Some algebraic invariants of the residue class rings of the edge ideals of perfect semiregular trees. (English) Zbl 1521.13017
Summary: Let \(S\) be a polynomial algebra over a field. If \(I\) is the edge ideal of a perfect semiregular tree, then we give precise formulas for values of depth, Stanley depth, projective dimension, regularity and Krull dimension of \(S/I\).
MSC:
| 13C15 | Dimension theory, depth, related commutative rings (catenary, etc.) |
| 13A70 | General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.) |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
Keywords:
depth; Krull dimension; monomial ideal; perfect \(n\)-ary tree; perfect semiregular tree; projective dimension; regularity; Stanley depthReferences:
| [1] | Alipour, A.; Tehranian, A., Depth and Stanley depth of edge ideals of star graphs, Int. J. Appl. Math. Stat, 56, 4, 63-69 (2017) |
| [2] | Bouchat, R. R., Free resolutions of some edge ideals of simple graphs, J. Commut. Algebra, 2, 1, 1-35 (2010) · Zbl 1238.13028 |
| [3] | Bruns, W.; Herzog, H. J., Cohen-Macaulay Rings (1998), Cambridge: Cambridge University Press, Cambridge · Zbl 0909.13005 |
| [4] | Caviglia, G.; Há, H. T.; Herzog, J.; Kummini, M.; Terai, N.; Trung, N. V., Depth and regularity modulo a principal ideal, J. Algebraic Comb, 49, 1, 1-20 (2019) · Zbl 1448.13028 · doi:10.1007/s10801-018-0811-9 |
| [5] | Cha, S. H., On integer sequences derived from balanced k-ary trees, In Proceedings of American Conference on Applied Mathematics, 377-381 (2012) |
| [6] | Cimpoeas, M., Several inequalities regarding Stanley depth, Romanian J. Math. Comput. Sci., 2, 1, 28-40 (2012) · Zbl 1313.13036 |
| [7] | CoCoATeam, CoCoA: A system for doing computations in commutative algebra |
| [8] | Dao, H.; Huneke, C.; Schweig, J., Bounds on the regularity and projective dimension of ideals associated to graphs, J. Algebraic Comb, 38, 1, 37-55 (2013) · Zbl 1307.13021 · doi:10.1007/s10801-012-0391-z |
| [9] | Din, N. U.; Ishaq, M.; Sajid, Z., Values and bounds for depth and Stanley depth of some classes of edge ideals, AIMS Math., 6, 8, 8544-8566 (2021) · Zbl 1484.13029 · doi:10.3934/math.2021496 |
| [10] | Dinu, R.; Ene, V.; Hibi, T., On the regularity of join-meet ideals of modular lattices, arXiv preprint arXiv:1806.05200 (2018) |
| [11] | Duval, A. M.; Goeckner, B.; Klivans, C. J.; Martin, J. L., A non-partitionable Cohen-Macaulay simplicial complex, Adv. Math, 299, 381-395 (2016) · Zbl 1341.05256 · doi:10.1016/j.aim.2016.05.011 |
| [12] | Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry, 150 (2013), New York: Springer, New York |
| [13] | Fakhari, S. S., On the Stanley depth of powers of edge ideals, J. Algebra, 489, 463-474 (2017) · Zbl 1388.13031 |
| [14] | Faridi, S.; Hersey, B., Resolutions of monomial ideals of projective dimension 1, Commun. Algebra, 45, 12, 5453-5464 (2017) · Zbl 1386.13037 · doi:10.1080/00927872.2017.1313422 |
| [15] | Fouli, L.; Morey, S., A lower bound for depths of powers of edge ideals, J. Algebraic Comb, 42, 3, 829-848 (2015) · Zbl 1330.05174 · doi:10.1007/s10801-015-0604-3 |
| [16] | Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry (2002) |
