# The Type Defect of a Simplicial Complex

## TL;DR

This paper introduces the 'type defect' invariant for simplicial complexes, revealing its properties, relation to Cohen-Macaulayness, and applications to chordal graphs, Betti number bounds, and higher-dimensional generalizations.

## Contribution

It defines the type defect invariant, explores its properties, and extends chordality and Betti number bounds to higher-dimensional complexes.

## Key findings

- Type defect is well-behaved under gluing complexes.
- Cohen-Macaulay complexes have non-positive type defect.
- Chordal graphs have non-negative type defect for all induced subgraphs.

## Abstract

Fix a field $k$. When $\Delta$ is a simplicial complex on $n$ vertices with Stanley-Reisner ideal $I_\Delta$, we define and study an invariant called the $\textit{type defect}$ of $\Delta$. Except when $\Delta$ is of a single simplex, the type defect of $\Delta$, $\textrm{td}(\Delta)$, is the difference $ \dim_k \textrm{Tor}_c^S(S/ I_\Delta,k) - c$, where $c$ is the codimension of $\Delta$ and $S = k[x_1, \ldots x_n]$. 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, $\Delta$ is Cohen-Macaulay if $\textrm{td}(\Delta) \leq 0$. On the other hand, if $\Delta$ is a simple graph (viewed as a one-dimensional complex), then $\textrm{td}(\Delta') \geq 0$ for every induced subgraph $\Delta'$ of $\Delta$ if and only if $\Delta$ is chordal. Requiring connected induced subgraphs to have type defect zero allows us to define a class of graphs that we call $\textit{treeish}$, 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.