An algebraic approach to finite projective planes

详细信息    查看全文

• 作者：David Cook II ; Juan Migliore ; Uwe Nagel…
• 关键词：Finite projective plane ; Linear space ; Weak Lefschetz property ; Strong Lefschetz property ; Minimal free resolution ; Monomial algebra ; Level algebra ; Stanley–Reisner ring ; Inverse system
• 刊名：Journal of Algebraic Combinatorics
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：43
• 期：3
• 页码：495-519
• 全文大小：500 KB
• 参考文献：1.Björner, A.: Nonpure shellability, \(f\) -vectors, subspace arrangements and complexity. In: Formal Power Series and Algebraic Combinatorics (1994); DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24 (1994), 25–54
  2.Boij, M., Migliore, J., Miró-Roig, R.M., Nagel, U., Zanello, F.: On the shape of a pure O-sequence, Mem. Amer. Math. Soc. 218(1024) (2012)
  3.Chen, C., Guo, A., Jin, X., Liu, G.: Trivariate monomial complete intersections and plane partitions. J. Commut. Algebra 3(4), 459–490 (2011)MathSciNet CrossRef MATH
  4.Cook II, D.: The Lefschetz properties of monomial complete intersections in positive characteristics. J. Algebra 369, 42–58 (2012)MathSciNet CrossRef MATH
  5.Cook II, D., Nagel, U.: The Weak Lefschetz property, monomial ideals, and lozenges. Ill. J. Math. 55(1), 377–395 (2011)MathSciNet MATH
  6.Cook II, D., Nagel, U.: Enumerations of lozenge tilings, lattice paths, and perfect matchings and the weak Lefschetz property, preprint. arXiv:​1305.​1314
  7.Eagon, J., Reiner, V.: Resolutions of Stanley–Reisner rings and Alexander duality. J. Pure Appl. Algebra 130(3), 265–275 (1998)MathSciNet CrossRef MATH
  8.Geramita, A.V.: Inverse Systems of Fat Points: Waring’s Problem, Secant Varieties of Veronese Varieties and Parameter Spaces for Gorenstein Ideals, The Curves Seminar at Queen’s, Vol. X, Queen’s Papers in Pure and Applied Mathematics no. 102 (1996)
  9.Gould, H.W.: Combinatorial Identities: Table I: Intermediate Techniques for Summing Finite Series. Available at http://​www.​math.​wvu.​edu/​~gould/​Vol.​4.​PDF . See also: Combinatorial Identities. A standardized set of tables listing 500 binomial coefficient summations, Henry W. Gould, Morgantown, W.Va., (1972)
  10.Harima, T., Migliore, J., Nagel, U., Watanabe, J.: The weak and strong Lefschetz properties for artinian \(K\) -Algebras. J. Algebra 262, 99–126 (2003)MathSciNet CrossRef MATH
  11.Herzog, J., Hibi, T.: Monomial ideals, Graduate Texts in Mathematics, vol. 260. Springer, London (2011)CrossRef MATH
  12.Iarrobino, A., Kanev, V.: Power sums, Gorenstein algebras, and determinantal loci. Lecture Notes in Mathematics, vol. 1721. Springer, Heidelberg (1999)MATH
  13.Kustin, A., Vraciu, A.: The Weak Lefschetz property for monomial complete intersections in positive characteristic. Trans. Am. Math. Soc. 366, 4571–4601 (2014)MathSciNet CrossRef MATH
  14.Kåhrström, J.: On Projective Planes. http://​kahrstrom.​com/​mathematics/​documents/​OnProjectivePlan​es
  15.Li, J., Zanello, F.: Monomial complete intersections, the Weak Lefschetz property and plane partitions. Discrete Math. 310(24), 3558–3570 (2010)MathSciNet CrossRef MATH
  16.Mermin, J., Peeva, I., Stillman, M.: Ideals containing the squares of the variables. Adv. Math. 217, 2206–2230 (2008)MathSciNet CrossRef MATH
  17.Migliore, J., Miró-Roig, R., Nagel, U.: Monomial ideals, almost complete intersections and the Weak Lefschetz property. Trans. Am. Math. Soc. 363, 229–257 (2011)MathSciNet CrossRef MATH
  18.Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra, Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)
  19.Moorhouse, G.E.: Incidence Geometry, course notes. http://​www.​uwyo.​edu/​moorhouse/​handouts/​incidence_​geometry
  20.Provan, J.S., Billera, L.J.: Decompositions of simplicial complexes related to diameters of convex polyhedra. Math. Oper. Res. 5, 576–594 (1980)MathSciNet CrossRef MATH
  21.Stanley, R.: Weyl groups, the hard Lefschetz theorem, and the Sperner property. SIAM J. Algebr. Discrete Methods 1, 168–184 (1980)MathSciNet CrossRef MATH
  22.Stanley, R.: Combinatorics and Commutative Algebra. Progress in Mathematics, vol. 41, 2nd edn. Birkhäuser, Boston (1996)MATH
  
• 作者单位：David Cook II (1)
  Juan Migliore (2)
  Uwe Nagel (3)
  Fabrizio Zanello (4)
  
  1. Department of Mathematics and Computer Science, Eastern Illinois University, Charleston, IL, 61920, USA
  2. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA
  3. Department of Mathematics, University of Kentucky, Lexington, KY, 40506, USA
  4. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Combinatorics
  Convex and Discrete Geometry
  Order, Lattices and Ordered Algebraic Structures
  Computer Science, general
  Group Theory and Generalizations
  
• 出版者：Springer U.S.
• ISSN：1572-9192

文摘

A finite projective plane, or more generally a finite linear space, has an associated incidence complex that gives rise to two natural algebras: the Stanley–Reisner ring \(R/I_\Lambda \) and the inverse system algebra \(R/I_\Delta \). We give a careful study of both of these algebras. Our main results are a full description of the graded Betti numbers of both algebras in the more general setting of linear spaces (giving the result for the projective planes as a special case), and a classification of the characteristics in which the inverse system algebra associated to a finite projective plane has the weak or strong Lefschetz Property.