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Keywords:
almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module
almost clean module; clean module; $d$-sequence; filter-regular sequence; pretty clean module
Summary:
Let $K$ be a field and $S=K[x_1,\ldots , x_n]$. Let $I$ be a monomial ideal of $S$ and $u_1,\ldots , u_r$ be monomials in $S$. We prove that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then $S/I$ is pretty clean if and only if $S/(I,u_1,\ldots , u_r)$ is pretty clean. Also, we show that if $u_1,\ldots , u_r$ form a filter-regular sequence on $S/I$, then Stanley's conjecture is true for $S/I$ if and only if it is true for $S/(I,u_1, \ldots , u_r)$. Finally, we prove that if $u_1,\ldots , u_r$ is a minimal set of generators for $I$ which form either a $d$-sequence, proper sequence or strong $s$-sequence (with respect to the reverse lexicographic order), then $S/I$ is pretty clean.
References:
[1] Apel, J.: On a conjecture of R. P. Stanley. I. Monomial ideals. J. Algebr. Comb. 17 (2003), 39-56. DOI 10.1023/A:1021912724441 | MR 1958008 | Zbl 1031.13003
[2] Apel, J.: On a conjecture of R. P. Stanley. II. Quotients modulo monomial ideals. J. Algebr. Comb. 17 (2003), 57-74. DOI 10.1023/A:1021916908512 | MR 1958009 | Zbl 1031.13004
[3] Lorestani, K. Borna, Sahandi, P., Sharif, T.: A note on the associated primes of local cohomology modules. Commun. Algebra 34 (2006), 3409-3412. DOI 10.1080/00927870600794115 | MR 2252680
[4] Dress, A.: A new algebraic criterion for shellability. Beitr. Algebra Geom. 34 (1993), 45-55. MR 1239277 | Zbl 0780.52012
[5] Herzog, J., Jahan, A. S., Yassemi, S.: Stanley decompositions and partitionable simplicial complexes. J. Algebr. Comb. 27 (2008), 113-125. DOI 10.1007/s10801-007-0076-1 | MR 2366164 | Zbl 1131.13020
[6] Herzog, J., Hibi, T.: Monomial Ideals. Graduate Texts in Mathematics 260 Springer, London (2011). MR 2724673 | Zbl 1206.13001
[7] Herzog, J., Popescu, D.: Finite filtrations of modules and shellable multicomplexes. Manuscr. Math. 121 (2006), 385-410. DOI 10.1007/s00229-006-0044-4 | MR 2267659 | Zbl 1107.13017
[8] Herzog, J., Restuccia, G., Tang, Z.: $s$-sequences and symmetric algebras. Manuscr. Math. 104 (2001), 479-501. DOI 10.1007/s002290170022 | MR 1836109 | Zbl 1058.13011
[9] Herzog, J., Vladoiu, M., Zheng, X.: How to compute the Stanley depth of a monomial ideal. J. Algebra 322 (2009), 3151-3169. DOI 10.1016/j.jalgebra.2008.01.006 | MR 2567414 | Zbl 1186.13019
[10] Popescu, D.: Stanley depth of multigraded modules. J. Algebra 321 (2009), 2782-2797. DOI 10.1016/j.jalgebra.2009.03.009 | MR 2512626 | Zbl 1179.13016
[11] Rauf, A.: Stanley decompositions, pretty clean filtrations and reductions modulo regular elements. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 50 (2007), 347-354. MR 2370321 | Zbl 1155.13311
[12] Sabzrou, H., Tousi, M., Yassemi, S.: Simplicial join via tensor product. Manuscr. Math. 126 (2008), 255-272. DOI 10.1007/s00229-008-0175-x | MR 2403189 | Zbl 1165.13003
[13] Jahan, A. Soleyman: Easy proofs of some well known facts via cleanness. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 54 (2011), 237-243. MR 2856300
[14] Jahan, A. Soleyman: Prime filtrations and primary decompositions of modules. Commun. Algebra 39 (2011), 116-124. DOI 10.1080/00927870903431225 | MR 2770881
[15] Jahan, A. Soleyman: Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality. Manuscr. Math. 130 (2009), 533-550. DOI 10.1007/s00229-009-0308-x | MR 2563149
[16] Jahan, A. Soleyman: Prime filtrations of monomial ideals and polarizations. J. Algebra 312 (2007), 1011-1032. DOI 10.1016/j.jalgebra.2006.11.002 | MR 2333198
[17] Jahan, A. Soleyman, Zheng, X.: Monomial ideals of forest type. Commun. Algebra 40 (2012), 2786-2797. DOI 10.1080/00927872.2011.585679 | MR 2968912 | Zbl 1254.13025
[18] Stanley, R. P.: Linear Diophantine equations and local cohomology. Invent. Math. 68 (1982), 175-193. DOI 10.1007/BF01394054 | MR 0666158 | Zbl 0516.10009
[19] Tang, Z.: On certain monomial sequences. J. Algebra 282 (2004), 831-842. DOI 10.1016/j.jalgebra.2004.08.027 | MR 2101086 | Zbl 1147.13304
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