N-fold module factorizations: triangle equivalences and recollements

As an extension of Eisenbud's matrix factorization into the non-commutative realm, X.W. Chen introduced the concept of module factorizations over an arbitrary ring. A theorem of Chen establishes a triangle equivalence between the stable category of module factorizations with Gorenstein projective components and the stable category of Gorenstein projective modules over a quotient ring. In this paper, we introduce $n$-fold module factorizations, which generalize both the commutative $n$-fold matrix factorizations and the non-commutative module factorizations. To adapt triangle equivalences in module factorizations to $n$-fold module factorizations, we identify suitable subcategories of module factorizations and rings for the $n$-analogue. We further provide the $n$-analogue of Chen's theorem on triangle equivalences. Additionally, we study recollements involving the stable categories of higher-fold module factorizations, revealing intriguing recollements within the stable categories of Gorenstein modules of specific matrix subrings.