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

Abstract

In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let ${\rm Y}:C\text{-Comod}⟶C^{\ast }\text{-Mod}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^{\ast }$-modules, and let $\Theta :C\text{-Contra}⟶C^{\ast }\text{-Mod}$ be the natural forgetful functor. We prove that the functor ${\rm Y}$ 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 ${\mathrm{Ext}}_C^n(k,k)$ is finite-dimensional for all $n\ge 0$. We call such coalgebras "weakly finitely Koszul".

LaTeX 2e with xy-pic; 44 pages, 3 commutative diagrams; v.2: Lemma 3.2 and Proposition 3.3 added, Sections 8-10 added, category-theoretic lemmas from Sections 4-7 moved to Appendix; v.3: Remark 8.2 inserted, references added and updated, several misprints corrected; v.4: several misprints corrected, the numbering of sections shifted to agree with the journal version

References in corpus (4)

• Koszul property and Bogomolov's conjecture
• Differential graded Koszul duality: an introductory survey
• Remarks on derived complete modules and complexes
• Unbounded derived categories of small and big modules: Is the natural functor fully faithful?