The type defect of a simplicial complex

Abstract

Fix a field k. When Δ is a simplicial complex on n vertices with Stanley–Reisner ideal IΔ, we define and study an invariant called the type defect of Δ. Except when Δ is a single simplex, the type defect of Δ, td(Δ), is the difference dim⁡TorcS(S/IΔ,k)−c, where c is the codimension of Δ and S=k[x1,…xn]. We show that this invariant admits surprisingly nice properties. For example, it is well-behaved when one glues two complexes together along a face. Furthermore, Δ is Cohen–Macaulay if td(Δ)≤0. On the other hand, if Δ is a simple graph (viewed as a one-dimensional complex), then td(Δ′)≥0 for every induced subgraph Δ′ of Δ if and only if Δ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call treeish, and which we generalize to simplicial complexes. We then extend some of our chordality results to higher dimensions, proving sharp lower bounds for most Betti numbers of ideals with linear resolution, and classifying when equalities occur. As an application, we prove sharp lower bounds for Betti numbers of graded ideals (not necessarily monomial) with linear resolution.