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

TL;DR

This paper characterizes when the inclusion of comodules and the forgetful functor of contramodules into modules are fully faithful on derived categories, linking this to a finiteness condition on Ext groups for conilpotent coalgebras.

Contribution

It establishes an equivalence between the full faithfulness of comodule and contramodule functors on derived categories and a finiteness condition on Ext groups, introducing the concept of weakly finitely Koszul coalgebras.

Findings

Full faithfulness of functors is characterized by Ext group finiteness.

Introduces the class of weakly finitely Koszul coalgebras.

Provides criteria linking comodules, contramodules, and module categories.

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".