On the degree two entry of a Gorenstein \(h\)-vector and a conjecture of Stanley. (English) Zbl 1148.13011
Let \(A = \bigoplus_{i \geq 0}A_i\) denote a standard graded Gorenstein \(k\)-algebra with \(A_0 = k\) a field. Then the Hilbert series \(F(R,t) = \sum_{i \geq 0} \dim_k A_i t^i\) provides the \(h\)-vector \((h_0,h_1,\ldots,h_e)\) of \(A.\) Therefore \(h_0 = h_e = 1\) and \(h_1 = \dim_k A_1 - \dim A =:r.\) Not so much is known about the degree two entry \(h_2.\) For \(r \geq 2\) and \(e \geq 4\) the authors define \(f(r,e)\) the least value of the degree two component \(h_2.\) The main result of the paper is a lower bound on \(f(r,e).\)
This follows by a clever combination of M. Green’s theorem [Algebraic curves and projective geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1389, 76–86 (1989; Zbl 0717.14002)], with the a classical result of Macaulay. In the case of \(e = 4,\) i.e. of the \(h\)-vector \((1,r,h_2,r,1),\) let \(f(r) = f(r,4).\) R. P. Stanley, see [Adv. Math. 28, 57–83 (1978; Zbl 0384.13012)], conjectured that \(\lim f(r)/r^{2/3} = 6^{2/3}.\) This is proved by the authors as an outcome of their lower bound on \(f(r).\) By recent work of M. Boij and the third author, see [Proc. Am. Math. Soc. 135, No. 9, 2713–2722 (2007; Zbl 1144.13011)], it turns out that the lower bound on \(f(r)\) is not sharp.
This follows by a clever combination of M. Green’s theorem [Algebraic curves and projective geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1389, 76–86 (1989; Zbl 0717.14002)], with the a classical result of Macaulay. In the case of \(e = 4,\) i.e. of the \(h\)-vector \((1,r,h_2,r,1),\) let \(f(r) = f(r,4).\) R. P. Stanley, see [Adv. Math. 28, 57–83 (1978; Zbl 0384.13012)], conjectured that \(\lim f(r)/r^{2/3} = 6^{2/3}.\) This is proved by the authors as an outcome of their lower bound on \(f(r).\) By recent work of M. Boij and the third author, see [Proc. Am. Math. Soc. 135, No. 9, 2713–2722 (2007; Zbl 1144.13011)], it turns out that the lower bound on \(f(r)\) is not sharp.
Reviewer: Peter Schenzel (Halle)
MSC:
| 13E10 | Commutative Artinian rings and modules, finite-dimensional algebras |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
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