Betti numbers of strongly color-stable ideals and squarefree strongly color-stable ideals
Satoshi Murai
Osaka University Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology Toyonaka Osaka 560-0043 Japan
DOI: 10.1007/s10801-007-0095-y
Abstract
In this paper, we will show that the color-squarefree operation does not change the graded Betti numbers of strongly color-stable ideals. In addition, we will give an example of a nonpure balanced complex which shows that colored algebraic shifting, which was introduced by Babson and Novik, does not always preserve the dimension of reduced homology groups of balanced simplicial complexes.
Pages: 383–398
Keywords: keywords colored algebraic shifting; balanced complexes; graded Betti numbers
Full Text: PDF
References
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2. Aramova, A., Crona, K., De Negri, E.: Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions. J. Pure Appl. Algebra 150(3), 215-235 (2000)
3. Aramova, A., Herzog, J., Hibi, T.: Gotzmann theorems for exterior algebras and combinatorics. J. Algebra 191, 174-211 (1997)
4. Aramova, A., Herzog, J., Hibi, T.: Shifting operations and graded Betti numbers. J. Algebra Comb. 12(3), 207-222 (2000)
5. Babson, E., Novik, I.: Face numbers and nongeneric initial ideals. Electron. J. Comb. 11(2), 23 (2006) Research Paper 25
6. Bigatti, A.M., Conca, A., Robbiano, L.: Generic initial ideals and distractions. Commun. Algebra 33(6), 1709-1732 (2005)
7. Björner, A., Frankl, P., Stanley, R.P.: The number of faces of balanced Cohen-Macaulay complexes and generalized Macaulay theorem. Combinatorica 7, 23-34 (1987)
8. Björner, A., Kalai, G.: An extended Euler-Poincaré theorem. Acta. Math. 161, 279-303 (1988)
9. Björner, A., Wachs, M.: Shellable nonpure complexes and posets. I. Trans. Am. Math. Soc. 348(4), 1299-1327 (1996)
10. Bruns, W., Herzog, J.: Cohen-Macaulay rings, Revised edn. Cambridge University Press, Cambridge (1998)
11. CoCoA Team: CoCoA: a system for doing computations in commutative algebra. Available at