Stanley’s theorem on codimension 3 Gorenstein ℎ-vectors

Zanello, Fabrizio

doi:10.1090/S0002-9939-05-08276-6

Stanley’s theorem on codimension 3 Gorenstein $h$-vectors
HTML articles powered by AMS MathViewer

by Fabrizio Zanello PDF

Proc. Amer. Math. Soc. 134 (2006), 5-8 Request permission

Abstract:

In this note we supply an elementary proof of the following well-known theorem of R. Stanley: the $h$-vectors of Gorenstein algebras of codimension 3 are SI-sequences, i.e. are symmetric and the first difference of their first half is an $O$-sequence.

References

David Bernstein and Anthony Iarrobino, A nonunimodal graded Gorenstein Artin algebra in codimension five, Comm. Algebra 20 (1992), no. 8, 2323–2336. MR 1172667, DOI 10.1080/00927879208824466
Mats Boij, Graded Gorenstein Artin algebras whose Hilbert functions have a large number of valleys, Comm. Algebra 23 (1995), no. 1, 97–103. MR 1311776, DOI 10.1080/00927879508825208
Mats Boij and Dan Laksov, Nonunimodality of graded Gorenstein Artin algebras, Proc. Amer. Math. Soc. 120 (1994), no. 4, 1083–1092. MR 1227512, DOI 10.1090/S0002-9939-1994-1227512-2
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
Y.H. Cho and A. Iarrobino: Inverse systems of zero-dimensional schemes in $\textbf {P}^n$, J. of Algebra, to appear.
Tadahito Harima, Characterization of Hilbert functions of Gorenstein Artin algebras with the weak Stanley property, Proc. Amer. Math. Soc. 123 (1995), no. 12, 3631–3638. MR 1307527, DOI 10.1090/S0002-9939-1995-1307527-7
A. Iarrobino and H. Srinivasan: Some Gorenstein Artin algebras of embedding dimension four, I: components of $PGOR(H)$ for $H=(1,4,7,...,1)$, J. of Pure and Applied Algebra, to appear.
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
J. Migliore and U. Nagel, Reduced arithmetically Gorenstein schemes and simplicial polytopes with maximal Betti numbers, Adv. Math. 180 (2003), no. 1, 1–63. MR 2019214, DOI 10.1016/S0001-8708(02)00079-8
Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57–83. MR 485835, DOI 10.1016/0001-8708(78)90045-2
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579