On a Conjecture of R.P. Stanley; Part I-Monomial Ideals
Joachim Apel
DOI: 10.1023/A:1021912724441
Abstract
In 1982 Richard P. Stanley conjectured that any finitely generated 
n -graded module M over a finitely generated 
n -graded K-algebra R can be decomposed in a direct sum M = 
i = 1 t 
i S i of finitely many free modules i S i which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the S i have to be subalgebras of R of dimension at least depth M.
n -graded module M over a finitely generated n -graded K-algebra R can be decomposed in a direct sum M = i = 1 t i S i of finitely many free modules i S i which have to satisfy some additional conditions. Besides homogeneity conditions the most important restriction is that the S i have to be subalgebras of R of dimension at least depth M. We will study this conjecture for the special case that R is a polynomial ring and M an ideal of R, where we encounter a strong connection to generalized involutive bases. We will derive a criterion which allows us to extract an upper bound on depth M from particular involutive bases. As a corollary we obtain that any monomial ideal M which possesses an involutive basis of this type satisfies Stanley”s Conjecture and in this case the involutive decomposition defined by the basis is also a Stanley decomposition of M. Moreover, we will show that the criterion applies, for instance, to any monomial ideal of depth at most 2, to any monomial ideal in at most 3 variables, and to any monomial ideal which is generic with respect to one variable. The theory of involutive bases provides us with the algorithmic part for the computation of Stanley decompositions in these situations.
Pages: 39–56
Keywords: monomial ideal; combinatorial decomposition; involutive basis
Full Text: PDF
References
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2. K. Baclawski and A.M. Garsia, “Combinatorial decompositions of a class of rings,” Adv. Math. 39 (1981), 155-184.
3. D. Bayer, I. Peeva, and B. Sturmfels, “Monomial resolutions,” Math. Res. Lett. 5 (1998), 31-46.
4. D. Eisenbud, “Commutative algebra with a view toward algebraic geometry,” in Graduate Texts in Mathematics, Vol. 150, Springer, New York, 1995.
5. R. Hemmecke, “The calix project,” available at http://www.hemmecke.de/calix.
6. M. Janet, Lecons sur les syst`emes d'equations aux dérivées partielles, Gauthier-Villars, Paris, 1929.
7. I. Kaplansky, Commutative Rings, Polygonal Publishing House, Washington,
1994. Originally published by Allyn and Bacon, Boston, 1970.
8. E. Miller, B. Sturmfels, and K. Yanagawa, “Generic and cogeneric monomial ideals,” J. Symb. Comp. 29 (2000), 691-708.
9. D. Rees, “A basis theorem for polynomial modules,” Proc. Cambridge Phil. Soc. 52 (1956), 12-16.
10. G.A. Reisner, “Cohen-Macaulay quotients of polynomial rings,” Adv. Math. 21 (1976), 30-49.
11. C.H. Riquir, Les systémes d'equations aux dérivées partielles, Gauthier-Villars, Paris, 1910.
12. R.P. Stanley, “Balanced Cohen-Macaulay complexes,” Trans. Amer. Math. Soc. 249 (1979), 139-157.
13. R.P. Stanley, “Linear Diophantine equations and local cohomology,” Invent. Math. 68 (1982), 175-193.
14. J.M. Thomas, “Riquier's existence theorems,” Ann. Math. 30(2) (1929), 285-310.
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