Extremal properties of Hilbert functions (Q1392974)

Macaulay characterized Hilbert functions of graded quotients of polynomial rings. Kruskal and Katona have characterized \(f\)-vectors of simplicial complexes or, equivalently, Hilbert functions of graded quotients of the exterior algebra. Gotzmann's persistence theorem states that a space of \(d\)-forms, which is extremal in Macaulay's theorem, generates an extremal space of \((d+1)\)-forms. Aramova, Herzog and Hibi have proved a persistence theorem for exterior algebras. Green characterized Hilbert functions of rings when moding out by a general linear form in a ring with given Hilbert function. In this paper the theorems mentioned above are generalized to modules over the polynomial ring or the exterior algebra.