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Stanley-Reisner Rings

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Abstract

This article is a try to describe the algebraic side of the story on Stanley-Reisner rings.

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References

  1. Adiprasito, K. A., Björner, A., and Goodarzi, A. Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 12, 3851–3865.

    Article  MathSciNet  MATH  Google Scholar 

  2. Alilooee, A. and Faridi, S. On the resolution of path ideals of cycles, Comm. Algebra 43 (2015), no. 12, 5413–5433.

    Article  MathSciNet  MATH  Google Scholar 

  3. Alilooee, A.; Faridi, S. Graded Betti numbers of path ideals of cycles and lines, J. Algebra Appl. 17 (2018), no. 1, 1850011, 17 pp.

    Google Scholar 

  4. Anick, D. J. A counterexample to a conjecture of Serre, Ann. of Math. (2) 115 (1982), no. 1, 1–33.

    Google Scholar 

  5. Anick, D. J. Comment: ”A counterexample to a conjecture of Serre”, Ann. of Math. (2) 116 (1982), no. 3, 661.

    Google Scholar 

  6. Aramova, A., Avramov, L., and Herzog, J. Resolutions of monomial ideals and cohomology over exterior algebras, Trans. Amer. Math. Soc. 352 (2000), no. 2, 579–594.

    Article  MathSciNet  MATH  Google Scholar 

  7. Aramova, A., De Negri, E., and Herzog, J. Lexsegment ideals with linear resolution, Illinois J. Math. 42 (1998), no. 3, 509–523.

    Article  MathSciNet  MATH  Google Scholar 

  8. Aramova, A., Herzog, J., and Hibi, T. Squarefree lexsegment ideals Math. Z. 228 (1998), 353–378.

    MathSciNet  MATH  Google Scholar 

  9. Aramova, A., Herzog, J., and Hibi, T. Weakly stable ideals, Osaka J. Math. 34 (1997), no. 4, 745–755.

    MathSciNet  MATH  Google Scholar 

  10. Aramova, A., Herzog, J., and Hibi, T. Gotzmann theorems for exterior algebras and combinatorics, J. Algebra 191 (1997), no. 1, 174–211.

    Article  MathSciNet  MATH  Google Scholar 

  11. Aramova, A., Herzog, J., and Hibi, T. Shifting operations and graded Betti numbers, J. Algebraic Combin. 12 (2000), no. 3, 207–222.

    Article  MathSciNet  MATH  Google Scholar 

  12. Backelin, J. Les anneaux locaux a relations monomiales ont des series de Poincare-Betti rationnelles, C. R. Acad. Sci. Paris Ser. I Math. 295 (1982), no. 11, 607–610.

    MathSciNet  MATH  Google Scholar 

  13. K. Baclawski, Cohen-Macaulay ordered sets, J. Algebra 63 (1980), no. 1, 226–258.

    Article  MathSciNet  MATH  Google Scholar 

  14. Banerjee, A., Beyarslan, S. K., and Ha, H.T. Regularity of edge ideals and their powers, Advances in algebra, 17–52, Springer Proc. Math. Stat., 277, Springer, Cham, 2019.

    Google Scholar 

  15. Bayer, D., Peeva, Ir., and Sturmfels, B. Monomial resolutions, Math. Res. Lett. 5 (1998), no. 1-2, 31–46.

    Google Scholar 

  16. Bayer, D. and Stillman, M. A criterion for detecting m-regularity, Invent. Math. 87 (1987), no. 1, 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  17. Bayer, D. and Sturmfels, B. Cellular resolutions of monomial modules, J. Reine Angew. Math. 502 (1998), 123–140.

    Article  MathSciNet  MATH  Google Scholar 

  18. Beyarslan, S., Hà, H., and Trung, T. Regularity of powers of forests and cycles, J. Algebraic Combin. 42 (2015), no. 4, 1077–1095.

    Article  MathSciNet  MATH  Google Scholar 

  19. Björner, A. Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc. 260 (1980), no. 1, 159–183.

    Article  MathSciNet  MATH  Google Scholar 

  20. Björner, A., Garsia, A. M., and Stanley, R. P. An introduction to Cohen-Macaulay partially ordered sets, Ordered sets (Banff, Alta., 1981), pp. 583–615, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., 83, Reidel, Dordrecht-Boston, Mass., 1982.

