Stanley decompositions and partitionable simplicial complexes
Jürgen Herzog1
, Ali Soleyman Jahan1
and Siamak Yassemi3
1Universität Duisburg-Essen Fachbereich Mathematik und Informatik Campus Essen 45117 Essen Germany
3University of Tehran Department of Mathematics P.O. Box 13145448 Tehran Iran
3University of Tehran Department of Mathematics P.O. Box 13145448 Tehran Iran
DOI: 10.1007/s10801-007-0076-1
Abstract
We study Stanley decompositions and show that Stanley's conjecture on Stanley decompositions implies his conjecture on partitionable Cohen-Macaulay simplicial complexes. We also prove these conjectures for all Cohen-Macaulay monomial ideals of codimension 2 and all Gorenstein monomial ideals of codimension 3.
Pages: 113–125
Keywords: keywords Stanley decompositions; partitionable simplicial complexes; pretty clean modules
Full Text: PDF
References
1. Ahmad, S., & Popescu, D. (2007). Sequentially Cohen-Macaulay monomial ideals of embedding dimension four. math.AC/0702569.
2. Anwar, I., & Popescu, D. (2007). Stanley conjecture in small embedding dimension. math.AC/0702728.
3. Apel, J. (2003). On a conjecture of R. P. Stanley; part II-quotients modulo monomial ideals. Journal of Algebraic Combinatorics, 17, 57-74.
4. Björner, A., & Wachs, M. (1997). Shellable nonpure complexes and posets. I. Transactions of American Mathematical Society, 349, 3945-3975.
5. Bruns, W., & Herzog, J. (1995). On multigraded resolutions. Mathematical Proceedings of the Cambridge Philosophical Society, 118, 234-251.
6. Bruns, W., & Herzog, J. (1996). Cohen-Macaulay rings, revised ed., Cambridge Univ. Press, Cambridge.
7. Conca, A., & Herzog, J. (2003). Castelnuovo-Mumford regularity of products of ideals. Collec\?tia Matematic\check a, 54, 137-152.
8. Dress, A. (1993). A new algebraic criterion for shellability. Beiträge zur Algebra und Geometrie, 34(1), 45-55.
9. Eagon, J., & Reiner, V. (1998). Resolutions of Stanley-Reisner rings and Alexander duality. Journal of Pure and Applied Algebra, 130, 265-275.
10. Hachimori, M. Decompositions of two-dimensional simplicial complexes. Discrete Mathematics, in press.
11. Herzog, J., Hibi, T., & Zheng, X. (2004). Monomial ideals whose powers have a linear resolution. Mathematica Scandinavica, 95(1), 23-32.
12. Herzog, J., Hibi, T., & Zheng, X. (2004). Dirac's theorem on chordal graphs and Alexander duality. European Journal of Combinatorics, 25(7), 949-960.
13. Herzog, J., & Popescu, D. (2006). Finite filtrations of modules and shellable multicomplexes. Manuscripta Mathematica, 121, 385-410.
14. Herzog, J., & Takayama, Y. (2002). Resolutions by mapping cones. Homology, Homotopy and Applications, 4, 277-294. The Roos Festschrift, Vol. 2(2).
15. Jahan, A. S. (2007). Prime filtrations of monomial ideals and polarizations. Journal of Algebra, 312(2), 1011-1032.
16. Stanley, R. P. (1975). Cohen-Macaulay rings and constructible polytopes. Bulletin of the American Mathematical Society, 81, 133-135.
17. Stanley, R. P. (1982). Linear Diophantine equations and local cohomology. Inventiones Mathematicae, 68, 175-193.
18. Stanley, R. P. (1983). Combinatorics and commutative algebra. Birkhäuser, Basel.
19. Stanley, R. P. (2000). Positivity problems and conjectures in algebraic combinatorics. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: frontiers and perspectives (pp. 295-319). Providence: American Mathematical Society.
20. Terai, N. (2000). Generalization of Eagon-Reiner theorem and h-vectors of graded rings. Preprint.
2. Anwar, I., & Popescu, D. (2007). Stanley conjecture in small embedding dimension. math.AC/0702728.
3. Apel, J. (2003). On a conjecture of R. P. Stanley; part II-quotients modulo monomial ideals. Journal of Algebraic Combinatorics, 17, 57-74.
4. Björner, A., & Wachs, M. (1997). Shellable nonpure complexes and posets. I. Transactions of American Mathematical Society, 349, 3945-3975.
5. Bruns, W., & Herzog, J. (1995). On multigraded resolutions. Mathematical Proceedings of the Cambridge Philosophical Society, 118, 234-251.
6. Bruns, W., & Herzog, J. (1996). Cohen-Macaulay rings, revised ed., Cambridge Univ. Press, Cambridge.
7. Conca, A., & Herzog, J. (2003). Castelnuovo-Mumford regularity of products of ideals. Collec\?tia Matematic\check a, 54, 137-152.
8. Dress, A. (1993). A new algebraic criterion for shellability. Beiträge zur Algebra und Geometrie, 34(1), 45-55.
9. Eagon, J., & Reiner, V. (1998). Resolutions of Stanley-Reisner rings and Alexander duality. Journal of Pure and Applied Algebra, 130, 265-275.
10. Hachimori, M. Decompositions of two-dimensional simplicial complexes. Discrete Mathematics, in press.
11. Herzog, J., Hibi, T., & Zheng, X. (2004). Monomial ideals whose powers have a linear resolution. Mathematica Scandinavica, 95(1), 23-32.
12. Herzog, J., Hibi, T., & Zheng, X. (2004). Dirac's theorem on chordal graphs and Alexander duality. European Journal of Combinatorics, 25(7), 949-960.
13. Herzog, J., & Popescu, D. (2006). Finite filtrations of modules and shellable multicomplexes. Manuscripta Mathematica, 121, 385-410.
14. Herzog, J., & Takayama, Y. (2002). Resolutions by mapping cones. Homology, Homotopy and Applications, 4, 277-294. The Roos Festschrift, Vol. 2(2).
15. Jahan, A. S. (2007). Prime filtrations of monomial ideals and polarizations. Journal of Algebra, 312(2), 1011-1032.
16. Stanley, R. P. (1975). Cohen-Macaulay rings and constructible polytopes. Bulletin of the American Mathematical Society, 81, 133-135.
17. Stanley, R. P. (1982). Linear Diophantine equations and local cohomology. Inventiones Mathematicae, 68, 175-193.
18. Stanley, R. P. (1983). Combinatorics and commutative algebra. Birkhäuser, Basel.
19. Stanley, R. P. (2000). Positivity problems and conjectures in algebraic combinatorics. In V. Arnold, M. Atiyah, P. Lax, & B. Mazur (Eds.), Mathematics: frontiers and perspectives (pp. 295-319). Providence: American Mathematical Society.
20. Terai, N. (2000). Generalization of Eagon-Reiner theorem and h-vectors of graded rings. Preprint.