Abstract
Let Δ be a simplicial complex on V = {x 1, . . . , x n }, with Stanley–Reisner ideal \({I_{\Delta}\subseteq R=k[x_1,\ldots, x_n]}\) . The goal of this paper is to investigate the class of artinian algebras \({A=A(\Delta,a_1,\ldots,a_n)= R/(I_{\Delta},x_1^{a_1},\ldots,x_n^{a_n})}\) , where each a i ≥ 2. By utilizing the technique of Macaulay’s inverse systems, we can explicitly describe the socle of A in terms of Δ. As a consequence, we determine the simplicial complexes, that we will call levelable, for which there exists a tuple (a 1, . . . , a n ) such that A(Δ, a 1, . . . , a n ) is a level algebra.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Aramova A., Herzog J., Hibi T.: Squarefree lexsegment ideals. Math. Z. 228, 353–378 (1998)
Boij, M.: Artin Level Algebras. Doctoral Dissertation. Royal Institute of Technology (KTH), Stockholm (1994)
Bruns W., Herzog J.: Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993)
Cocoa Team, CoCoA: a system for doing Computations in Commutative Algebra. Available at http://cocoa.dima.unige.it
Faridi S.: The facet ideal of a simplicial complex. Manuscripta Math. 109, 159–174 (2002)
Geramita, A.V.: Inverse systems of fat points: waring’s problem, secant varieties and veronese varieties and parametric spaces of Gorenstein ideals. In: Queen’s Papers in Pure and Applied Mathematics, No. 102. The Curves Seminar at Queen’s, vol. X, pp. 3–114 (1996)
Geramita, A.V., Harima, T., Migliore, J., Shin, Y.S.: The Hilbert function of a level algebra. Mem. Am. Math. Soc. 186(872) vi+139 pp (2007)
Herzog J., Hibi T.: Upper bounds for the number of facets of a simplicial complex. Proc. Am. Math. Soc. 125, 1579–1583 (1997)
Herzog J., Hibi T.: Level rings arising from meet-distributive meet-semilattices. Nagoya Math. J. 181, 29–39 (2006)
Herzog J., Hibi T., Vladoiu M.: Ideals of fiber type and polymatroids. Osaka J. Math. 42(4), 807–829 (2005)
Hibi T.: Level rings and algebras with straightening laws. J. Algebra 117(2), 343–362 (1988)
Hibi T.: Plane graphs and Cohen-Macaulay posets. Eur. J. Combin. 10(6), 537–542 (1989)
Iarrobino, A., Kanev, V.: Power sums, Gorenstein algebras, and determinantal loci. Springer Lectures Notes in Mathematics, No. 1721. Springer, Heidelberg (1999)
Klivans, A.R., Shpilka, A.: Learning arithmetic circuits via partial derivatives. In: Proc. 16th Annual Conference on Computational Learning Theory, Morgan Kaufmann Publishers, pp. 463–476 (2003)
Nisam, N., Wigderson, A.: Lower bounds on arithmetic circuits via partial derivatives. Comput. Complexity 6(3), 217–234 (1996/1997)
Simis A., Vasconcelos W., Villarreal R.: On the ideal theory of graphs. J. Algebra 167, 389–416 (1994)
Stanley, R.: Cohen-Macaulay complexes, In: Higher combinatorics (Proc. NATO Advanced Study Inst., Berlin, 1976). NATO Adv. Study Inst. Ser., Ser. C: Math. and Phys. Sci., vol. 31, pp. 51–62. Reidel, Dordrecht (1977)
Stanley R.: Hilbert functions of graded algebras. Adv. Math. 28, 57–83 (1978)
Stanley, R.: Combinatorics and commutative algebra, 2nd edn. Progress in Mathematics, vol. 41. Birkhäuser Boston, Inc., Boston (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Van Tuyl, A., Zanello, F. Simplicial complexes and Macaulay’s inverse systems. Math. Z. 265, 151–160 (2010). https://doi.org/10.1007/s00209-009-0507-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-009-0507-x
Keywords
- Simplicial complex
- Macaulay’s inverse systems
- Stanley–Reisner ideal
- Monomial algebra
- Level algebra
- Socle-vector
- Edge ideal