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arXiv · 2301.09561

Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors

Leonid Positselski
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Abstract

In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let $\Upsilon\colon C\textsf{-Comod}\longrightarrow C^*\textsf{-Mod}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $\Theta\colon C\textsf{-Contra}\longrightarrow C^*\textsf{-Mod}$ be the natural forgetful functor. We prove that the functor $\Upsilon$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $\Theta$ 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".

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Leonid Positselski. 2023-01-23. Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors. https://doi.org/10.1017/s001708952500014x

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