My research is in commutative algebra. I am mainly interested in maximal Cohen-Macaulay (MCM) modules over Cohen-Macaulay rings. In particular, my thesis is focused on studying matrix factorizations and MCM modules over hypersurface rings. If you are not familiar with the concept of a matrix factorization, here is a short explanation. They are an amazingly simple yet powerful tool originating in Eisenbud’s study of free resolutions of modules over hypersurface rings. Lately, more and more of my attention has been aimed at understanding their role in the representation theory of hypersurface rings.
In this paper, Graham and I study the connection between the d-fold branched cover of a hypersurface ring and the category of matrix factorizations with d factors. We show that, from a representation theory perspective, these ideas are closely related. Namely, the category of MCM modules over the branched cover is representation finite if and only if the same is true of the category of matrix factorizations with d factors.
This paper is a study of the category of matrix factorizations with more than two factors. I start by describing a naturally occurring exact structure which turns out to be Frobenius. I determine the indecomposable projective-inejctive objects and I give explicit formulas for syzygies, cosyzygies, and cones.
The majority of the rest of the paper is focused on establishing two module theoretic descriptions of the category of matrix factorizations with d factors. This builds upon work of Knörrer and Solberg.