nLab derived category

Idea

The derived (infinity,1)-category or derived category of an abelian category $\mathcal{A}$ is the setting for homological algebra in $\mathcal{A}$: the (infinity,1)-categorical localization of the category of chain complexes in $\mathcal{A}$ at the class of quasi-isomorphisms. The derived category is a fundamental example of a stable (infinity,1)-category. In the case that $\mathcal{A} \simeq$ $R Mod$ (cf. the Freyd-Mitchell embedding theorem), the stable Dold-Kan correspondence says that the derived $\infty$-category of $\mathcal{A}$ is equivalently the stable $\infty$-category of $H R$-module spectra.

The derived $\infty$-category is presented by various dg-model structures on the category of chain complexes, as described at model structures on chain complexes. As such it has also a natural incarnation as a pretriangulated dg-category, which might be called the derived dg-category.

Like any stable (infinity,1)-category, the homotopy category of the derived (infinity,1)-category admits a canonical triangulated category structure. Often in the literature, the term derived category refers to the homotopy category, viewed only as a triangulated category. The loss of information can often be problematic, but for many purposes is not important.

In what follows, we will describe only the homotopy category. See (infinity,1)-category of chain complexes for the full (infinity,1)-category.

Associated to $\mathcal{A}$ is

• the category of chain complexes $Ch_\bullet(\mathcal{A})$ in $\mathcal{A}$ which is naturally a homotopical category;

• the “homotopy category of chain complexes” $K(\mathcal{A})$;

• the stable ∞-category $K_\infty(\mathcal{A})$ of chain complexes in $C$.

The derived category $D(C)$ of $\mathcal{A}$ is equivalently

• the 1-categorical homotopy category of $Ch_\bullet(\mathcal{A})$ with respect to the quasi-isomorphisms;

• the (∞,1)-categorical homotopy category of $K_\infty(\mathcal{A})$.

In either case, this means that under the canonical localization functor

$$Q : Ch_\bullet(\mathcal{A}) \to D(\mathcal{A})$$

the quasi-isomorphisms of chain complexes become true isomorphisms and that $D(\mathcal{A})$ is universal with respect to this property.

Hence the derived category is an approximation to the full simplicial localization of $K(\mathcal{A})$. It is or can be equipped with several further properties and structure that give a more accurate approximation. Notably every derived category is a triangulated category, which is a way of remembering the suspension and de-suspension operations on its objects – the suspension of chain complexes – hence its “stability”.

History

Derived categories were introduced by Jean-Louis Verdier in his thesis under the supervision of Alexandre Grothendieck. It was originally used to extend Serre duality to a relative context. See Hartshorne‘s lecture notes “Residues and duality”.

Definition

Let $\mathcal{A}$ be an abelian category and $K(\mathcal{A})$ its category of chain complexes modulo chain homotopy (the “homotopy category of chain complexes”).

Equip $K(\mathcal{A})$ with the structure of a homotopical category by declaring the weak equivalences to be the quasi-isomorphisms: those morphisms $f : V \to W$ which induce isomorphisms in homology, $H(f) : H(V) \stackrel{\simeq}{\to} H(W)$.

Definition

The derived category $D(\mathcal{A})$ is the homotopy category of $K(\mathcal{A})$ with respect to these weak equivalences.

Analogously, for $K^{+,-,b}(\mathcal{A})$ denoting the full subcategory on the chain complexes bounded above, bounded below, or bounded, respectively (see at category of chain complexes), one writes

$$D^{+,-,b}(\mathcal{A}) \hookrightarrow D(\mathcal{A})$$

for the correspponding full subcategory of the derived category.

Equivalent characterizations and constructions

There are various ways to construct or express the derived category more explicitly in terms of various special objects or morphisms in the category of chain complexes.

• In terms of localization at a null system

• In terms of resolutions.

In terms of localization at a null system

The “homotopy category of chain complexes” $K(\mathcal{A})$ is already a triangulated category. The derived category can be obtained as the construction of a homotopy category of a triangulated category with respect to a null system.

Definition

Let

$$N(\mathcal{A}) \subset K(\mathcal{A})$$

and analogously

$$N^{+,-n,b}(\mathcal{A}) \subset K^{+,-,b}(\mathcal{A})$$

be the full subcategory of $K(C)$ or on $K^{+,-,b}$, respectively, on those chain complexes $V$ whose chain homology vanishes in every degree, $H_\bullet(V) = 0$.

Proposition

A chain map $f_\bullet : V_\bullet \to W_\bullet$ is a quasi-isomorphism precisely there exists a distinguished triangle in $K(\mathcal{A})$ of the form

$$V \stackrel{f}{\to} W \to cone(f)$$

with the mapping cone $cone(f) \in N(\mathcal{C})$.

