The unimodality of pure \(O\)-sequences of type two in four variables. (English) Zbl 1332.05157

Summary: Since the 1970’s, great interest has been taken in the study of pure \(O\)-sequences, which, due to Macaulay’s theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay’s theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into two classes of ideals, prove that they are strictly unimodal and show that one of the classes is licci.


05E40 Combinatorial aspects of commutative algebra
13C40 Linkage, complete intersections and determinantal ideals
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
Full Text: DOI Euclid


[1] M. Boij, J. Migliore, R. Miró-Roig, U. Nagel and F. Zanello, On the shape of a pure \(O\)-sequence , Mem. Amer. Math. Soc. 218 (2012), no. 2024. · Zbl 1312.13001
[2] H. Brenner and A. Kaid, Syzygy Bundles on \(\mathbb{P}\) and the weak Lefschetz Ppoperty , Illinois J. Math. 51 (2007), 1299-1308. · Zbl 1148.13007
[3] M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes , Trans. Amer. Math. Soc. 349 (1997), 3925-3943. · Zbl 0889.52013
[4] A.V. Geramita, Inverse systems of fat points : Waring’s problem, secant varieties and Veronese varieties and parametric spaces of Gorenstein ideals , Queen’s Papers Pure Appl. Math. 102 , 3-114. · Zbl 0864.14031
[5] A.V. Geramita, T. Harima, J. Migliore and Y.S. Shin, The Hilbert function of a level algebra , Mem. Amer. Math. Soc. 186 (2007), No. 872. · Zbl 1121.13019
[6] T. Harima, J. Migliore, U. Nagel and J. Watanabe, The weak and strong Lefschetz properties for Artinian \(K\)-algebras , J. Alg. 262 (2003), 99-126. · Zbl 1018.13001
[7] T. Hausel, Quaternionic geometry of matroids , Cent. Europ. J. Math. 3 (2005), 26-38. · Zbl 1079.52009
[8] J. Herzog and T. Hibi, Monomial ideals , Springer-Verlag, London 2011. · Zbl 1206.13001
[9] T. Hibi, What can be said about pure \(O\)-sequences? , J. Comb. Theor. 50 (1989), 319-322. · Zbl 0667.05004
[10] C. Huneke, and B. Ulrich, Liaison of monomial ideals , Bull. Lond. Math. Soc. 39 (2007), 384-392. · Zbl 1114.13011
[11] A. Iarrobino and V. Kanev, Power sums, Gorenstein algebras, and determinantal loci , Lect. Notes Math. 1721 , Springer-Verlag, Berlin, 1999. · Zbl 0942.14026
[12] P. Kaski, P. Östergård and R.J. Patric, Classification algorithms for codes and designs , Algor. Comp. Math. 15 , Springer-Verlag, Berlin, 2006. · Zbl 1089.05001
[13] J. Kleppe, R. Miró-Roig, J. Migliore, U. Nagel and C. Peterson, Gorenstein liaison, complete intersection liaison invariants and unobstructedness , Mem. Amer. Soc. 154 (2001), No. 732. · Zbl 1006.14018
[14] C. Merino, S.D. Noble, M. Ramírez-Ibáñez and R. Villarroel-Flores, On the structure of the \(h\) -vector of a paving matroid, Europ. J. Comb. 33 (2012), 1787-1799. · Zbl 1248.05033
[15] J. Migliore, Introduction to liaison theory and deficiency modules , Birkhäuser Boston, Progr. Math. 165 , 1998. · Zbl 0921.14033
[16] S. Oh, Generalized permutohedra, \(h\)-vectors of cotransversal matroids and pure \(O\)-sequences , preprint. Available on the arXiv at http://arxiv.org/abs/1005.5586. · Zbl 1295.52017
[17] L. Reid, L. Roberts and M. Roitman, On complete intersections and their Hilbert functions , Canad. Math. Bull. 34 (1991), 525-535. · Zbl 0757.13005
[18] J. Schweig, On the \(h\)-vector of a lattice path matroid , Electr. J. Comb. 17 (2010), N3. · Zbl 1267.05057
[19] R. Stanley, Cohen-Macaulay complexes , in Higher combinatorics , M. Aigner, ed., Reidel, Dordrecht, 1977.
[20] R. Stanley, Weyl groups , The hard Lefschetz theorem, and the Sperner property , SIAM J. Alg. Discr. Meth. 1 (1980), 168-184. · Zbl 0502.05004
[21] E. Stokes, The \(h\)-vectors of \(1\)-dimensional matroid complexes and a conjecture of Stanley , preprint. Available on the arXiv at http://arxiv.org/abs/0903.3569. · Zbl 06936362
[22] J. Watanabe, The Dilworth number of Artinian rings and finite posets with rank function , Comm. Alg. Combin. Adv. Stud. Pure Math. 11 , Kinokuniya Co. North Holland, Amsterdam, 1987. · Zbl 0648.13010
[23] F. Zanello, A non-unimodal codimesion three level \(h\)-vector , J. Alg. 305 (2006), 949-956. · Zbl 1110.13012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.