Hilbert functions over the exterior algebra and f-vectors in higher rank I: Amata-Crupi monomial modules, Kozlov polytopes, and r-vectors of simplicial complexes

Abstract

Let $E$ be the exterior algebra on $n$ generators over a field, and let $F$ be a graded free $E$-module with $r$ generators, of degrees $d_1 \le \dots \le d_r$. We determine the convex hull of the set of Hilbert functions of the quotients $F/M$, where $M$ runs over the monomial submodules of Amata and Crupi. The hull is the Minkowski sum of $r$ shifted copies of one and the same simplex: the Kozlov simplex, which Kozlov proved in 1997 to be the convex hull of the $f$-vectors of simplicial complexes on $n$ vertices. For $r=1$ the statement is Kozlov's theorem, so this is a rank-$r$ generalization of it. We give a self-contained proof, including a short proof of Kozlov's theorem from the local Lubell-Yamamoto-Meshalkin inequality, and an explicit facet description of the Kozlov simplex that appears not to be recorded in the literature. Along the way we isolate the class of submodules for which the argument works, the ideal-direct-sum modules, and show by example that the Hilbert function of a graded submodule outside that class need not be a shifted sum of $f$-vectors at all. The combinatorial content is stated separately, in language that uses no algebra, as a result on $r$-vectors of simplicial complexes and their $ff$-vectors. Finally we describe the vertex structure of the Minkowski sum -- which sums of vertices of the summands survive as vertices -- proving the answer for the all-ones leg vector at every rank and for the Kozlov leg vector at rank two.