Date of Award

Spring 5-15-2022

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Leuschke, Graham J.

Keywords

branched cover, epimorphism category, hypersurface ring, matrix factorization, maximal Cohen-Macaulay modules

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

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 $(\phi_1,\phi_2,\dots,\phi_d)$, 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 $d>2$.

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.

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 $f+z^2 \in S[[ z ]]$, 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.

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.

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.

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