Matrix Factorizations With More Than Two Factors

<p>Let $S$ be a regular local ring and $f$ a non-zero non-invertible element of $S$. In this thesis, we study the notion of a matrix factorization of $f$ with $d\ge 2$ factors, that is, we consider tuples of square matrices (\phi₁,\phi₂,\dots,\phi_d), with entries in $S$, such that their product is $f$ times an identity matrix of the appropriate size. These objects have been studied thoroughly in the case $d=2$ and were originally introduced by Eisenbud in his study of free resolutions of modules over hypersurface rings. Many of the results given in this thesis are extensions of well-known results in the $d=2$ case while others give new and unexpected properties which only arise when $d>2$.</p><p>First we investigate the structure of the category of matrix factorizations with $d\ge 2$ factors in Chapter 2. We show that the stable category of $d$-fold matrix factorizations is naturally triangulated and we give an explicit formula for the relevant suspension functor. In Chapters 3 and 4 we give two different module-theoretic descriptions of this category, which turn out to be equivalent under mild assumptions, extending results of Solberg and Kn\"orrer to the case of $d\ge 2$ factors.</p><p>The primary motivation for Chapter 4 is a theorem due to Kn\"orrer which states that the category of $2$-fold matrix factorizations of $f$ has finite representation type if and only if the same is true of f+z² \in S[[ z ]], where $z$ is an indeterminate. We consider an analogue of this statement in the case of the equation f+z^d \in S[[ z ]], $d\ge 2$. In particular, we show that there are, up to isomorphism, only finitely many indecomposable $d$-fold matrix factorizations of $f$ if and only if the hypersurface ring defined by f+z^d has finite Cohen-Macaulay representation type.</p><p>In Chapter 5, we provide a generalization of Eisenbud's fundamental theorem on the connection between matrix factorizations of $f$ and maximal Cohen-Macaulay modules over the hypersurface ring defined by $f$. Namely, we give a correspondence between $d$-fold matrix factorizations of $f$ and sequences of $d-1$ surjective homomorphisms between the aforementioned modules.</p><p>Finally, Chapter 6 contains a formula for a tensor product of $d$-fold matrix factorizations in the sense of Yoshino as well as some criteria for decomposability of the construction.</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2022

Author dc:creator

• Tribone, Tim

Contributors dc:contributor

• Leuschke, Graham J.

• branched cover
• epimorphism category
• hypersurface ring
• matrix factorization
• maximal Cohen-Macaulay modules
• Mathematics
• Physical Sciences and Mathematics

Repository record dc:identifier

https://surface.syr.edu/etd/1436

OAI identifier oai:identifier

oai:surface.syr.edu:etd-2437