    Google Scholar 

  21. Björner, A. and Tancer, M. Note: Combinatorial Alexander duality – a short and elementary proof, Discrete Comput. Geom. 42 (2009), no. 4, 586–593.

    Article  MathSciNet  MATH  Google Scholar 

  22. Björner, A. and Wachs, M. On lexicographically shellable posets, Trans. Amer. Math. Soc. 277 (1983), no. 1, 323–341.

    Article  MathSciNet  MATH  Google Scholar 

  23. Björner, A. and Wachs, M. Shellable nonpure complexes and posets, I. Trans. Amer. Math. Soc. 348 (1996), no. 4, 1299–1327.

    Article  MathSciNet  MATH  Google Scholar 

  24. Björner, A. and Wachs, M. Shellable nonpure complexes and posets, II. Trans. Amer. Math. Soc. 349 (1997), no. 10, 3945–3975.

    Article  MathSciNet  MATH  Google Scholar 

  25. Björner, A., Wachs, M., and Welker, V. On sequentially Cohen-Macaulay complexes and posets, Israel J. Math. 169 (2009), 295–316.

    MathSciNet  MATH  Google Scholar 

  26. Bocci, C., Cooper, S., Guardo, E., Harbourne, B., Janssen, M., Nagel, U., Seceleanu, A., Van Tuyl, A., and Vu, T. The Waldschmidt constant for squarefree monomial ideals, J. Algebraic Combin. 44 (2016), no. 4, 875–904.

    Article  MathSciNet  MATH  Google Scholar 

  27. Bouchat, R.R.., Hà, H., and O’Keefe, A. Path ideals of rooted trees and their graded Betti numbers J. Combin. Theory Ser. A 118 (2011), no. 8, 2411–2425.

    Article  MathSciNet  MATH  Google Scholar 

  28. Bouchat, R., Hà, H., O’Keefe, A. Corrigendum to ”Path ideals of rooted trees and their graded Betti numbers”, J. Combin. Theory Ser. A 119 (2012), no. 7, 1610–1611.

    Article  MathSciNet  MATH  Google Scholar 

  29. Bruggesser H. and Mani P. Shellable decompositions of cells and spheres, Math. Scand. 29 (1971), 197–205.

    Article  MathSciNet  MATH  Google Scholar 

  30. Bruns, W. and Herzog, J. Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp.

    Google Scholar 

  31. Bruns, W. and Hibi, T. Cohen-Macaulay partially ordered sets with pure resolutions, European J. Combin. 19 (1998), no. 7, 779–785.

    Article  MathSciNet  MATH  Google Scholar 

  32. Bruns, W. and Hibi, T. Stanley-Reisner rings with pure resolutions, Comm. Algebra 23 (1995), no. 4, 1201–1217.

    Article  MathSciNet  MATH  Google Scholar 

  33. Charalambous, H. and Evans, E. G. Resolutions obtained by iterated mapping cones, J. Algebra 176 (1995), no. 3, 750–754.

    Article  MathSciNet  MATH  Google Scholar 

  34. Conca, A. and De Negri, E. M-sequences, graph ideals, and ladder ideals of linear type, J. Algebra 211 (1999), no. 2, 599–624.

    Article  MathSciNet  MATH  Google Scholar 

  35. Conca, A. and Herzog, J. Castelnuovo-Mumford regularity of products of ideals, Collect. Math. 54 (2003), no. 2, 137–152.

    MathSciNet  MATH  Google Scholar 

  36. Connon, E. and Faridi, S. Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution, J. Combin. Theory Ser. A 120 (2013), no. 7, 1714–1731.

    Article  MathSciNet  MATH  Google Scholar 

  37. Corso, A. and Nagel, U. Monomial and toric ideals associated to Ferrers graphs, Trans. Amer. Math. Soc. 361 (2009), no. 3, 1371–1395

    Article  MathSciNet  MATH  Google Scholar 

  38. Dochtermann, A. and Engström, A. Algebraic properties of edge ideals via combinatorial topology, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 2, 24 pp.

