Derived category $D(\mathcal{A})$ — localisation at quasi-isomorphisms

Intuition Beginner

A complex can contain extra pieces that cancel out and do not affect cohomology. If we only care about cohomology, then two complexes with the same cohomology should often count as equivalent.

The derived category is built to make that happen. It starts with complexes, identifies chain-homotopic maps, and then formally turns every cohomology-preserving map into an isomorphism.

Those cohomology-preserving maps are called quasi-isomorphisms.

The result is a category where one can do homological algebra without constantly choosing particular resolutions. Injective and projective resolutions become tools for computing objects in the derived category, not extra structure that changes the answer.

Visual Beginner

Localization turns quasi-isomorphisms into reversible arrows.

Worked example Beginner

An injective resolution of a sheaf replaces that sheaf by a longer complex of injective sheaves. The resolution has the same cohomology information as the original sheaf placed in one degree.

In the ordinary category of complexes, the sheaf and its injective resolution are different objects. In the derived category, the quasi-isomorphism from the sheaf to the resolution becomes an isomorphism.

What this tells us: derived categories make resolutions legitimate replacements rather than merely computational tricks.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathcal{A}$ be an abelian category. A cochain map $$ f^\bullet\to D^\bullet $$ is a quasi-isomorphism if the induced maps $$ H^n(f)^n(C^\bullet)\to H^n(D^\bullet) $$ are isomorphisms for every $n$.

The derived category of $\mathcal{A}$ is the localization $$ D(\mathcal A)=K(\mathcal A)[\mathrm{qis}^{-1}], $$ where $K(\mathcal{A})$ is the homotopy category of complexes and $\mathrm{qis}$ is the class of quasi-isomorphisms.

The localization functor $$ Q(\mathcal A)\to D(\mathcal A) $$ has the universal property that every functor $F:K(\mathcal{A})\to \mathcal{C}$ sending quasi-isomorphisms to isomorphisms factors uniquely through $Q$, up to unique isomorphism.

The bounded variants are $$ D^+(\mathcal A),\qquad D^-(\mathcal A),\qquad D^b(\mathcal A), $$ obtained from bounded-below, bounded-above, and bounded complexes.

The derived category is triangulated. Its distinguished triangles are the images of distinguished triangles from $K(\mathcal{A})$ under localization.

Counterexamples to common slips

• A quasi-isomorphism need not be a homotopy equivalence.
• The derived category is not the category of cohomology objects.
• Localization changes morphisms; morphisms in $D(\mathcal{A})$ can be represented by roofs, not only by single chain maps.

Key theorem with proof Intermediate+

Theorem (injective-resolution model). If $\mathcal{A}$ has enough injectives and suitable bounded-below injective resolutions, then the natural functor $$ K^+(\operatorname{Inj}\mathcal A)\to D^+(\mathcal A) $$ is an equivalence.

Proof. Every bounded-below complex admits a quasi-isomorphism into a bounded-below complex of injective objects. This is the complex-level version of resolving an object by injectives, built degree by degree and controlled by enough injectives.

Essential surjectivity follows: every object of $D^{+}(\mathcal{A})$ is represented by some bounded-below complex, and that complex is isomorphic in the derived category to its injective resolution.

For full faithfulness, maps between injective resolutions are already correctly computed in the homotopy category. If a map of injective complexes becomes zero after localization, then the usual lifting and homotopy arguments show it was already null-homotopic. Likewise, roofs through quasi-isomorphisms can be replaced by actual maps between injective complexes.

Thus localization adds no new morphisms between injective complexes and every derived object has an injective representative. $\square$

Bridge. This theorem explains why classical injective resolutions compute derived functors. The next unit 04.03.12 packages that computation as total right and left derived functors between derived categories.

Exercises Intermediate+

Advanced results Master

The passage $$ \mathrm{Ch}(\mathcal A)\to K(\mathcal A)\to D(\mathcal A) $$ has two stages. First, quotient by chain homotopy. Second, localize at quasi-isomorphisms. The first stage removes chain-level null differences; the second stage declares cohomologically equivalent complexes to be isomorphic.

Gabriel-Zisman localization supplies the general categorical mechanism. A morphism in a localized category can be represented by a roof $$ X\xleftarrow{s}X'\xrightarrow{f}Y $$ with $s$ in the class being inverted. In the derived category, $s$ is a quasi-isomorphism. Under calculus-of-fractions hypotheses, roofs can be composed by common refinements.

The cone criterion gives a useful test: $$ f\text{ is a quasi-isomorphism}\quad\Longleftrightarrow\quad \operatorname{Cone}(f)\text{ is acyclic}. $$ This criterion ties localization to the triangulated structure: the maps inverted in $D(\mathcal{A})$ are exactly those whose cones become zero objects.

Derived categories are triangulated because localization preserves the cone-triangle structure in the appropriate Verdier sense. The distinguished triangles in $D(\mathcal{A})$ are those isomorphic to images of cone triangles from $K(\mathcal{A})$.

