Matrix factorizations for self-orthogonal categories of modules

For a commutative ring [Formula: see text] and self-orthogonal subcategory [Formula: see text] of [Formula: see text], we consider matrix factorizations whose modules belong to [Formula: see text]. Let [Formula: see text] be a regular element. If [Formula: see text] is [Formula: see text]-regular for every [Formula: see text], we show there is a natural embedding of the homotopy category of [Formula: see text]-factorizations of [Formula: see text] into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if [Formula: see text] is the category of projective or flat-cotorsion [Formula: see text]-modules. Dually, using divisibility in place of regularity, we observe there is a parallel equivalence when [Formula: see text] is the category of injective [Formula: see text]-modules.