Tensor products of $d$-fold matrix factorizations

TL;DR

This paper explores the tensor product of d-fold matrix factorizations over commutative rings, analyzing their decompositions and applying findings to construct indecomposable modules over specific hypersurface domains.

Contribution

It extends the theory of tensor products of matrix factorizations to d-fold cases and investigates their decomposability, with applications to module construction.

Findings

Characterization of when tensor products decompose non-trivially

Construction of indecomposable maximal Cohen-Macaulay modules

Development of methods for analyzing d-fold matrix factorizations

Abstract

Consider a pair of elements $f$ and $g$ in a commutative ring $Q$. Given a matrix factorization of $f$ and another of $g$, the tensor product of matrix factorizations, which was first introduced by Kn\"orrer and later generalized by Yoshino, produces a matrix factorization of the sum $f+g$. We will study the tensor product of $d$-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen-Macaulay and Ulrich modules over hypersurface domains of a certain form.