Resolution models make derived categories computable. If $\mathcal{A}$ has enough injectives, bounded-below derived categories can be computed by homotopy categories of injective complexes. If $\mathcal{A}$ has enough projectives, bounded-above derived categories can be computed by projective complexes. These equivalences are the technical bridge between classical homological algebra and derived-category language.

For algebraic geometry, $D^b(\mathrm{Coh}X)$, $D^{+}(\mathrm{Sh}X)$, and derived categories of quasi-coherent sheaves become the ambient categories in which $R\Gamma$, $R{f}_{\ast }$, $R\mathcal{H}om$, tensor products, duality, and perverse sheaves live.

Synthesis. The derived category is the category of complexes with quasi-isomorphisms treated as equivalences. It keeps the triangulated exactness of cones while making resolutions canonical replacements. This is the formal home of modern cohomological algebra.

Full proof set Master

Proposition 1 (quasi-isomorphisms are detected by cones). A cochain map $f:C^{\bullet }\to D^{\bullet }$ is a quasi-isomorphism if and only if $\mathrm{Cone}(f)$ is acyclic.

Proof. The cone long exact sequence from 01.02.32 is $$ \cdots\to H^n(C)\xrightarrow{H^n(f)}H^n(D)\to H^n(\operatorname{Cone}(f))\to H^{n+1}(C)\to\cdots. $$ If $H^n(f)$ is an isomorphism for every $n$, exactness forces every $H^n(\mathrm{Cone}(f))$ to vanish. Conversely, if cone cohomology vanishes, exactness forces every $H^n(f)$ to be both injective and surjective. $\square$

Proposition 2 (acyclic complexes become zero in the derived category). If $C^{\bullet }$ is acyclic, then it is isomorphic to the zero object in $D(\mathcal{A})$.

Proof. The unique map $0\to C^{\bullet }$ is a quasi-isomorphism because both source and target have zero cohomology in every degree. Since quasi-isomorphisms are inverted in $D(\mathcal{A})$, this map becomes an isomorphism. Therefore $C^{\bullet }$ is isomorphic to zero in the derived category. $\square$

Proposition 3 (objects of $\mathcal{A}$ embed into $D(\mathcal{A})$). There is a natural functor $\mathcal{A}\to D(\mathcal{A})$ sending an object to the complex concentrated in degree $0$.

Proof. Send $A\in \mathcal{A}$ to the complex with $A$ in degree $0$ and zero elsewhere. A morphism $A\to B$ gives a chain map between the corresponding concentrated complexes. Passing through $K(\mathcal{A})$ and then the localization defines the desired functor. This is how ordinary objects are viewed inside the derived category. $\square$

Connections Master

• Chain homotopy and $K(\mathcal{A})$ 01.02.31. The derived category starts from the homotopy category.

• Mapping cone 01.02.32. The cone detects quasi-isomorphisms and supplies distinguished triangles.

• Triangulated categories 04.03.10. Derived categories are triangulated categories.

• Derived functors and Ext 04.03.06. Classical Ext becomes Hom in the derived category.

• Derived functors via derived categories 04.03.12. Total derived functors are functors between derived categories.

• Derived GIT and magic windows 04.10.15. Modern algebraic geometry uses bounded derived categories as geometric invariants.

Historical & philosophical context Master

Verdier introduced derived categories to make derived functors and duality intrinsic rather than dependent on chosen resolutions ^[Verdier]. His construction used localization of homotopy categories at quasi-isomorphisms.

Gabriel and Zisman supplied the general calculus of fractions and localization framework ^{[Gabriel-Zisman]}. That machinery explains why morphisms in derived categories are naturally represented by roofs.

Hartshorne's Residues and Duality brought Verdier's derived-category language into algebraic geometry, where it became indispensable for Grothendieck duality ^[Hartshorne]. Gelfand-Manin and Weibel later made the derived-category formulation standard in homological algebra textbooks ^{[Gelfand-Manin] [Weibel]}.

Philosophically, the derived category says that the true object is not a particular resolution but the cohomological information represented by any quasi-isomorphic complex. It is a category built to remember exactly the information that derived functors can see.

Bibliography Master

@book{Verdier1996DerivedCategory,
  author = {Verdier, Jean-Louis},
  title = {Des categories derivees des categories abeliennes},
  series = {Asterisque},
  volume = {239},
  publisher = {Societe Mathematique de France},
  year = {1996}
}

@book{GabrielZisman1967CalculusFractions,
  author = {Gabriel, Peter and Zisman, Michel},
  title = {Calculus of Fractions and Homotopy Theory},
  series = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume = {35},
  publisher = {Springer},
  year = {1967}
}

@book{GelfandManinDerivedCategory,
  author = {Gelfand, Sergei I. and Manin, Yuri I.},
  title = {Methods of Homological Algebra},
  publisher = {Springer},
  year = {1996}
}

@book{WeibelDerivedCategory,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  year = {1994}
}

@book{Hartshorne1966ResiduesDualityDerived,
  author = {Hartshorne, Robin},
  title = {Residues and Duality},
  series = {Lecture Notes in Mathematics},
  volume = {20},
  publisher = {Springer},
  year = {1966}
}