    Google Scholar 

  39. Dress, A. A new algebraic criterion for shellability, Beiträge Algebra Geom. 34 (1993), no. 1, 45–55.

    MathSciNet  MATH  Google Scholar 

  40. Dupont, L. and Villarreal, R. Symbolic Rees algebras, vertex covers and irreducible representations of Rees cones, Algebra Discrete Math. 10 (2010), no. 2, 64–86.

    MathSciNet  MATH  Google Scholar 

  41. Duval, A. M. Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes, Electron. J. Combin. 3 (1996), no. 1, Research Paper 21, approx. 14 pp.

    Google Scholar 

  42. Eagon J.A. and Reiner V. Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra 130 (1998), no. 3, 265–275.

    Article  MathSciNet  MATH  Google Scholar 

  43. Eliahou S. and Kervaire, M. Minimal resolutions of some monomial ideals, J. Algebra 129 (1990), 1–5.

    Article  MathSciNet  MATH  Google Scholar 

  44. Emtander, E. A class of hypergraphs that generalizes chordal graphs, Math. Scand. 106 (2010), no. 1, 50–66.

    Article  MathSciNet  MATH  Google Scholar 

  45. Estrada, M. and Villarreal, R. Cohen-Macaulay bipartite graphs, Arch. Math. (Basel) 68 (1997), no. 2, 124–128

    Article  MathSciNet  MATH  Google Scholar 

  46. Faridi, S. The facet ideal of a simplicial complex, Manuscripta Math. 109 (2002), no. 2, 159–174.

    Article  MathSciNet  MATH  Google Scholar 

  47. Faridi, S. Monomial ideals via square-free monomial ideals, Commutative algebra 85–114, Lect. Notes Pure Appl. Math. 244, Chapman and Hall, Boca Raton, 2006.

    Google Scholar 

  48. Faridi, S. Simplicial trees are sequentially Cohen-Macaulay, J. Pure Appl. Algebra 190 (2004), no. 1-3, 121–136.

    Article  MathSciNet  MATH  Google Scholar 

  49. Faridi, S. Cohen-Macaulay properties of square-free monomial ideals, J. Combin. Theory Ser. A 109 (2005), no. 2, 299–329.

    Article  MathSciNet  MATH  Google Scholar 

  50. Folkman, J. The homology groups of a lattice, J. Math. Mech. 15 (1966) 631–636.

    MathSciNet  MATH  Google Scholar 

  51. Francisco, C. A., Hà, H. T., and Mermin, J. Powers of square-free monomial ideals and combinatorics, Commutative algebra, 373–392, Springer, New York, 2013.

    Google Scholar 

  52. Francisco, C. A., Hà, H. T. and Van Tuyl, A. Splittings of monomial ideals, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3271–3282.

    Article  MathSciNet  MATH  Google Scholar 

  53. Francisco, C., Hà, H., and Van Tuyl, A. Colorings of hypergraphs, perfect graphs, and associated primes of powers of monomial ideals, J. Algebra 331 (2011), 224–242.

    Article  MathSciNet  MATH  Google Scholar 

  54. Francisco, C. and Van Tuyl, A. Some families of componentwise linear monomial ideals, Nagoya Math. J. 187 (2007), 115–156.

    MathSciNet  MATH  Google Scholar 

  55. Francisco, C. A. and Van Tuyl, A. Sequentially Cohen-Macaulay edge ideals, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2327–2337.

    Article  MathSciNet  MATH  Google Scholar 

  56. Fröberg, R. On Stanley-Reisner rings, In S. Balcerzyk et al (eds.) Topics in algebra, Part 2. Polish Scientific Publishers, 1990.

    Google Scholar 

  57. Fröberg, R. Determination of a class of Poincaré series, Math. Scand. 37 (1975), no. 1, 29–39.

    Article  MathSciNet  MATH  Google Scholar 

  58. Fröberg, R. A study of graded extremal rings and of monomial rings, Math. Scand. 51 (1982), no. 1, 22–34.

    Article  MathSciNet  MATH  Google Scholar 

  59. Galligo, A. À propos du théorème de préparation de Weierstrass, Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau), pp. 543–579) Lecture Notes in Math., Vol. 409, Springer, Berlin, 1974.

