## All Theses

#### Title

The Scarf Complex of Weighted Edge Ideals

8-2021

Thesis

#### Degree Name

Master of Science (MS)

#### Department

School of Mathematical and Statistical Sciences

#### Committee Member

Keri Sather-Wagstaff

James Coykendall

Kevin James

#### Abstract

Let $R$ be a commutative ring with identity. A free resolution is a sequence of $R$-module homomorphisms of the form \begin{center} \begin{tikzcd} X = \cdots \arrow[r,"\partial^{\beta_{i+1}}"] & R^{\beta_i} \arrow[r, "\partial_i"] & \cdots \arrow[r,"\partial_2"] & R^{\beta_1} \arrow[r, "\partial_1"] & R^{\beta_0} \arrow[r,"\partial_0"] & 0. \end{tikzcd} \end{center} where $R^{\beta_i}$ is a free $R$-module and $\ker(\partial^X_i) = \im( \partial^X_{i+1})$ for all $i > 0$. When we only have the containment $\im \partial^X_{i+1} \subseteq \ker(\partial^X_i)$ we call $X$ a chain complex. Computing these resolutions in a traditional fashion tends to be rather expensive, though some classes of resolutions can be constructed explicitly.

In 1998, Bayer, Peeva and Sturmfels proved that every labeled simplicial complex has an associated chain complex and gave a combinatorial/topological criterion for the chain complex to be a resolution~\cite{MR1618363}. One important example is the Scarf complex. We will explore when the Scarf complex is a resolution.

COinS