[Paper Review] Exact DG-categories and fully faithful triangulated inclusion functors

Leonid Positselski|arXiv (Cornell University)|Oct 15, 2021

Algebraic structures and combinatorial models4 citations

TL;DR

This paper introduces an 'almost involution' construction on DG-categories that recovers the abelian category of graded modules from a DG-ring, enabling a unified framework for exact DG-categories and derived categories of the second kind. The key contribution is a general theory of fully faithful triangulated functors between derived categories of the second kind, proven via homological algebra over abelian DG-categories and applied to complexes, matrix factorizations, and CDG-modules.

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.

Motivation & Objective

• To develop a category-theoretic framework for exact DG-categories using an 'almost involution' construction.
• To recover the abelian category of graded modules from a DG-ring via a new DG-category construction.
• To establish a general setting for derived categories of the second kind, including coderived and contraderived categories.
• To prove full faithfulness theorems for triangulated functors induced by fully exact DG-subcategory inclusions.
• To unify and generalize existing results on derived categories across complexes, matrix factorizations, and quasi-coherent CDG-modules.

Proposed method

• The paper constructs a new DG-category $\mathbf{A}^\natural$ from a given DG-category $\mathbf{A}$, using a duality-like 'almost involution' that recovers the graded structure.
• It defines abelian and exact DG-categories via the interplay between $\mathsf{Z}^0(\mathbf{A})$ and $\mathsf{Z}^0(\mathbf{A}^\natural)$, the categories of closed morphisms of degree 0.
• The construction uses free generation of DG-modules from graded modules, equipped with canonical contracting homotopies.
• It introduces a triple of adjoint functors between $\mathsf{Z}^0(\mathbf{A})$ and $\mathsf{Z}^0(\mathbf{A}^\natural)$, interpreting 'forgetting the differential' and 'freely generating a DG-module'.
• The theory is applied to DG-modules, curved DG-modules, and CDG-modules over schemes, with a focus on derived categories of the second kind.
• Key results are proven using homological dimension assumptions and fp-projective dimension conditions on $\mathsf{Z}^0(\mathbf{A}^\natural)$.

Experimental results

Research questions

• RQ1 How can one recover the abelian category of graded modules over a graded ring from a DG-ring using DG-categorical constructions?
• RQ2 What is the correct abstract definition of an exact DG-category that generalizes known examples like complexes and factorization categories?
• RQ3 Under what conditions is the inclusion of a fully exact DG-subcategory fully faithful on derived categories of the second kind?
• RQ4 How do the coderived and contraderived categories defined by Positselski and Becker compare in terms of output?
• RQ5 Can the finite fp-projective dimension assumption be relaxed in full faithfulness theorems for coderived categories?

Key findings

• The construction $\mathbf{A}^\natural$ recovers the abelian category of graded $R^*$-modules as $\mathsf{Z}^0(\mathbf{A}^\natural)$, where $\mathbf{A} = \boldsymbol{R}^\bullet\mathbf{--mod}$.
• For an abelian DG-category $\mathbf{A}$, the functor $\natural\natural: \mathbf{A} \to \mathbf{A}^{\natural\natural}$ is a DG-equivalence, showing the construction is self-dual in this case.
• The inclusion $\boldsymbol{R}^{\blackdiamond}\mathbf{--mod}_{\mathbf{fp}} \to \boldsymbol{R}^{\blackdiamond}\mathbf{--mod}$ induces a fully faithful triangulated functor between $\mathsf{D}^{\mathsf{abs}}(\boldsymbol{R}^{\blackdiamond}\mathbf{--mod}_{\mathbf{fp}})$ and $\mathsf{D}^{\mathsf{co}}(\boldsymbol{R}^{\blackdiamond}\mathbf{--mod})$.
• Under $\aleph_n$-Noetherianity, the fp-projective dimension of $\mathsf{Z}^0(\mathbf{A}^\natural)$ is bounded by $n$, satisfying the assumptions of the main theorems.
• The results of [50] allow dropping the finite fp-projective dimension assumption on $\mathsf{Z}^0(\mathbf{A})$ when using Becker’s coderived category, improving the main theorems.
• Positselski’s and Becker’s coderived categories agree when $\mathsf{Z}^0(\mathbf{A}^\natural)$ has finite fp-projective dimension, suggesting potential equivalence in broader settings.