The monomorphism category of Gorenstein projective modules and comparision with the category of matrix factorization

Let ($S, \mathfrak{n})$ be a commutative noetherian local ring and let $\omega\in\mathfrak{n}$ be non-zero divisor. This paper is concerned with the category of monomorphisms between finitely generated Gorenstein projective S-modules, such that their cokernels are annihilated by $\omega$. We will observe that this category, which will be denoted by Mon$(\omega,\mathcal{G})$, is an exact category in the sense of Quillen. More generally, it is proved that Mon$(\omega,\mathcal{G})$ is a Frobenius category. Surprisingly, it is shown that not only the category of matrix factorizations embeds into Mon$(\omega,\mathcal{G})$, but also its stable category as well as the singularity category of the factor ring $R = S/(\omega)$, can be realized as triangulated subcategories of the stable category of Mon$(\omega,\mathcal{G})$.