Proposition

The derived category is equivalently the localization of $K(\mathcal{A})$ at the null system $N(\mathcal{A})$.

$$D(\mathcal{A}) \simeq K(\mathcal{A})/N(\mathcal{A}) \,.$$

This perspective is discussed in (Kashiwara-Schapira, section 13) and (Schapira, section 6.2, 72).

In terms of injective and projective resolutions

In the case that the underlying abelian category $\mathcal{A}$ has enough injectives or enough projectives, the hom sets in the derived category may equivalently be obtained as homotopy-classes of chain maps from projective resolutions to injective resolutions of chain complexes.

In view of the existence of the injective and projective model structure on chain complexes this is a special case of the general fact that homotopy categories of model categories may be obtained by forming homotopy classes of maps in the model category from cofibrant resolutions to fibrant resolutions. For more details on the model-category point of view, see e.g. Appendix C of Toën, Vaquié. But here we spell out an direct discussion of this fact for chain complexes.

Definition

Write $K^+(\mathcal{I}_{\mathcal{A}}) \hookrightarrow K^+(\mathcal{A})$ for the full subcategory of the homotopy category of chain complexes bounded above on those that are degreewise injective objects.

Dually, let $K^-(\mathcal{P}_{\mathcal{A}}) \hookrightarrow K^-(\mathcal{A})$ for the full subcategory of the homotopy category of chain complexes bounded below on those that are degreewise projective objects.

Theorem

If $\mathcal{A}$ has enough injectives then the canonical functor

$$K^+(\mathcal{I}_{\mathcal{A}}) \to D^+(\mathcal{A})$$

is an equivalence of categories.

Dually, if $\mathcal{A}$ has enough projectives then the canonical functor

$$K^-(\mathcal{P}_{\mathcal{A}}) \to D^-(\mathcal{A})$$

is an equivalence of categories.

For instance (Schapira, cor. 7.2.3).

Related concepts

• derived functor
• triangulated categories

References

The original reference:

• Verdier, Jean-Louis, Des Catégories Dérivées des Catégories Abéliennes, Astérisque 239 (1996) [doi:10.24033/ast.364, numdam:AST_1996__239__R1_0, pdf]

• Sergei Gelfand, Yuri Manin, Section III of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [doi:10.1007/978-3-662-12492-5]

Systematic discussion from the point of view of localization and homotopy theory:

• Masaki Kashiwara, Pierre Schapira, section 13 of Categories and Sheaves (2006)

and, similarly, in section 7 of

• Pierre Schapira, Categories and homological algebra (2011) (pdf)

For the model-category version of this story for an arbitrary ringed site, including details about the tensor, internal hom, and $(-)_!$ and $(-)^*$, see Appendix C of

• Bertrand Toën, Michel Vaquié, Algebraization of complex analytic varieties and derived categories (arXiv:math/0703555v1)

A detailed treatment of derived categories (including of DG modules over DG rings), with applications to noncommutative algebra, is in the book

• Amnon Yekutieli, Derived Categories, Cambridge University Press (2019) [doi:10.1017/9781108292825, arXiv:1610.09640]

A pedagogical introduction:

• R. P. Thomas, Derived categories for the working mathematician (arXiv:math.AG/0001045)

A good survey of the more general topic of derived categories is

• Bernhard Keller, Derived categories and their uses (ps)

See in particular also the list of references given there.

Other lecture notes include

• Theo Bühler, An introduction to the derived category (pdf).

and for applications to coherent sheaves,

• Franco Rota, An introduction to the derived category of coherent sheaves (pdf)

For a discussion in the context of (∞,1)-categories and in particular stable (∞,1)-categories see section 13, p. 53 of

• Jacob Lurie, Stable ∞-Categories

On applications of derived categories in algebraic geometry:

• Dmitri Orlov, Derived categories of coherent sheaves and equivalences between them, Uspekhi Mat. Nauk, 2003, Vol. 58, issue 3(351), pp. 89–172, English translation (PDF)

• Aleksei Bondal, Dmitri Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compositio Mathematica 125 (03), 327-344. See also Bondal-Orlov reconstruction theorem.

• Daniel Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry, Oxford University Press, USA, 2006.

• Igor Dolgachev, Derived categories.

• Andrei Caldararu, Derived categories of coherent sheaves: a skimming. Lecture notes from Algebraic Geometry: Presentations by Young Researchers in Snowbird, Utah, July 2004 [arXiv:math/0501094]

On the algebraic K-theory of rings being encoded in the respective derived categories:

• Daniel Dugger, Brooke Shipley, K-theory and derived equivalences, Duke Math. J. 124 3 (2004) 587-617 [arXiv:math/0209084, doi:10.1215/S0012-7094-04-12435-2]