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Abstract Fix a field k. When Δ is a simplicial complex on n vertices with Stanley–Reisner ideal I Δ , 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 ⁡ Tor c S (S / I Δ , k) − c , where c is the codimension of Δ and S = k [ x 1 , ... x n ]. 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. [ABSTRACT FROM AUTHOR]

Abstract: We construct several pairwise-incomparable bounds on the projective dimensions of edge ideals. Our bounds use combinatorial properties of the associated graphs. In particular, we draw heavily from the topic of dominating sets. Through Hochsterʼs Formula, we recover and strengthen existing results on the homological connectivity of graph independence complexes. [Copyright &y& Elsevier]