Triangulated categories arising from n-fold matrix factorizations

Abstract

Let $\mathcal{A}$ be an additive category and let $T:\mathcal{A}\to \mathcal{A}$ be an additive functor equipped with a natural transformation $\ddot{I}‰:{\mathrm{Id}}_{\mathcal{A}}\to T$. We prove that the homotopy category of $n$-fold matrix factorizations of $\ddot{I}‰$, denoted ${\mathrm{HFact}}_n(\mathcal{A},T,\ddot{I}‰)$, admits a natural structure of a right triangulated category. In particular, when $T$ is an automorphism, the homotopy category ${\mathrm{HFact}}_n(\mathcal{A},T,\ddot{I}‰)$ becomes triangulated. Furthermore, if $\mathcal{A}$ is a Frobenius exact category and $T$ is an autoequivalence, we obtain that the category ${\mathrm{Fact}}_n(\mathcal{A},T,\ddot{I}‰)$ of $n$-fold $(\mathcal{A},T)$-factorizations of $\ddot{I}‰$ is a Frobenius exact category. Consequently, the stable category of the Frobenius exact category ${\mathrm{Fact}}_n(\mathcal{A},T,\ddot{I}‰)$ is a triangulated category.

29 pages