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N-Fold Matrix Factorizations
The study of matrix factorizations began when they were introduced by Eisenbud; they have since been an important topic in commutative algebra. Results by Eisenbud, Buchweitz, and Yoshino relate matrix factorizations to maximal Cohen-Macaulay modules over hypersurface rings. There are many important properties of the category of matrix factorizations, as well as tensor product and hom constructions. More recently, Backelin, Herzog, Sanders, and Ulrich used a generalization of matrix factorizations -- so called N-fold matrix factorizations -- to construct Ulrich modules over arbitrary hypersurface rings. In this dissertation we build up the theory of N-fold matrix factorizations, proving analogues of many known properties of the classical setting. We also obtain tensor product and internal hom constructions using a special type of roots of unity and combinatorial results from Heller and Stephan. Finally, we prove generalizations of two of Eisenbud's landmark results for the classical setting in the context of 3-fold matrix factorizations.
Hopkins, Eric, "N-Fold Matrix Factorizations" (2021). ETD collection for University of Nebraska - Lincoln. AAI28713565.