Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866918367643828224 |
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| author | Positselski, Leonid |
| author_facet | Positselski, Leonid |
| contents | In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let $Υ\colon C\textsf{-Comod}\longrightarrow C^*\textsf{-Mod}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $Θ\colon C\textsf{-Contra}\longrightarrow C^*\textsf{-Mod}$ be the natural forgetful functor. We prove that the functor $Υ$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $Θ$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\operatorname{Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge0$. We call such coalgebras "weakly finitely Koszul". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09561 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors Positselski, Leonid Rings and Algebras Category Theory In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let $Υ\colon C\textsf{-Comod}\longrightarrow C^*\textsf{-Mod}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $Θ\colon C\textsf{-Contra}\longrightarrow C^*\textsf{-Mod}$ be the natural forgetful functor. We prove that the functor $Υ$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $Θ$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\operatorname{Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge0$. We call such coalgebras "weakly finitely Koszul". |
| title | Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors |
| topic | Rings and Algebras Category Theory |
| url | https://arxiv.org/abs/2301.09561 |