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Hilbert polynomials and minimum Hilbert functions. (English) Zbl 0593.13009
Curves Semin. at Queen’s, Vol. 2, Kingston/ Can. 1981-82, Queen’s Pap. Pure Appl. Math. 61, Exp. F, 21 p. (1982).
[For the entire collection see Zbl 0584.00014.]
In this note I first characterize the Hilbert polynomials of standard G- algebras by using Macaulay’s characterization of the Hilbert functions of such algebras. The equivalence between this characterization and that obtained by R. Hartshorne (by different methods) in Publ. Math., Inst. Hautes Etud. Sci. 29, 5-48 (1966; Zbl 0171.415) is made explicit. Then I note that every Hilbert polynomial of a standard G-algebra is the Hilbert polynomial of a reduced standard G-algebra, namely the homogeneous co-ordinate ring of Hartshorne’s tight fan. I conclude by showing that the Hilbert function of the tight fan is the smallest Hilbert function among those with the same Hilbert polynomial.

13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14C05 Parametrization (Chow and Hilbert schemes)