An Orlov theorem for matrix factorizations with multiple factors

We prove a generalization of Orlov’s theorem for matrix factorizations with $n$ steps. Let $X$ be a regular scheme, $W\text{\hspace{0.17em}⁣}:X\to {\mathbb{A}}^1$ a flat morphism and $D:=W^{-1}(0)$ its central fiber. We construct an appropriate triangulated category of matrix factorizations with $n$-steps and show that it is equivalent to the singularity category of the root stack $\sqrt[n]{(X,D)}$. We also show that this category admits a semiorthogonal decomposition into $n-1$ copies of the usual (absolute derived) category of matrix factorizations with $2$ steps.