.2013.02.017Suche in Google Scholar
[30] M. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20 (1984), 279–289. 10.4310/jdg/1214439000Suche in Google Scholar
[31] M. Green and R. Lazarsfeld, Some results on the syzygies of finite sets and algebraic curves, Compos. Math. 67 (1988), no. 3, 301–314. Suche in Google Scholar
[32] E. Gross, S. Petrović and D. Stasi, Goodness-of-fit for log-linear network models: Dynamic Markov bases using hypergraphs, Ann. Inst. Statist. Math. (2016), 10.1007/s10463-016-0560-2. 10.1007/s10463-016-0560-2Suche in Google Scholar
[33] J. Haglund, Rook theory and hypergeometric series, Adv. in Appl. Math. 17 (1996), 408–459. 10.1006/aama.1996.0017Suche in Google Scholar
[34] H. Hara, S. Aoki and A. Takemura, Running Markov chain without Markov basis, Harmony of Gröbner Bases and the Modern Industrial Society, World Science Publisher, Hackensack (2012), 45–62. 10.1142/9789814383462_0005Suche in Google Scholar
[35] J. Herzog and T. Hibi, Distributive lattices, bipartite graphs and Alexander duality, J. Algebraic Combin. 22 (2005), 289–302. 10.1007/s10801-005-4528-1Suche in Google Scholar
[36] J. Herzog and T. Hibi, Monomial Ideals, Grad. Texts in Math. 260, Springer, London, 2011. 10.1007/978-0-85729-106-6Suche in Google Scholar
[37] J. Herzog and N. V. Trung, Gröbner bases and multiplicity of determinantal and Pfaffian ideals, Adv. Math. 96 (1992), 1–37. 10.1016/0001-8708(92)90050-USuche in Google Scholar
[38] C. Huneke and I. Swanson, Integral Closure of Ideals, Rings, and Modules, London Math. Soc. Lecture Note Ser. 336, Cambridge University Press, Cambridge, 2006. Suche in Google Scholar
[39] T. Jozefiak, P. Pragacz and J. Weyman, Resolutions of determinantal varieties and tensor complexes associated with symmetric and antisymmetric matrices, Astérisque 87–88 (1981), 109–189. Suche in Google Scholar
[40] M. Kateri, Contingency Table Analysis: Methods and Implementation Using R, Stat. Ind. Technol., Birkhäuser, New York, 2014. 10.1007/978-0-8176-4811-4Suche in Google Scholar
[41] J. Kleppe, R. Miró-Roig, J. Migliore, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. Amer. Math. Soc. 732 (2001), 1–116. 10.1090/memo/0732Suche in Google Scholar
[42] A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), 1245–1318. 10.4007/annals.2005.161.1245Suche in Google Scholar
[43] C. Krattenthaler and S. G. Mohanty, On lattice path counting by major and descents, European J. Combin. 14 (1993), 43–51. 10.1006/eujc.1993.1007Suche in Google Scholar
[44] C. Krattenthaler and M. Prohaska, A remarkable formula for counting non-intersecting lattice paths in a ladder with respect to turns, Trans. Amer. Math. Soc. 351 (1999), 1015–1042. 10.1090/S0002-9947-99-01884-XSuche in Google Scholar
[45] C. Krattenthaler and M. Rubey, A determinantal formula for the Hilbert series of one-sided ladder determinantal rings, Algebra, Arithmetic and Geometry with Applications (West Lafayette 2000), Springer, Berlin (2004), 525–551. 10.1007/978-3-642-18487-1_30Suche in Google Scholar
[46] D. M. Kulkarni, Counting of paths and coefficients of Hilbert polynomial of a determinantal ideal, Discrete Math. 154 (1996), 141–151. 10.1016/0012-365X(94)00345-JSuche in Google Scholar
[47] J. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progr. Math. 165, Birkhäuser, Basel, 1998. 10.1007/978-1-4612-1794-7Suche in Google Scholar
[48] E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Grad. Texts in Math. 227, Springer, New York, 2005. Suche in Google Scholar
[49] U. Nagel, Castelnuovo’s regularity and Hilbert functions, Compos. Math. 76 (1990), 265–275. 10.1007/978-94-009-0685-3_13Suche in Google Scholar
[50] U. Nagel, Comparing Castelnuovo–Mumford regularity and extended degree: The borderline cases, Trans. Amer. Math. Soc. 357 (2005), 3585–3603. 10.1090/S0002-9947-04-03595-0Suche in Google Scholar
[51] U. Nagel and V. Reiner, Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16 (2009), no. 2, Research Paper R3. 10.37236/69Suche in Google Scholar
