Betti numbers of lex ideals over some Macaulay-Lex rings
Jeff Mermin
and Satoshi Murai
DOI: 10.1007/s10801-009-0192-1
Abstract
Let A= K[ x 1,\cdots , x n ] be a polynomial ring over a field K and M a monomial ideal of A. The quotient ring R= A/ M is said to be Macaulay-Lex if every Hilbert function of a homogeneous ideal of R is attained by a lex ideal. In this paper, we introduce some new Macaulay-Lex rings and study the Betti numbers of lex ideals of those rings. In particular, we prove a refinement of the Frankl-Füredi-Kalai Theorem which characterizes the face vectors of colored complexes. Additionally, we disprove a conjecture of Mermin and Peeva that lex-plus- M ideals have maximal Betti numbers when A/ M is Macaulay-Lex.
Pages: 299–318
Keywords: keywords lex ideals; graded Betti numbers; Hilbert functions; colored simplicial complexes
Full Text: PDF
References
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2. Bigatti, A.: Upper bounds for the Betti numbers of a given Hilbert function. Comm. Algebra 21, 2317-2334 (1993)
3. Björner, A., Frankl, P., Stanley, R.P.: The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem. Combinatorica 7, 23-34 (1987)
4. Clements, G.F., Lindström, B.: A generalization of a combinatorial theorem of Macaulay. J. Combinatorial Theory 7, 230-238 (1969)
5. CoCoA Team: CoCoA: a system for doing Computations in Commutative Algebra. Available at
6. Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Grad. Texts in Math., vol.
150. Springer, New York (1995)
7. Francisco, C.A., Richert, B.P.: Lex-plus-powers ideals. In: Peeva, I. (ed.) Syzygies and Hilbert functions. Lect. Notes Pure Appl. Math., vol. 254, pp. 113-144. CRC Press, Boca Raton (2007) J Algebr Comb (2010) 31: 299-318
8. Frankl, P., Füredi, Z., Kalai, G.: Shadows of colored complexes. Math. Scand. 63, 169-178 (1988)
9. Grayson, D., Stillman, M.: Macaulay 2, a software system for research in algebraic geometry. Available at
10. Herzog, J., Takayama, Y.: Resolutions by mapping cones. Homology Homotopy Appl. 4, 277-294 (2002)
11. Hochster, M.: Cohen-Macaulay rings, combinatorics, and simplicial complexes. In: McDonald, B.R., Morris, R. (eds.) Ring theory, II, Proc. Second Conf., Univ. Oklahoma, Norman, Oklahoma. Lecture Notes in Pure and Appl. Math., vol. 26, pp. 171-223. Dekker, New York (1977)
12. Hulett, H.: Maximum Betti numbers of homogeneous ideals with a given Hilbert function. Comm. Algebra 21, 2335-2350 (1993)
13. London, E.: A new proof of the colored Kruskal-Katona theorem. Discrete Math. 126, 217-223 (1994)
14. Macaulay, F.S.: Some properties of enumeration in the theory of modular systems. Proc. London Math. Soc. 26, 531-555 (1927)
15. Mermin, J.: Compressed ideals. Bull. London Math. Soc. 40, 77-87 (2008)
16. Mermin, J., Murai, S.: The lex-plus-powers conjecture holds for pure powers (2008, submitted)
17. Mermin, J., Peeva, I.: Lexifying ideals. Math. Res. Letters 13, 409-422 (2006)
18. Mermin, J., Peeva, I.: Hilbert functions and lex ideals. J. Algebra 313, 642-656 (2007)
19. Mermin, J., Peeva, I., Stillman, M.: Ideals containing the squares of the variables. Adv. Math. 217, 2206-2230 (2008)
20. Murai, S.: Borel-plus-powers monomial ideals. J. Pure Appl. Algebra 212, 1321-1336 (2008)
21. Murai, S.: Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals.