# Matrix factorizations for self-orthogonal categories of modules

## TL;DR

This paper explores matrix factorizations within self-orthogonal module categories over commutative rings, establishing equivalences with totally acyclic complexes under certain regularity conditions.

## Contribution

It introduces a framework connecting matrix factorizations and totally acyclic complexes for self-orthogonal categories, extending known results to broader module classes.

## Key findings

- Embedding of homotopy categories under regularity conditions
- Equivalence for projective or flat-cotorsion modules
- Dual equivalence for injective modules

## Abstract

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, we show there is a natural embedding of the homotopy category of $\mathsf{C}$-factorizations of $f$ into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if $\mathsf{C}$ is the category of projective or flat-cotorsion $S$-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when $\mathsf{C}$ is the category of injective $S$-modules.