Shifting Operations and Graded Betti Numbers
Annetta Aramova
, Jürgen Herzog
and Takayuki Hibi
DOI: 10.1023/A:1011238406374
Abstract
The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.
Pages: 207–222
Keywords: algebraic shifting; shifted complexes; generic initial ideals; extremal Betti numbers
Full Text: PDF
References
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2. A. Aramova and J. Herzog, “Almost regular sequences and Betti numbers,” Amer. J. Math. 122 (2000), 689-719.
3. A. Aramova, J. Herzog, and T. Hibi, “Squarefree lexsegment ideals,” Math. Z. 228 (1998), 353-378.
4. A. Aramova, J. Herzog, and T. Hibi, “Ideals with stable Betti numbers,” Adv. Math. 152 (2000), 72-77.
5. A. Aramova, J. Herzog, and T. Hibi, “Gotzmann theorems for exterior algebras combinatorics,” J. Alg. 191 (1997), 174-211.
6. D. Bayer, H. Charalambous, and S. Popescu, “Extremal Betti numbers and Applications to Monomial Ideals,” J. Alg. 221 (1999), 497-512.
7. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised edition, Cambridge,
1996. ARAMOVA, HERZOG AND HIBI
8. A. Bj\ddot orner and G. Kalai, “An extended Euler-Poincaré theorem,” Acta Math. 161 (1988), 279-303.
9. D. Eisenbud, Commutative Algebra, with a View Towards Algebraic Geometry, Graduate Texts Math., Springer, 1995.
10. S. Eliahou and M. Kervaire, “Minimal resolutions of some monomial ideals,” J. Alg. 129 (1990), 1-25.
11. M. Green, “Generic initial ideals,” in Proc. CRM-96, Six Lectures on Commutative Algebra, Barcelona, Spain, Vol. 166, pp. 119-186, Birkh\ddot auser, 1998.
12. T. Hibi, Algebraic Combinatorics on Convex Polytopes, CarslawPublications, Glebe, N.S.W., Australia, 1992.
13. J. Herzog and N. Terai, “Stable properties of algebraic shifting,” Results Math. 35 (1999), 260-265.
14. M. Hochster, “Cohen-Macaulay rings, combinatorics, and simplicial complexes,” in Proc. Ring Theory II, Lect. Notes in Pure and Appl Math., Vol. 26, pp. 171-223, Deeker, New York, 1977.
15. G. Kalai, “Algebraic shifting,” Unpublished manuscript, 1993.
16. G. Kalai, “The diameter of graphs of convex polytopes and f -vector theory,” in Proc. Applied Geometry and Discrete Mathematics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 4, pp. 387-411, Amer. Math. Soc., 1991.
17. R.P. Stanley, Combinatorics and Commutative Algebra, Birkh\ddot auser, 1983.