View source for Derived category

The notion of a derived category has been introduced by J.-L. Verdier in his 1963 notes [[#References|[a7]]]. This facilitated a proof of a duality theorem of A. Grothendieck (cf. [[#References|[a5]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312801.png" /> be an [[Additive category|additive category]] equipped with an additive automorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312802.png" />, called the translation functor. A triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312803.png" /> is a sextuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312804.png" /> of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312806.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312807.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312808.png" /> and morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d0312809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128011.png" />. One often uses

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128012.png" /></td> </tr></table>

to denote such a triangle. It is obvious what it meant by a morphism of triangles. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128013.png" /> equipped with a family of triangles, the distinguished triangles, is called a triangulated category if the axioms TR1)–TR4) in [[#References|[a7]]] are satisfied.

Writing briefly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128014.png" /> for a triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128015.png" />, these axioms are as follows.

TR1) Each triangle isomorphic to a distinguished triangle is distinguished. For each morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128016.png" /> there is a distinguished triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128017.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128018.png" /> is distinguished.

TR2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128019.png" /> is distinguished if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128020.png" /> is distinguished.

TR3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128022.png" /> are distinguished and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128023.png" /> is a morphism, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128024.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128025.png" /> is a morphism of triangles.

TR4) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128028.png" /> be three distinguished triangles with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128031.png" />. Then there exists two morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128035.png" /> are morphisms of triangles and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128036.png" /> is a distinguished triangle.

An additive functor between two triangulated categories is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128038.png" />-functor (or exact functor) if it commutes with the translation functor and preserves distinguished triangles.

To get some feeling for these axioms and the terminology it is (perhaps) useful to keep the example below in mind: the category of complexes over an Abelian category (and algebraic mapping cones, the corresponding long exact sequences, and connecting homomorphisms of long exact sequences). One often writes a distinguished triangle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128039.png" /> as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128040.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128041.png" /> is thought of as a  "morphism of degree 1"  from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128042.png" /> (which, by definition, is the same thing as a morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128043.png" />). Whence the terminology  "triangulated category" . Writing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128044.png" /> for the group of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128045.png" /> one finds straightforwardly from TR1)–TR3) for each distinguished triangle and object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128046.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128047.png" /> long exact sequences of groups

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128048.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128051.png" /></td> </tr></table>

The next step, still inspired by cohomology and complexes, is to  "localize suitably" , i.e.  "to find a categorical setting in which morphisms which induce isomorphisms in cohomology can be inverted and thus become isomorphisms" .

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128052.png" /> be a triangulated category. A collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128053.png" /> of morphism s in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128054.png" /> is called a multiplicative system if it satisfies properties (FR1)–(FR5) (given in [[#References|[a7]]]).

(FR1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128056.png" /> are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128057.png" />, then so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128058.png" />. All identity morphisms are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128059.png" />.

(FR2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128060.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128062.png" />, then there are an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128063.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128064.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128065.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128066.png" />, and (symmetrically) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128067.png" /> is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128069.png" />, then there are an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128070.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128071.png" /> and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128072.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128073.png" />.

(FR3) For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128074.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128075.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128077.png" />.

(FR4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128078.png" />, then also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128079.png" />.

(FR5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128081.png" /> are two distinguished triangles and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128082.png" /> is a morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128083.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128084.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128085.png" />, then there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128086.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128087.png" /> is a morphism of distinguished triangles.

Axioms (FR1) and (FR2), and to a lesser extent (FR3), are  "general"  in the setting of categories of fractions (cf. (the comments to) [[Localization in categories|Localization in categories]]). The other two are special for this particular setting of triangulated categories.

The localization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128088.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128089.png" /> is a category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128090.png" /> together with the canonical functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128091.png" /> such that the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128092.png" /> has the universal property: Any functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128093.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128094.png" /> is an isomorphism for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128095.png" /> factors uniquely through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128096.png" />.

Such a pair exists and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128097.png" /> carries a unique structure of a triangulated category such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128098.png" /> is exact. Note that the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d03128099.png" /> are the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280100.png" /> and that a morphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280101.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280102.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280103.png" /> may be represented by a diagram <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280104.png" /> of morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280105.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280106.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280107.png" /> be an [[Abelian category|Abelian category]]. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280108.png" /> the additive category of complexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280109.png" />. The translation functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280110.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280111.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280112.png" />, and one often writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280113.png" /> instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280114.png" /> [[#References|[a1]]]. Denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280115.png" /> the additive category whose objects are the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280116.png" /> and whose morphisms are homotopy equivalence classes of morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280117.png" />. Call a triangle distinguished if it is isomorphic to a triangle of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280118.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280119.png" /> denotes the maping cone (cf. [[Mapping-cone construction|Mapping-cone construction]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280120.png" />. Similarly one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280121.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280122.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280123.png" />), the category of bounded below (respectively, bounded above, respectively, bounded) complexes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280124.png" />. A complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280125.png" /> is bounded above if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280126.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280127.png" /> large enough, etc.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280128.png" />. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280129.png" /> is called a quasi-isomorphism if it induces an isomorphism on cohomology. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280130.png" /> be the collection of all quasi-isomorphisms. The localized category (cf. [[Localization in categories|Localization in categories]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280131.png" /> is called the derived category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280132.png" />. Similarly one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280133.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280134.png" />, respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280135.png" />). Every short [[Exact sequence|exact sequence]] gives rise to a distinguished triangle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280136.png" />.

Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280137.png" /> has enough injectives (cf. [[Injective object|Injective object]]). Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280138.png" /> the collection of injective objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280139.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280140.png" /> be the triangulated subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280141.png" /> consisting of bounded below complexes of injective objects in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280142.png" />. The canonical functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280143.png" /> induces an equivalence of categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280144.png" />. A similar discussion applies to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280145.png" /> in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280146.png" /> has enough projectives (cf. [[Projective object of a category|Projective object of a category]]).

Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280147.png" /> be an Abelian category and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280148.png" /> be a thick Abelian subcategory. Define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280149.png" /> as the full triangulated subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280150.png" /> consisting of the complexes whose cohomology objects are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280151.png" />, and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280152.png" />. This is the full subcategory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280153.png" /> consisting of those complexes whose cohomology objects are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280154.png" />.

==The derived functor.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280155.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280156.png" /> be Abelian categories. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280157.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280158.png" />-functor (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280159.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280160.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280161.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280162.png" />, or b). One says that the right derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280163.png" /> (respectively, left derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280164.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280165.png" /> exists if the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280166.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280167.png" />) from the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280168.png" />-functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280169.png" /> to the category of sets is representable (cf. [[Representable functor|Representable functor]]). In that case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280170.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280171.png" />) is, by definition, a representative. For every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280172.png" /> one puts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280173.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280174.png" />).

Concerning existence one has the following. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280175.png" /> is a triangulated subcategory such that: 1) every object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280176.png" /> admits a quasi-isomorphism into (respectively, from) an object of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280177.png" />; and 2) for every acyclic object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280178.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280179.png" /> is acyclic. (An acyclic complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280180.png" /> is one whose cohomology is zero.) Then the right derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280181.png" /> (respectively, left derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280182.png" />) exists and for every object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280183.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280184.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280185.png" />).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280186.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280187.png" /> be Abelian categories and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280188.png" /> be an additive left exact (respectively, right exact) functor (cf. [[Exact functor|Exact functor]]). Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280189.png" /> has enough injective (respectively, projective) objects. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280190.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280191.png" />) exists. The functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280192.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280193.png" />) coincides with the usual <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280194.png" />-th right (respectively, left) derived functor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280195.png" />.

The most important property is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280196.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280197.png" /> be additive left exact functors between Abelian categories. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280198.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280199.png" /> have enough injective objects. Assume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280200.png" /> sends injective objects into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280201.png" />-acyclic objects. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280202.png" />. A similar statement holds for left derived functors. See also [[Derived functor|Derived functor]].

==Verdier duality.==
The concept of derived categories is very well suited to state and prove a result on duality by Verdier (cf. [[#References|[a8]]]). For related topics such as [[Alexander duality|Alexander duality]] and [[Poincaré duality|Poincaré duality]] see also [[#References|[a6]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280204.png" /> be topological spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280205.png" /> be a [[Noetherian ring|Noetherian ring]]. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280207.png" /> are locally compact and of finite dimension. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280208.png" /> be the Abelian category of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280209.png" />-modules. This category has enough injective objects. Denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280210.png" /> the derived category. Consider a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280211.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280212.png" /> be the functor direct image with proper support. This is an additive left exact functor.

Verdier duality. There exists an additive functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280213.png" /> and a natural isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280214.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280215.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280216.png" />.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280217.png" /> and put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280218.png" />. This is called the dualizing sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280219.png" />. For any object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280220.png" /> the Verdier dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280221.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280222.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.A. Beilinson,   J. Bernstein,   P. Deligne,   "Faisceaux pervers"  ''Astérisque. Analyse et topologie sur les espaces singuliers (I)'' , '''100'''  (1982)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Borel,   et al.,   "Intersection cohomology" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Deligne,   "Cohomology à supports propres" , ''Sem. Geom. Alg. 4. Exp. 17'' , ''Lect. notes in math.'' , '''305''' , Springer  (1973)  pp. 82–115</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P.-P. Grivel,   "Catégories derivées et foncteurs derivés"  A. Borel (ed.)  et al. (ed.) , ''Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031280/d031280223.png" />-modules'' , Acad. Press  (1987)  pp. 1–108</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Hartshorne,   "Residues and duality" , Springer  (1966)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Iversen,   "Cohomology of sheaves" , Springer  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.-L. Verdier,   "Categories derivées, Etat 0" , ''Sem. Geom. Alg. 4 1/2. Cohomologie etale'' , ''Lect. notes in math.'' , '''569''' , Springer  (1977)  pp. 262–311</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.-L. Verdier,   "Dualité dans la cohomologie des espaces localement compacts" , ''Sem. Bourbaki. Exp 300''  (1965–1966)</TD></TR></table>