An Orlov theorem for matrix factorizations with multiple factors

TL;DR

This paper generalizes Orlov's theorem to matrix factorizations with multiple steps, establishing an equivalence with the singularity category of a root stack and revealing a semiorthogonal decomposition structure.

Contribution

It introduces a new triangulated category for multi-step matrix factorizations and proves its equivalence to the singularity category of a root stack, extending Orlov's theorem.

Findings

Equivalence between the new matrix factorization category and the singularity category of the root stack.

Existence of a semiorthogonal decomposition into multiple copies of the usual matrix factorization category.

Generalization of Orlov's theorem to n-step matrix factorizations.

Abstract

We prove a generalization of Orlov's theorem for matrix factorizations with $n$ steps. Let $X$ be a regular scheme, $W\colon X\to \mathbb{A}^1$ 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.