| [17] | Hà, H. T.; Van Tuyl, A., Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Comb, 27, 2, 215-245 (2008) · Zbl 1147.05051 · doi:10.1007/s10801-007-0079-y |
| [18] | Herzog, J., A generalization of the Taylor complex construction, Commun. Algebra, 35, 5, 1747-1756 (2007) · Zbl 1121.13013 · doi:10.1080/00927870601139500 |
| [19] | Herzog, J.; Takayama, Y., Resolutions by mapping cones, Homol. Homotopy Appl., 4, 2, 277-294 (2002) · Zbl 1028.13008 · doi:10.4310/HHA.2002.v4.n2.a13 |
| [20] | Herzog, J.; Vladoiu, M.; Zheng, X., How to compute the Stanley depth of a monomial ideal, J. Algebra, 322, 9, 3151-3169 (2009) · Zbl 1186.13019 · doi:10.1016/j.jalgebra.2008.01.006 |
| [21] | Hirano, A.; Matsuda, K., Matching numbers and dimension edge ideals, Graphs Comb, 37, 3, 761-774 (2021) · Zbl 1470.05128 · doi:10.1007/s00373-021-02277-x |
| [22] | Iqbal, Z.; Ishaq, M.; Aamir, M., Depth and Stanley depth of the edge ideals of square paths and square cycles, Commun. Algebra, 46, 3, 1188-1198 (2018) · Zbl 1421.13004 · doi:10.1080/00927872.2017.1339068 |
| [23] | Ishaq, M., Values and bounds for the Stanley depth, Carpathian J. Math, 27, 2, 217-224 (2011) · Zbl 1249.13017 |
| [24] | Ishaq, M.; Qureshi, M. I., Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings, Commun. Algebra, 41, 3, 1107-1116 (2013) · Zbl 1278.13025 · doi:10.1080/00927872.2011.630708 |
| [25] | Kalai, G.; Meshulam, R., Intersections of Leray complexes and regularity of monomial ideals, J. Comb. Theory, Ser. A, 113, 1586-1592 (2006) · Zbl 1105.13026 |
| [26] | Katzman, M., Characteristic-independence of Betti numbers of graph ideals, J. Comb. Theory, Ser. A, 113, 3, 435-454 (2006) · Zbl 1102.13024 |
| [27] | Morey, S., Depths of powers of the edge ideal of a tree, Commun. Algebra, 38, 11, 4042-4055 (2010) · Zbl 1210.13020 · doi:10.1080/00927870903286900 |
| [28] | Morey, S.; Villarreal, R. H., Edge ideals: Algebraic and combinatorial properties, Prog. Commut. Algebra, 1, 85-126 (2012) · Zbl 1246.13001 |
| [29] | Olteanu, A., Edge ideals of squares of trees, Osaka J. Math, 59, 2, 369-386 (2022) · Zbl 1487.05280 |
| [30] | Rauf, A., Depth and Stanley depth of multigraded modules, Commun. Algebra, 38, 2, 773-784 (2010) · Zbl 1193.13025 · doi:10.1080/00927870902829056 |
| [31] | Stanley, R. P., Linear Diophantine equations and local cohomology, Invent. Math, 68, 2, 175-193 (1982) · Zbl 0516.10009 |
| [32] | Ştefan, A., Stanley depth of powers of the path ideal, arXiv preprint arXiv:1409.6072 (2014) |
| [33] | Uribe-Paczka, M. E.; Van Tuyl, A., The regularity of some families of circulant graphs, Mathematics, 7, 7, 657 (2019) · doi:10.3390/math7070657 |
| [34] | Villarreal, R. H., Monomial Algebras, 238 (2001), New York: Marcel Dekker, Inc, New York · Zbl 1002.13010 |
| [35] | Woodroofe, R., Matchings, coverings, and Castelnuovo-Mumford regularity, J. Commut. Algebra, 6, 2, 287-304 (2014) · Zbl 1330.13040 |
| [36] | Zheng, X., Resolutions of facet ideals, Commun. Algebra, 32, 2, 2301-2324 (2004) · Zbl 1089.13014 · doi:10.1081/AGB-120037222 |
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