    Google Scholar 

  60. Garsia, A. M. Combinatorial methods in the theory of Cohen-Macaulay rings, Adv. in Math. 38 (1980), no. 3, 229–266.

    Article  MathSciNet  MATH  Google Scholar 

  61. Gasharov, V., Peeva, I., and Welker, V. The lcm-lattice in monomial resolutions, Math. Res. Lett. 6 (1999), no. 5-6, 521–532.

    Article  MathSciNet  MATH  Google Scholar 

  62. Goto, S., and Takayama, Y. Stanley-Reisner ideals whose powers have finite length cohomologies, Proc. Amer. Math. Soc. 135 (2007), no. 8, 2355–2364.

    Article  MathSciNet  MATH  Google Scholar 

  63. Hà, H. T. Regularity of squarefree monomial ideals, Connections between algebra, combinatorics, and geometry, 251–276, Springer Proc. Math. Stat., 76, Springer, New York, 2014.

    Google Scholar 

  64. Hà, H. T., and Morey, S. Embedded associated primes of powers of square-free monomial ideals, J. Pure Appl. Algebra 214 (2010), no. 4, 301–308.

    Article  MathSciNet  MATH  Google Scholar 

  65. Hà, H. T. and Trung, N. V. Membership criteria and containments of powers of monomial ideals, Acta Math. Vietnam. 44 (2019), no. 1, 117–39.

    Article  MathSciNet  MATH  Google Scholar 

  66. Hà, H.T. and Van Tuyl, A. Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), no. 2, 215–245.

    Article  MathSciNet  MATH  Google Scholar 

  67. Hà, H.T. and Van Tuyl, A. Resolutions of square-free monomial ideals via facet ideals: a survey, Algebra, geometry and their interactions, 91–117,Contemp. Math., 448, Amer. Math. Soc., Providence, RI, 2007.

    Google Scholar 

  68. Hà, H. T. and Van Tuyl, A. Splittable ideals and the resolutions of monomial ideals, J. Algebra 309 (2007), no. 1, 405–425.

    Article  MathSciNet  MATH  Google Scholar 

  69. Hà, H. T. and Woodroofe, R. Results on the regularity of square-free monomial ideals, Adv. in Appl. Math. 58 (2014), 21–36.

    Article  MathSciNet  MATH  Google Scholar 

  70. Hà, H. T. and Woodroofe, R. Corrigendum to ”Results on the regularity of square-free monomial ideals”, Adv. in Appl. Math. 86 (2017), 175–177.

    Article  MathSciNet  MATH  Google Scholar 

  71. He, J. and Van Tuyl, A. Algebraic properties of the path ideal of a tree, Comm. Algebra 38 (2010), no. 5, 1725–1742.

    Article  MathSciNet  MATH  Google Scholar 

  72. Herzog, J. Generic initial ideals and graded Betti numbers, Computational commutative algebra and combinatorics (Osaka, 1999), 75–120, Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, 2002.

    Google Scholar 

  73. Herzog, J. and Hibi, T. Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153.

    MathSciNet  MATH  Google Scholar 

  74. Herzog, J. and Hibi, T. Monomial ideals. Graduate Texts in Mathematics, 260. Springer-Verlag London, Ltd., London, 2011. xvi+305 pp.

    Google Scholar 

  75. Herzog, J. and Hibi, T. Distributive lattices, bipartite graphs and Alexander duality., J. Algebraic Combin. 22 (2005), no. 3, 289–302.

    Article  MathSciNet  MATH  Google Scholar 

  76. Herzog, J. and Hibi, T. Cohen-Macaulay polymatroidal ideals, European J. Combin. 27 (2006), no. 4, 513–517.

    Article  MathSciNet  MATH  Google Scholar 

  77. Herzog, J., Hibi, T., Murai, S., and Takayama, Y. Componentwise linear ideals with minimal or maximal Betti numbers, Ark. Mat. 46 (2008), no. 1, 69–75.

    Article  MathSciNet  MATH  Google Scholar 

  78. Herzog, J., Hibi, T., and Trung, N.V. Symbolic powers of monomial ideals and vertex cover algebras, Adv. Math. 210 (2007), no. 1, 304–322.

    Article  MathSciNet  MATH  Google Scholar 

  79. Herzog, J., Hibi, Ta., and Zheng, X. Monomial ideals whose powers have a linear resolution, Math. Scand. 95 (2004), no. 1, 23–32.

