On derived categories of module categories over multiring categories

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 $F:{\mathbf{D}}^b(\mathcal{C})\to {\mathbf{D}}^b(\mathcal{D})$, 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.