Date of Award


Document Type


Degree Name

Master of Science (MS)


School of Mathematical and Statistical Sciences

Committee Member

Keri Sather-Wagstaff

Committee Member

James Coykendall

Committee Member

Kevin James


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.



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