Hilbert Functions and Level Algebras

Abstract

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 V̄ 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=dimKSoc(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 OZ of a punctual subscheme Z⊂Pn,n=r−1, in terms of the Hilbert function HZ and j,t. This bound was known and known to be sharp in the case Z 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 Z of P2 with Hilbert function HZ=(1,3,4,5,6,6,…), this result shows that the sharpness of the natural upper bound in the case where Z is Gorenstein and t=1 does not extend to level algebra quotients of type 2, even when Z 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. Gotzmann's [1978, Math. Z.158, 61–70] results on the Hilbert scheme.