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

Leonid Positselski

TL;DR

The paper characterizes when the comodule inclusion Υ: C{--Comod} → C^{*}{--Mod} and the contramodule forgetful Θ: C{--Contra} → C^{*}{--Mod} induce fully faithful triangulated functors on derived categories, tying this to the finite dimensionality of Ext_C^n(k,k) for all n. It introduces the notion of weakly finitely Koszul coalgebras and establishes equivalences between bounded/below and bounded/above derived full faithfulness and Ext-vanishing conditions, as well as equivalences for contramodules via Eklof and duality arguments. The work also analyzes the half-bounded derived categories, co-Noetherian and cocoherent conilpotent coalgebras, and culminates with the cocommutative coconilpotent case, where C^* is a complete Noetherian local ring and the rational/discrete modules provide a concrete bridge between comodules/contramodules and C^*-modules. Overall, it provides a precise, cohomological criterion for full faithfulness and connects homological dimensions to derived-category behavior in the comod/contra-C^*-module setting.

Abstract

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".

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

TL;DR

The paper characterizes when the comodule inclusion Υ: C{--Comod} → C^{*}{--Mod} and the contramodule forgetful Θ: C{--Contra} → C^{*}{--Mod} induce fully faithful triangulated functors on derived categories, tying this to the finite dimensionality of Ext_C^n(k,k) for all n. It introduces the notion of weakly finitely Koszul coalgebras and establishes equivalences between bounded/below and bounded/above derived full faithfulness and Ext-vanishing conditions, as well as equivalences for contramodules via Eklof and duality arguments. The work also analyzes the half-bounded derived categories, co-Noetherian and cocoherent conilpotent coalgebras, and culminates with the cocommutative coconilpotent case, where C^* is a complete Noetherian local ring and the rational/discrete modules provide a concrete bridge between comodules/contramodules and C^*-modules. Overall, it provides a precise, cohomological criterion for full faithfulness and connects homological dimensions to derived-category behavior in the comod/contra-C^*-module setting.

Abstract

In this paper we consider a conilpotent coalgebra over a field . Let be the natural functor of inclusion of the category of -comodules into the category of -modules, and let be the natural forgetful functor. We prove that the functor induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the -vector space is finite-dimensional for all . We call such coalgebras "weakly finitely Koszul".
Paper Structure (11 sections, 44 theorems, 105 equations)

This paper contains 11 sections, 44 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on Coalgebras, Comodules, and Contramodules
  3. Conilpotent Coalgebras and Minimal Resolutions
  4. The Left and Right Comodule and Contramodule Ext Comparison
  5. Weakly Finite Koszulity Implies Comodule Ext Isomorphism
  6. Comodule Ext Isomorphism Implies Weakly Finite Koszulity
  7. Weakly Finite Koszulity Implies Contramodule Ext Isomorphism
  8. Contramodule Ext Isomorphism Implies Weakly Finite Koszulity
  9. Half-Bounded Derived Full-and-Faithfulness
  10. Co-Noetherian and Cocoherent Conilpotent Coalgebras
  11. Cocommutative Conilpotent Coalgebras

Key Result

Theorem 1.1

Let $C$ be a conilpotent coalgebra over a field $k$ and $n\ge1$ be an integer. Then the following five conditions are equivalent: If any one of the equivalent conditions (i--v) holds, then the map contramodule-ext-map is an isomorphism for all left $C$-contramodules $P$ and $Q$ and all $0\le i\le n-2$.

Theorems & Definitions (93)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 83 more