Free resolutions of lex-ideals over a Koszul toric ring
HTML articles powered by AMS MathViewer
- by Satoshi Murai PDF
- Trans. Amer. Math. Soc. 363 (2011), 857-885 Request permission
Abstract:
In this paper, we study the minimal free resolution of lex-ideals over a Koszul toric ring. In particular, we study in which toric ring $R$ all lex-ideals are componentwise linear. We give a certain necessity and sufficiency condition for this property, and show that lex-ideals in a strongly Koszul toric ring are componentwise linear. In addition, it is shown that, in the toric ring arising from the Segre product $\mathbb {P}^1 \times \cdots \times \mathbb {P}^1$, every Hilbert function of a graded ideal is attained by a lex-ideal and that lex-ideals have the greatest graded Betti numbers among all ideals having the same Hilbert function.References
- Annetta Aramova, Jürgen Herzog, and Takayuki Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), no. 1, 174–211. MR 1444495, DOI 10.1006/jabr.1996.6903
- Luchezar L. Avramov and Irena Peeva, Finite regularity and Koszul algebras, Amer. J. Math. 123 (2001), no. 2, 275–281. MR 1828224
- Eric Babson and Isabella Novik, Face numbers and nongeneric initial ideals, Electron. J. Combin. 11 (2004/06), no. 2, Research Paper 25, 23. MR 2195431
- Anna Maria Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2317–2334. MR 1218500, DOI 10.1080/00927879308824679
- Anders Björner and Michelle L. Wachs, Shellable nonpure complexes and posets. I, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327. MR 1333388, DOI 10.1090/S0002-9947-96-01534-6
- Giulio Caviglia, The pinched Veronese is Koszul, J. Algebraic Combin. 30 (2009), no. 4, 539–548. MR 2563140, DOI 10.1007/s10801-009-0176-1
- Hara Charalambous and E. Graham Evans Jr., Resolutions obtained by iterated mapping cones, J. Algebra 176 (1995), no. 3, 750–754. MR 1351361, DOI 10.1006/jabr.1995.1270
- G. F. Clements and B. Lindström, A generalization of a combinatorial theorem of Macaulay, J. Combinatorial Theory 7 (1969), 230–238. MR 246781
- Aldo Conca, Jürgen Herzog, and Takayuki Hibi, Rigid resolutions and big Betti numbers, Comment. Math. Helv. 79 (2004), no. 4, 826–839. MR 2099124, DOI 10.1007/s00014-004-0812-2
- Aldo Conca, Ngô Viêt Trung, and Giuseppe Valla, Koszul property for points in projective spaces, Math. Scand. 89 (2001), no. 2, 201–216. MR 1868173, DOI 10.7146/math.scand.a-14338
- David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
- Shalom Eliahou and Michel Kervaire, Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), no. 1, 1–25. MR 1037391, DOI 10.1016/0021-8693(90)90237-I
- Peter Frankl, Zoltán Füredi, and Gil Kalai, Shadows of colored complexes, Math. Scand. 63 (1988), no. 2, 169–178. MR 1018807, DOI 10.7146/math.scand.a-12231
- Ralph Fröberg, Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39. MR 404254, DOI 10.7146/math.scand.a-11585
- Vesselin Gasharov, Green and Gotzmann theorems for polynomial rings with restricted powers of the variables, J. Pure Appl. Algebra 130 (1998), no. 2, 113–118. MR 1635075, DOI 10.1016/S0022-4049(97)00089-3
- V. Gasharov, N. Horwitz and I. Peeva, Hilbert functions over toric rings, Michigan Math. Journal 57 (2008), 339–357.
- V. Gasharov, S. Murai and I. Peeva, Hilbert schemes and maximal Betti numbers over Veronese rings, Math. Z., to appear.
- Gerd Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978), no. 1, 61–70 (German). MR 480478, DOI 10.1007/BF01214566
- Mark L. Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119–186. MR 1648665
- D. Grayson and M. Stillman, Macaulay 2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu.Macaulay2/
- Jürgen Herzog and Takayuki Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153. MR 1684555, DOI 10.1017/S0027763000006930
- Jürgen Herzog, Takayuki Hibi, and Gaetana Restuccia, Strongly Koszul algebras, Math. Scand. 86 (2000), no. 2, 161–178. MR 1754992, DOI 10.7146/math.scand.a-14287
- Jürgen Herzog and Yukihide Takayama, Resolutions by mapping cones, Homology Homotopy Appl. 4 (2002), no. 2, 277–294. The Roos Festschrift volume, 2. MR 1918513, DOI 10.4310/hha.2002.v4.n2.a13
- Heather A. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra 21 (1993), no. 7, 2335–2350. MR 1218501, DOI 10.1080/00927879308824680
- Ali Soleyman Jahan and Xinxian Zheng, Ideals with linear quotients, J. Combin. Theory Ser. A 117 (2010), no. 1, 104–110. MR 2557882, DOI 10.1016/j.jcta.2009.03.012
- F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc. 26 (1927), 531–555.
- Jeff Mermin and Satoshi Murai, Betti numbers of lex ideals over some Macaulay-Lex rings, J. Algebraic Combin. 31 (2010), no. 2, 299–318. MR 2592080, DOI 10.1007/s10801-009-0192-1
- Jeffrey Mermin and Irena Peeva, Hilbert functions and lex ideals, J. Algebra 313 (2007), no. 2, 642–656. MR 2329561, DOI 10.1016/j.jalgebra.2007.03.036
- Jeffrey Mermin, Irena Peeva, and Mike Stillman, Ideals containing the squares of the variables, Adv. Math. 217 (2008), no. 5, 2206–2230. MR 2388092, DOI 10.1016/j.aim.2007.11.014
- S. Murai and I. Peeva, Hilbert schemes and Betti numbers over a Clements-Lindström ring, submitted.
- Satoshi Murai and Pooja Singla, Rigidity of linear strands and generic initial ideals, Nagoya Math. J. 190 (2008), 35–61. MR 2423828, DOI 10.1017/S0027763000009557
- Keith Pardue, Deformation classes of graded modules and maximal Betti numbers, Illinois J. Math. 40 (1996), no. 4, 564–585. MR 1415019
- Irena Peeva, Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3503–3507. MR 2084070, DOI 10.1090/S0002-9939-04-07517-3
- G. Baley Price, On the completeness of a certain metric space with an application to Blaschke’s selection theorem, Bull. Amer. Math. Soc. 46 (1940), 278–280. MR 2010, DOI 10.1090/S0002-9904-1940-07195-2
Additional Information
- Satoshi Murai
- Affiliation: Department of Mathematics, Graduate School of Science, Kyoto University, Sakyou-ku, Kyoto 606-8502, Japan
- Address at time of publication: Department of Mathematical Science, Faculty of Science, Yamaguchi University, 1677-1 Yoshida, Yamaguchi 753-8512, Japan
- MR Author ID: 800440
- Email: murai@math.kyoto-u.ac.jp, murai@yamaguchi-u.ac.jp
- Received by editor(s): December 8, 2008
- Received by editor(s) in revised form: April 2, 2009
- Published electronically: September 16, 2010
- Additional Notes: The author was supported by JSPS Research Fellowships for Young Scientists.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 857-885
- MSC (2010): Primary 13D02; Secondary 05D05, 05E40, 13D40, 13F45, 16S37
- DOI: https://doi.org/10.1090/S0002-9947-2010-05074-3
- MathSciNet review: 2728587