Triangulated categories arising from n-fold matrix factorizations
Yixia Zhang, Panyue Zhou
TL;DR
This work extends the theory of matrix factorizations to arbitrary additive categories with a functor $T$ and a natural transformation $\\omega$, defining $n$-fold factorizations and their homotopy category $\\mathrm{HFact}_{n}(\\mathcal{A},T,\\omega)$. For even $n$, it constructs a suspension $\\Sigma$ and a mapping-cone framework that endows $\\mathrm{HFact}_{n}(\\mathcal{A},T,\\omega)$ with a natural right triangulated structure, becoming triangulated when $T$ is an automorphism. Furthermore, when $\\mathcal{A}$ is Frobenius exact and $T$ is an autoequivalence, $\\mathrm{Fact}_{n}(\\mathcal{A},T,\\omega)$ is Frobenius exact and its stable category is triangulated, unifying various noncommutative and categorical generalizations of matrix factorizations. The results extend prior work on $n$-fold and twisted factorizations, providing explicit constructions for suspensions and cones in a purely additive setting without DG enhancements, and establishing a robust bridge to Frobenius exact categories and their stable triangulated categories.
Abstract
Let $\mathcal{A}$ be an additive category and let $T\colon \mathcal{A}\rightarrow \mathcal{A}$ be an additive functor equipped with a natural transformation $ω\colon \mathrm{Id}_{\mathcal{A}}\rightarrow T$. We prove that the homotopy category of $n$-fold matrix factorizations of $ω$, denoted ${\rm HFact}_{n}(\mathcal{A},T,ω)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\rm HFact}_{n}(\mathcal{A},T,ω)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $ω$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\rm Fact}_{n}(\mathcal{A},T,ω)$ is a triangulated category.