On derived categories of module categories over multiring categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909983111643136 |
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| author | Yu, Jing |
| author_facet | Yu, Jing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03128 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On derived categories of module categories over multiring categories Yu, Jing Representation Theory Category Theory 18G80, 18M05, 16T05 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. |
| title | On derived categories of module categories over multiring categories |
| topic | Representation Theory Category Theory 18G80, 18M05, 16T05 |
| url | https://arxiv.org/abs/2601.03128 |