    Google Scholar 

  80. Herzog, J., Hibi, T., and Zheng, X. Cohen-Macaulay chordal graphs, J. Combin. Theory Ser. A 113 (2006), no. 5, 911–916.

    Article  MathSciNet  MATH  Google Scholar 

  81. Herzog, J. and Kühl, M. On the Betti numbers of finite pure and linear resolutions, Commun. Algebra 12 (1984), 1627–1646.

    Article  MathSciNet  MATH  Google Scholar 

  82. Herzog, J. and Popescu, D. Finite filtrations of modules and shellable multicomplexes, Manuscripta Math. 121 (2006), no. 3, 385–410.

    Article  MathSciNet  MATH  Google Scholar 

  83. Herzog, J., Reiner, V., and Welker, V. Componentwise linear ideals and Golod rings, Michigan Math. J. 46 (1999), no. 2, 211–223.

    Article  MathSciNet  MATH  Google Scholar 

  84. Herzog, J. and Takayama, Y. Resolutions by mapping cones, The Roos Festschrift volume, 2. Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277–294.

    Google Scholar 

  85. Hibi, T. Buchsbaum complexes with linear resolutions, J. Algebra 179 (1996), no. 1, 127–136.

    Article  MathSciNet  MATH  Google Scholar 

  86. Hochster, M. Cohen-Macaulay rings, combinatorics, and simplicial complexes, Ring theory, II (Proc. Second Conf., Univ. Oklahoma, Norman, Oklahoma, (1975), 171–223.

    Google Scholar 

  87. Jahan, A. S. Prime filtrations of monomial ideals and polarizations, J. Algebra 312 (2007), no. 2, 1011–1032.Ki-SM

    Google Scholar 

  88. Jöllenbeck, M. and Welker, V. Minimal resolutions via algebraic discrete Morse theory, Mem. Amer. Math. Soc. 197 (2009), no. 923, vi+74 pp.

    Google Scholar 

  89. Kalai, G. Algebraic shifting, Computational commutative algebra and combinatorics (Osaka, 1999), 121–163, Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, 2002.

    Google Scholar 

  90. Khosh-Ahang, F. and Moradi, S. Regularity and projective dimension of the edge ideal of C5-free vertex decomposable graphs, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1567–1576.

    Article  MathSciNet  MATH  Google Scholar 

  91. Kiani, D. and Saeedi Madani, S. The edge ideals of complete multipartite hypergraphs, Comm. Algebra 43 (2015), no. 7, 3020–3032.

    Article  MathSciNet  MATH  Google Scholar 

  92. Kind, B. and Kleinschmidt, P. Schälbare Cohen-Macauley-Komplexe und ihre Parametrisierung, (German) Math. Z. 167 (1979), no. 2, 173–179.

    MathSciNet  MATH  Google Scholar 

  93. Lyubeznik, G. A new explicit finite free resolution of ideals generated by monomials in an R-sequence, J. Pure Appl. Algebra 51 (1988), no. 1-2, 193–195.

    Article  MathSciNet  MATH  Google Scholar 

  94. Lyubeznik, G. On the arithmetical rank of monomial ideals, J. Algebra 112 (1988), no. 1, 86–89.

    Article  MathSciNet  MATH  Google Scholar 

  95. Mahmoudi,, M. Mousivand, A., Crupi, M., Rinaldo, G., Terai, N., and Yassemi, S. Vertex decomposability and regularity of very well-covered graphs, J. Pure Appl. Algebra 215 (2011), no. 10, 2473–2480.

    Article  MathSciNet  MATH  Google Scholar 

  96. Mermin, J. Three simplicial resolutions, Progress in commutative algebra 1, 127–141, de Gruyter, Berlin, 2012.

    Google Scholar 

  97. Miller, E., and Sturmfels, B. Combinatorial commutative algebra, Graduate Texts in Mathematics, 227. Springer-Verlag, New York, 2005. xiv+417 pp.

    Google Scholar 

  98. Minh, N. C. and Trung, N. Vi. Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals, Adv. Math. 226 (2011), no. 2, 1285–1306.

    Google Scholar 

  99. Minh, N. C. and Trung, N.V. Corrigendum to ”Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals” , Adv. Math. 228 (2011), no. 5, 2982–2983.

