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Exact DG-categories and fully faithful triangulated inclusion functors

Leonid Positselski

TL;DR

The paper builds a comprehensive, unified framework for abelian and exact differential graded categories (DG-categories) by introducing the almost-involution construction A^natural and its iterates. This yields a robust, general setting for derived categories of the second kind (absolute, coderived, contraderived) and clarifies when inclusions of subcategories induce fully faithful functors or equivalences. Core results include the Finite Resolution Dimension Theorem, the Full-and-Faithfulness Theorem, and the Compact Generation Theorem, which together show how resolution-dimension and (co)limits govern equivalences and generation in coderived/contraderived categories. The framework subsumes complexes, CDG-modules, matrix-factorizations, and quasi-coherent CDG-objects over schemes, unifying their derived-category theories under exact DG-category formalism. This provides a natural maximal setting for derived categories of the second kind and clarifies when substructures preserve exact/DG-data, with broad implications for homological algebra and algebraic geometry.

Abstract

We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring $R^*$ from the DG-category of DG-modules over a DG-ring $(R^*,d)$. This provides an appropriate technical background for the definition and discussion of abelian and exact DG-categories. In the setting of exact DG-categories, derived categories of the second kind are defined in the maximal natural generality. We develop the related abstract category-theoretic language and use it to formulate and prove several full-and-faithfulness theorems for triangulated functors induced by the inclusions of fully exact DG-subcategories. Such functors are fully faithful for derived categories of the second kind more often than for the conventional derived categories. Examples and applications range from the categories of complexes in abelian/exact categories to matrix factorization categories, and from curved DG-modules over curved DG-rings to quasi-coherent CDG-modules over quasi-coherent CDG-quasi-algebras over schemes.

Exact DG-categories and fully faithful triangulated inclusion functors

TL;DR

The paper builds a comprehensive, unified framework for abelian and exact differential graded categories (DG-categories) by introducing the almost-involution construction A^natural and its iterates. This yields a robust, general setting for derived categories of the second kind (absolute, coderived, contraderived) and clarifies when inclusions of subcategories induce fully faithful functors or equivalences. Core results include the Finite Resolution Dimension Theorem, the Full-and-Faithfulness Theorem, and the Compact Generation Theorem, which together show how resolution-dimension and (co)limits govern equivalences and generation in coderived/contraderived categories. The framework subsumes complexes, CDG-modules, matrix-factorizations, and quasi-coherent CDG-objects over schemes, unifying their derived-category theories under exact DG-category formalism. This provides a natural maximal setting for derived categories of the second kind and clarifies when substructures preserve exact/DG-data, with broad implications for homological algebra and algebraic geometry.

Abstract

We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring from the DG-category of DG-modules over a DG-ring . This provides an appropriate technical background for the definition and discussion of abelian and exact DG-categories. In the setting of exact DG-categories, derived categories of the second kind are defined in the maximal natural generality. We develop the related abstract category-theoretic language and use it to formulate and prove several full-and-faithfulness theorems for triangulated functors induced by the inclusions of fully exact DG-subcategories. Such functors are fully faithful for derived categories of the second kind more often than for the conventional derived categories. Examples and applications range from the categories of complexes in abelian/exact categories to matrix factorization categories, and from curved DG-modules over curved DG-rings to quasi-coherent CDG-modules over quasi-coherent CDG-quasi-algebras over schemes.

Paper Structure

This paper contains 61 sections, 149 theorems, 155 equations, 1 figure.

Table of Contents

  1. Graded Categories and DG-Categories
  2. Grading group datum
  3. Graded categories
  4. DG-categories
  5. Thematic Examples
  6. Complexes in an additive category
  7. Curved DG-modules
  8. Quasi-coherent graded quasi-algebras
  9. Quasi-coherent CDG-quasi-algebras
  10. Factorization categories
  11. The Almost Involution on DG-Categories
  12. CDG-modules revisited
  13. Main Construction
  14. Properties of the main construction
  15. Compatibility with DG-functors
  16. ...and 46 more sections

Key Result

Lemma 2.1

Let $A$ and $B$ be two quasi-modules over $R$. Then $A\otimes_RB$ is also a quasi-module over $R$.

Figures (1)

  • Figure 1: The composition of morphisms vanishes in $\mathsf H^0(\mathbf E)$

Theorems & Definitions (317)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 307 more