[52] U. Nagel and W. Robinson, Determinantal ideals to symmetrized skew shapes, in preparation. Suche in Google Scholar
[53] U. Nagel and T. Römer, Glicci simplicial complexes, J. Pure Appl. Algebra 212 (2008), 2250–2258. 10.1016/j.jpaa.2008.03.005Suche in Google Scholar
[54] D. G. Northcott, A generalization of a theorem on the contents of polynomials, Proc. Camb. Phil. Soc. 55 (1959), 282–288. 10.1017/S030500410003406XSuche in Google Scholar
[55] S. Petrović, A survey of discrete methods in (algebraic) statistics for networks, preprint (2015), http://arxiv.org/abs/1510.02838. 10.1090/conm/685/13714Suche in Google Scholar
[56] S. Petrović, A. Rinaldo and S. E. Fienberg, Algebraic statistics for a directed random graph model with reciprocation, Algebraic Methods in Statistics and Probability II, Contemp. Math. 516, American Mathematical Society, Providence (2010), 261–283. 10.1090/conm/516/10180Suche in Google Scholar
[57] C. Polini, B. Ulrich and M. Vitulli, The core of zero-dimensional monomial ideals, Adv. Math. 211 (2007), 72–93. 10.1016/j.aim.2006.07.020Suche in Google Scholar
[58] F. Rapallo, Markov bases and structural zeros, J. Symbolic Comput. 41 (2006), 164–172. 10.1016/j.jsc.2005.04.002Suche in Google Scholar
[59] F. Rapallo and R. Yoshida, Markov bases and subbases for bounded contingency tables, Ann. Inst. Statist. Math. 62 (2010), no. 4, 785–805. 10.1007/s10463-010-0289-2Suche in Google Scholar
[60]
M. Rubey,
The h-vector of a ladder determinantal ring cogenerated by
[61] A. Simis, W. V. Vasconcelos and R. Villarreal, On the ideal theory of graphs, J. Algebra 167 (1994), 389–416. 10.1006/jabr.1994.1192Suche in Google Scholar
[62] B. Smith, A formula for the core of certain strongly stable ideals, J. Algebra 347 (2011), 40–52. 10.1016/j.jalgebra.2011.08.031Suche in Google Scholar
[63] B. Sturmfels, Gröbner Bases and Convex Polytopes, Univ. Lecture Ser. 8, American Mathematical Society, Providence, 1996. 10.1090/ulect/008Suche in Google Scholar
[64] N. V. Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), 229–236. 10.1090/S0002-9939-1987-0902533-1Suche in Google Scholar
[65] A. Varvak, Rook numbers and the normal ordering problem, J. Combin. Theory Ser. A 112 (2005), 292–307. 10.1016/j.jcta.2005.07.012Suche in Google Scholar
[66] R. Villarreal, Monomial Algebras, Monogr. Textb. Pure Appl. Math. 238, Marcel Dekker, New York, 2001. 10.1201/9780824746193Suche in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Hurewicz fibrations, almost submetries and critical points of smooth maps
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds
- Diffusion semigroup on manifolds with time-dependent metrics
- Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
- A computational approach to Milnor fiber cohomology
- Metrical universality for groups
- The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
- Slender domains and compact domains
- Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
- A note on local Hardy spaces
- On the value group of a model of Peano Arithmetic
- Extremal values of the (fractional) Weinstein functional on the hyperbolic space
- Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
- Unitals in shift planes of odd order
Artikel in diesem Heft
- Frontmatter
- Hurewicz fibrations, almost submetries and critical points of smooth maps
- Lower bounds for regular genus and gem-complexity of PL 4-manifolds
- Diffusion semigroup on manifolds with time-dependent metrics
- Blow-up algebras, determinantal ideals, and Dedekind–Mertens-like formulas
- A computational approach to Milnor fiber cohomology
- Metrical universality for groups
- The second and third moment of L(1/2,χ) in the hyperelliptic ensemble
- Slender domains and compact domains
- Topological 2-generation of automorphism groups of countable ultrahomogeneous graphs
- A note on local Hardy spaces
- On the value group of a model of Peano Arithmetic
- Extremal values of the (fractional) Weinstein functional on the hyperbolic space
- Maximal subsemigroups of some semigroups of order-preserving mappings on a countably infinite set
- Unitals in shift planes of odd order
