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pretriangulated dg-category

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Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Stable Homotopy theory

Contents

Idea

Pretriangulated dg-categories over a commutative ring kk are, roughly speaking, dg-categories whose homotopy category is canonically triangulated. These form a model for stable k-linear (∞,1)-categories, in a sense which is made precise below (at least in characteristic zero). In other words pretriangulated dg-categories can be viewed as enhanced triangulated categories. For this reason some authors call them stable dg-categories.

History

The notion of pretriangulated dg-category goes back to (Bondal-Kapranov 1990).

Goncalo Tabuada demonstrated the existence of a model structure on the category of small dg-categories, the quasi-equiconic model structure on dg-categories?, where the fibrant objects are the pretriangulated dg-categories. See (Tabuada 07, Theorem 2.2 and Proposition 2.10). This model structure can be Bousfield localized to the Morita model structure on dg-categories?, where the fibrant objects are the idempotent complete pretriangulated dg-categories. In (Cohn 13) it is shown that the associated (infinity,1)-category is equivalent to the (infinity,1)-category of stable k-linear (∞,1)-categories.

Definitions

Pretriangulated dg-categories

Let AA be a dg-category and P(A)P(A) the dg-category of dg-presheaves or right dg-modules over AA. The Yoneda embedding induces a fully faithful functor h:ho(A)ho(P(A))h : ho(A) \hookrightarrow ho(P(A)) on the homotopy categories. The category ho(P(A))ho(P(A)) has a canonical triangulated structure (which can be written down directly).

Definition

The dg-category AA is called pretriangulated if the functor hh is stable under the suspension functor (and its inverse), and under taking mapping cones in ho(P(A))ho(P(A)).

Proposition

A dg-category AA is pretriangulated if and only if it is a fibrant object in the quasi-equiconic model structure on dg-categories?.

See (Tabuada 07, Proposition 2.10).

Strongly pretriangulated dg-categories

Let AA be a dg-category.

Definition

The nn-translation of an object XAX \in A is an object X[n]AX[n] \in A representing the functor

Hom(,X)[n]. \Hom(\cdot, X)[n].

The cone of a closed morphism f:XYf : X \to Y of degree zero is an object Cone(f)A\Cone(f) \in A representing the functor

Cone(Hom(,X)f *Hom(,Y)), \Cone(\Hom(\cdot, X) \stackrel{f_*}{\to} \Hom(\cdot, Y)),

which is a mapping cone in chain complexes.

Definition

The dg-category AA is called strongly pretriangulated if it admits a zero object, all translations of all objects, and all cones of all morphisms.

Definition

Let AA be a dg-category. A strongly pretriangulated envelope of AA is the data of a strongly pretriangulated dg-category tri(A)tri(A) and a fully faithful functor Atri(A)A \hookrightarrow tri(A) such that any functor u:ABu: A \to B to a strongly pretriangulated dg-category BB factors uniquely through a functor tri(A)Btri(A) \to B.

A strongly pretriangulated envelope Atri(A)A \hookrightarrow tri(A) always exists, and may be constructed by taking tri(A)tri(A) to be the full triangulated subcategory of ho(P(A))ho(P(A)) generated by the representable presheaves, and Atri(A)A \hookrightarrow tri(A) to be the functor induced by the Yoneda embedding. Here P(A)P(A) denotes the dg-category of dg-presheaves on AA. There is also another construction using twisted complexes, see Bondal-Kapranov.

Now we have the following characterization of pretriangulated dg-categories.

Proposition

Let AA be a dg-category and Atri(A)A \hookrightarrow tri(A) be a strongly pretriangulated envelope of AA. AA is pretriangulated if and only if the induced fully faithful functor ho(A)ho(tri(A))ho(A) \hookrightarrow ho(tri(A)) is essentially surjective (and hence an equivalence of categories).

As an immediate corollary, note that for a pretriangulated dg-category AA, its homotopy category ho(A)ho(A) inherits a canonical triangulated structure.

Properties

Proposition

Let u:ABu : A \to B be a functor between two dg-categories. If AA and BB are pretriangulated then the induced functor ho(u):ho(A)ho(B)ho(u): ho(A) \to ho(B) is triangulated. Further, uu is a quasi-equivalence? if and only if ho(u)ho(u) is a triangulated equivalence.

Applications

The use of pretriangulated dg-categories as enhanced triangulated categories has been especially successful in the study of the various triangulated categories of sheaves on algebraic varieties.

In the below paper it is shown that the triangulated categories of quasicoherent sheaves on quasiprojective varieties and some of their cousins (like categories of perfect complexes on quasiprojective varieties) have unique dg-enhancements. Fernando Muro has developed an obstruction theory for the existance and measuring non-uniqueness of dg-enhancements in more general settings (unpublished).

Similarly, pretriangulated dg-categories have proven to give a good model for derived noncommutative algebraic geometry in the sense of Maxim Kontsevich. See there for relevant references. In this connection see also the work of Goncalo Tabuada who has developed a theory of noncommutative motives in this framework.

References

The model structure presenting pretriangulated dg-categories is discussed in

For a summary of the various model structures on dg-categories?, see Section 2 of the paper

The relation to stable (infinity,1)-categories is discussed in

Revised on August 25, 2014 05:17:00 by Adeel Khan (132.252.62.217)