Homogeneous complete intersection Hodge algebras on simplicial complexes (Q1343887)

Let \(\Delta\) be a simplicial complex, \(k\) a field, and \(k[ \Delta]\) the Stanley-Reisner ring of \(\Delta\). A \(k\)-algebra \(R\) is called a Hodge algebra if: (H0) The vertex set of \(\Delta\) is a poset with respect to some ordering \(\leq\); (H1) \(R\) admits as \(k\)-basis the set of nonzero monomials in \(k[ \Delta]\) (the standard monomials); (H2) If \(L\) is a generator for the defining ideal of \(k[ \Delta]\) and \(L= \sum a_ i M_ i\) is the unique linear combination in standard monomials, there exists \(y\in \text{Supp } M_ i\) such that \(y<x\). The author classifies the simplicial complexes on which there exists a homogeneous complete intersection Hodge algebra of dimension \(\leq 3\). A homogeneous Hodge algebra has the same Hilbert series as its associated Stanley-Reisner ring. The Hilbert series of homogeneous complete intersections are well known, and the Hilbert series of a Stanley-Reisner ring is easily expressed in the \(f\)-vector of \(\Delta\). This gives a necessary condition. Examples are provided for the two possible cases in dimension \(=1\), the five cases in dimension \(=2\), and 25 cases in dimension \(=3\), which gives sufficiency.