    Article  MathSciNet  MATH  Google Scholar 

  100. Moradi, S. and Khosh-Ahang, F. On vertex decomposable simplicial complexes and their Alexander duals, Math. Scand. 118 (2016), no. 1, 43–56.

    Article  MathSciNet  MATH  Google Scholar 

  101. Morales, M., Nasrollah Nejad, A., Yazdan Pour, A.A., and Zaare-Nahandi, R. Monomial ideals with 3-linear resolutions, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014), no. 4, 877–891.

    Google Scholar 

  102. Morey, S. and Villarreal, R. Edge ideals: algebraic and combinatorial properties, Progress in commutative algebra 1, 85–126, de Gruyter, Berlin, 2012.

    Google Scholar 

  103. Murai, S. and Hibi, T. Algebraic shifting and graded Betti numbers, Trans. Amer. Math. Soc. 361 (2009), no. 4, 1853–1865.

    Article  MathSciNet  MATH  Google Scholar 

  104. Nagel, U. and Reiner, V. Betti numbers of monomial ideals and shifted skew shapes, Electron. J. Combin. 16 (2009), no. 2, Special volume in honor of Anders Björner, Research Paper 3, 59 pp.

    Google Scholar 

  105. Nagel, U., Römer, T., and Vinai, N.P. Algebraic shifting and exterior and symmetric algebra methods, Comm. Algebra 36 (2008), no. 1, 208–231.

    Article  MathSciNet  MATH  Google Scholar 

  106. Peeva, I. Graded syzygies, Algebra and Applications, 14. Springer-Verlag London, Ltd., London, 2011. xii+302 pp.

    Google Scholar 

  107. Pournaki, M. R., Seyed Fakhari, S. A., Terai, N., and Yassemi, S. Survey article: Simplicial complexes satisfying Serre’s condition: a survey with some new results, J. Commut. Algebra 6 (2014), no. 4, 455–483.

    Article  MathSciNet  MATH  Google Scholar 

  108. Reisner, G.A. Cohen-Macaulay quotients of polynomial rings, Advances in Math. 21 (1976), no. 1, 30–49.

    Article  MathSciNet  MATH  Google Scholar 

  109. Rinaldo, G., Terai, N., and Yoshida, K. Cohen-Macaulayness for symbolic power ideals of edge ideals, J. Algebra 347 (2011), 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  110. Römer, T. Homological properties of bigraded algebras, Illinois J. Math. 45 (2001), no. 4, 1361–1376.

    Article  MathSciNet  MATH  Google Scholar 

  111. Serre, J.-P. Algebre locale. Multiplicities Cours au College de France, 1957–1958, redige par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, 11 Springer-Verlag, Berlin-New York 1965 vii+188 pp.

    Google Scholar 

  112. Saeedi Madani, S., Kiani, D., and Terai, N. Sequentially Cohen-Macaulay path ideals of cycles, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 54 (102) (2011), no. 4, 353–363.

    Google Scholar 

  113. Sharifan, L. and Varbaro, M. Graded Betti numbers of ideals with linear quotients, Matematiche (Catania) 63 (2008), no. 2, 257–265 (2009).

    Google Scholar 

  114. Simis, A., Vasconcelos, W., and Villarreal, R. On the ideal theory of graphs, J. Algebra 167 (1994), no. 2, 389–416

    Article  MathSciNet  MATH  Google Scholar 

  115. Stanley, R.P. The upper bound conjecture and Cohen-Macaulay rings, Studies in Appl. Math. 54 no.2, (1975) 135–142.

    Article  MathSciNet  MATH  Google Scholar 

  116. Stanley, R.P. Combinatorics and commutative algebra, Second edition, Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. x+164 pp.

    Google Scholar 

  117. Stanley, R. P., Cohen-Macaulay complexes, Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, (1976), pp. 51–62. NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., 31 Reidel, Dordrecht, 1977.

    Google Scholar 

  118. Stanley, R. P. Balanced Cohen-Macaulay complexes, Trans. Amer. Math. Soc. 249 (1979), no. 1, 139–157.

    Article  MathSciNet  MATH  Google Scholar 

  119. Stanley, R. Cohen-Macaulay rings and constructible polytopes, Bull. Amer. Math. Soc. 81 (1975), 133–135.

    Article  MathSciNet  MATH  Google Scholar 

  120. Stückrad, J. and Vogel, W. Buchsbaum rings and applications. An interaction between algebra, geometry and topology, Springer-Verlag, Berlin, 1986. 286 pp.

