Date of Award
8-2021
Document Type
Thesis
Degree Name
Master of Science (MS)
Department
School of Mathematical and Statistical Sciences
Committee Member
Keri Sather-Wagstaff
Committee Member
James Coykendall
Committee Member
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.
Recommended Citation
Visser, James McKay, "The Scarf Complex of Weighted Edge Ideals" (2021). All Theses. 3600.
https://open.clemson.edu/all_theses/3600