On derived categories of module categories over multiring categories
Jing Yu
TL;DR
The work addresses when derived equivalences that respect a monoidal module action force equivalence of the underlying abelian module categories inside tensor categories. It builds a framework of triangulated module categories over multiring and tensor categories, proving a Künneth-type decomposition for cohomology under compatible t-structures and establishing a reconstruction theorem: a monoidal triangulated functor that induces an equivalence on ambient derived subcategories yields an equivalence of the original module categories. Key contributions include the general reconstruction theorem, a derived Künneth formula for triangulated module categories, and localization results linking Serre quotients with Verdier quotients, with applications to smash product algebras. The results extend prior work on derived and Morita theory in tensor categories to the setting of module categories and their derived categories, offering tools to study derived equivalences in rich, non-semisimple contexts and informing local characterizations of monoidal triangulated equivalences.
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.