    Google Scholar 

  121. Taylor, D. K. Ideals generated by monomials in an R-sequence, Thesis (Ph.D.)–The University of Chicago. 1966. ProQuest LLC

    Google Scholar 

  122. Terai, N. Alexander duality in Stanley-Reisner rings, Affine algebraic geometry, 449–462, Osaka Univ. Press, Osaka, 2007.

    Google Scholar 

  123. Terai, N. and Hibi, T. Alexander duality theorem and second Betti numbers of Stanley-Reisner rings, Adv. Math. 124 (1996), no. 2, 332–333.

    Article  MathSciNet  MATH  Google Scholar 

  124. Terai, N. and Trung, N.V. Cohen-Macaulayness of large powers of Stanley-Reisner ideals, Adv. Math. 229 (2012), no. 2, 711–730.

    Article  MathSciNet  MATH  Google Scholar 

  125. Terai, N. and Yoshida, K. Buchsbaum Stanley-Reisner rings and Cohen-Macaulay covers, Comm. Algebra 34 (2006), no. 7, 2673–2681.

    Article  MathSciNet  MATH  Google Scholar 

  126. Terai, N. and Yoshida, K. Locally complete intersection Stanley-Reisner ideals, Illinois J. Math. 53 (2009), no. 2, 413–429.

    Article  MathSciNet  MATH  Google Scholar 

  127. Trung, N. V. and Tuan, T. M. Equality of ordinary and symbolic powers of Stanley-Reisner ideals, J. Algebra 328 (2011), 77–93.

    Article  MathSciNet  MATH  Google Scholar 

  128. Van Tuyl, A. A beginner’s guide to edge and cover ideals, in Monomial ideals, computations and applications, 63–94, Lecture Notes in Math. 2083, Springer, Heidelberg, 2013.

    Google Scholar 

  129. Van Tuyl, A. Edge ideals using Macaulay2, Monomial ideals, computations and applications, 95–105, Lecture Notes in Math. 2083, Springer, Heidelberg, 2013.

    Google Scholar 

  130. Van Tuyl, A. Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity, Arch. Math. (Basel) 9 (2009), no. 5, 451–459.

    Article  MathSciNet  MATH  Google Scholar 

  131. Van Tuyl, A. and Villarreal, R.H. Shellable graphs and sequentially Cohen-Macaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), no. 5, 799–814.

    Article  MathSciNet  MATH  Google Scholar 

  132. Varbaro, M. Symbolic powers and matroids, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2357–2366.

    Article  MathSciNet  MATH  Google Scholar 

  133. Villarreal, R. Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), no. 3, 277–293.

    Article  MathSciNet  MATH  Google Scholar 

  134. Villarreal, R. Monomial algebras., Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015.

    MATH  Google Scholar 

  135. Villarreal, R. Rees algebras of edge ideals, Comm. Algebra 23 (1995), no. 9, 3513–3524.

    Article  MathSciNet  MATH  Google Scholar 

  136. Villarreal, R. Unmixed bipartite graphs, Rev. Colombiana Mat. 41 (2007), no. 2, 393–395.

    MathSciNet  MATH  Google Scholar 

  137. Woodroofe, R. Chordal and sequentially Cohen-Macaulay clutters, Electron. J. Combin. 18 (2011), no. 1, Paper 208, 20 pp.

    Google Scholar 

  138. Woodroofe, R. Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3235–3246

    Article  MathSciNet  MATH  Google Scholar 

  139. Zheng, X. Resolutions of facet ideals, Comm. Algebra 32 (2004), no. 6, 2301–2324

    Article  MathSciNet  MATH  Google Scholar 

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  1. Matematiska Institutionen, Stockholms Universitet, Stockholm, Sweden

    Ralf Fröberg

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  1. Department of Mathematics, Cornell University, Ithaca, NY, USA

    Irena Peeva

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Fröberg, R. (2021). Stanley-Reisner Rings. In: Peeva, I. (eds) Commutative Algebra. Springer, Cham. https://doi.org/10.1007/978-3-030-89694-2_10

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