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.
","abstract_html":"<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,\\phid), 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+zd \\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+zd 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>","abstract_has_math":true,"creators":["Tribone, Tim"],"institution":null,"degree_name":"Doctor of Philosophy (PhD)","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Leuschke, Graham J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2022,"date_issued":"2022-05-15T07:00:00Z","date_published":"2022-05-15T07:00:00Z","updated_at":"2026-07-24T04:56:14Z","subjects":["branched cover","epimorphism category","hypersurface ring","matrix factorization","maximal Cohen-Macaulay modules","Mathematics","Physical Sciences and Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://surface.syr.edu/etd/1436","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Leuschke, Graham J."]},{"key":"dc:creator","label":"Author","values":["Tribone, Tim"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["branched cover","epimorphism category","hypersurface ring","matrix factorization","maximal Cohen-Macaulay modules","Mathematics","Physical Sciences and Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://surface.syr.edu/etd/1436"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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.
"]},{"key":"dc:title","label":"Title","values":["Matrix Factorizations With More Than Two Factors"]}]}],"canonical_facts":{"dc:contributor":["Leuschke, Graham J."],"dc:creator":["Tribone, Tim"],"dc:description.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.
"],"dc:identifier":["https://surface.syr.edu/etd/1436"],"dc:subject":["branched cover","epimorphism category","hypersurface ring","matrix factorization","maximal Cohen-Macaulay modules","Mathematics","Physical Sciences and Mathematics"],"dc:title":["Matrix Factorizations With More Than Two Factors"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-24T04:56:14Z"}