Hilbert Functions and Level Algebras

The authors consider certain quotients A = R/I of the polynomial ring R = K[x1,…,xr] over an arbitrary field K. They first determine upper and lower bounds on the Hilbert functions of any algebra having the form A = R/V, where is the largest ideal of R agreeing in degrees at least j with the ideal (V) generated by a vector subspace V Rj of degree j forms: these bounds extend the Macaulay bounds (Theorem 1.4). A level Artinian quotient of A = R/I has socle Soc(A) = (0 : M), M = (x1,…,xr) in a single degree j. The authors next determine the extremal Hilbert functions Hmax(t, j, r) and Hmin(t, j, r) that occur for level graded Artin algebra quotients of R having socle degree j and “type” t = dimK Soc(A), and they describe the extremal strata (Theorem 1.8). They next give a natural upper bound for the Hilbert function of level algebra quotients of the coordinate ring of a punctual subscheme n, n = r − 1, in terms of the Hilbert function H and j, t. This bound was known and known to be sharp in the case is locally Gorenstein and t = 1. Finally, they show that there are no level algebras of Hilbert function T = (1, 3, 4, 5, 6,…,2) (Proposition 2.7). Given that there are smooth punctual subschemes of 2 with Hilbert function H = (1, 3, 4, 5, 6, 6,…), this result shows that the sharpness of the natural upper bound in the case where is Gorenstein and t = 1 does not extend to level algebra quotients of type 2, even when is smooth. The extremality results for the Hilbert functions of level algebras, and some counterexamples, use the extremality theorem of F. H. S. Macaulay [1927, Proc. London Math. Soc.26, 531–555] and G. Gotzmannʹs [1978, Math. Z.158, 61–70] results on the Hilbert scheme.