On derived categories of module categories over multiring categories

TL;DR

This paper establishes conditions under which equivalences of derived categories of certain subcategories imply equivalences of the original module categories, with applications to smash product algebras and localization theory.

Contribution

It proves that derived category equivalences induced by monoidal functors imply original category equivalences, extending understanding of module categories over multiring categories.

Findings

Derived category equivalences imply original category equivalences.

Application to smash product algebras.

Investigation of localization theory in module categories.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be subcategories of tensor categories $\mathcal{C}$ and $\mathcal{D}$, respectively, both of which are abelian categories with finitely many isomorphism classes of simple objects. We prove that if their derived categories $\mathbf{D}^b(\mathcal{A})$ and $\mathbf{D}^b(\mathcal{B})$ are left triangulated tensor ideals and are equivalent as triangulated $\mathbf{D}^b(\mathcal{C})$-module categories via an equivalence induced by a monoidal triangulated functor $F:\mathbf{D}^b(\mathcal{C})\rightarrow \mathbf{D}^b(\mathcal{D})$, then the original module categories $\mathcal{A}$ and $\mathcal{B}$ are themselves equivalent. We then apply this result to smash product algebras. Furthermore, the localization theory of module categories and triangulated module categories is investigated.