Derived category

The notion of a derived category has been introduced by J.-L. Verdier in his 1963 notes [a7]. This facilitated a proof of a duality theorem of A. Grothendieck (cf. [a5]). Let $ {\mathcal C} $ be an additive category equipped with an additive automorphism $ T $, called the translation functor. A triangle in $ {\mathcal C} $ is a sextuple $ ( X , Y , Z , u , v , w ) $ of objects $ X $, $ Y $, $ Z $ in $ {\mathcal C} $ and morphism $ u : X \rightarrow Y $, $ v : Y \rightarrow Z $, $ w : Z \rightarrow T ( X) $. One often uses

$$ X \rightarrow ^ { u } Y \rightarrow ^ { v } Z \rightarrow ^ { w } T ( X) $$

to denote such a triangle. It is obvious what it meant by a morphism of triangles. The category $ {\mathcal C} $ equipped with a family of triangles, the distinguished triangles, is called a triangulated category if the axioms TR1)–TR4) in [a7] are satisfied.

Writing briefly $ ( u , v , w ) $ for a triangle $ X \rightarrow ^ {u} Y \rightarrow ^ {v} Z \rightarrow ^ {w} T ( X) $, these axioms are as follows.

TR1) Each triangle isomorphic to a distinguished triangle is distinguished. For each morphism $ u $ there is a distinguished triangle $ ( u , v , w ) $; $ ( 1 _ {X} , 0 , 0 ) $ is distinguished.

TR2) $ ( u , v , w ) $ is distinguished if and only if $ ( v , w , - T ( u) ) $ is distinguished.

TR3) If $ ( u , v , w ) $, $ ( u ^ \prime , v ^ \prime , w ^ \prime ) $ are distinguished and $ ( f , g ) : u \rightarrow u ^ \prime $ is a morphism, then there is an $ h $ such that $ ( f , g , h ) $ is a morphism of triangles.

TR4) Let $ ( u , i , a ) $, $ ( v , j , b ) $, $ ( w , k , c ) $ be three distinguished triangles with $ w = v u $, $ u : X \rightarrow Y $, $ v : Y \rightarrow Z $. Then there exists two morphisms $ f $, $ g $ such that $ ( 1 _ {X} , v , f ) $, $ ( u , 1 _ {Z} , g ) $ are morphisms of triangles and such that $ ( f , g , T ( i) b ) $ is a distinguished triangle.

An additive functor between two triangulated categories is called a $ \delta $-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 $ ( u , v , w ) $ as

$$ \begin{array}{rcc} {} & Z &{} \\ {} _ { \mathop{\rm deg} ( w) = 1 } \swarrow & &\nwarrow _ {v} \\ X &\rightarrow _ { u } & Y \\ \end{array} $$

where $ w $ is thought of as a "morphism of degree 1" from $ Z \rightarrow X $ (which, by definition, is the same thing as a morphism $ Z \rightarrow T ( X) $). Whence the terminology "triangulated category" . Writing $ \Hom ^ {i} ( X , Y ) $ for the group of morphisms $ {\mathcal C} ( X , T ^ {i} Y ) $ one finds straightforwardly from TR1)–TR3) for each distinguished triangle and object $ M $ of $ {\mathcal C} $ long exact sequences of groups

$$ \dots \rightarrow \Hom ^ {i} ( M , X ) \rightarrow \Hom ^ {i} ( M , Y ) \rightarrow $$

$$ \rightarrow \ \Hom ^ {i} ( M , Z ) \rightarrow \Hom ^ {i+ 1} ( M , X ) \rightarrow \dots , $$

$$ \dots \rightarrow \Hom ^ {i} ( Z , M ) \rightarrow \Hom ^ {i} ( Y , M ) \rightarrow $$

$$ \rightarrow \ \Hom ^ {i} ( X , M ) \rightarrow \Hom ^ {i+ 1} ( Z , M ) \rightarrow \dots . $$

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 $ {\mathcal C} $ be a triangulated category. A collection $ S $ of morphism s in $ {\mathcal C} $ is called a multiplicative system if it satisfies properties (FR1)–(FR5) (given in [a7]).

(FR1) If $ s : Y \rightarrow X $ and $ t : Z \rightarrow Y $ are in $ S $, then so is $ st $. All identity morphisms are in $ S $.

(FR2) If $ s : Y \rightarrow X $ is in $ S $ and $ f : X ^ \prime \rightarrow X $, then there are an $ s ^ \prime : Y ^ \prime \rightarrow X ^ \prime $ in $ S $ and a $ g : Y ^ \prime \rightarrow Y $ such that $ f s ^ \prime = s g $, and (symmetrically) if $ s : Y \rightarrow X $ is in $ S $ and $ f : Y \rightarrow Y ^ \prime $, then there are an $ s ^ \prime : Y ^ \prime \rightarrow X ^ \prime $ in $ S $ and a $ g : X \rightarrow X ^ \prime $ such that $ s ^ \prime f = g s $.

(FR3) For all $ f , g : X \rightarrow Y $ there are $ s, t \in S $ such that $ s f = s g $, $ f t = g t $.

(FR4) If $ s \in S $, then also $ T ( s) \in S $.

(FR5) If $ ( u , v , w ) $ and $ ( u ^ \prime , v ^ \prime , w ^ \prime ) $ are two distinguished triangles and $ ( s , t ) $ is a morphism from $ u $ to $ u ^ \prime $ with $ s , t \in S $, then there is an $ r \in S $ such that $ ( s , t , r ) $ 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). The other two are special for this particular setting of triangulated categories.

The localization of $ {\mathcal C} $ with respect to $ S $ is a category $ {\mathcal C} _ {S} $ together with the canonical functor $ Q : {\mathcal C} \rightarrow {\mathcal C} _ {S} $ such that the pair $ ( {\mathcal C} _ {S} , Q ) $ has the universal property: Any functor $ F : {\mathcal C} \rightarrow {\mathcal D} $ such that $ F( s) $ is an isomorphism for all $ s \in S $ factors uniquely through $ Q $.

Such a pair exists and, moreover, $ {\mathcal C} _ {S} $ carries a unique structure of a triangulated category such that $ Q $ is exact. Note that the objects of $ {\mathcal C} _ {S} $ are the objects of $ {\mathcal C} $ and that a morphism from $ X $ to $ Y $ in $ {\mathcal C} _ {S} $ may be represented by a diagram $ X \leftarrow ^ {s} Z \rightarrow ^ {f} Y $ of morphisms in $ {\mathcal C} $ such that $ s \in S $.

Let $ {\mathcal A} $ be an Abelian category. Denote by $ \mathsf{ C } ( {\mathcal A} ) $ the additive category of complexes of $ {\mathcal A} $. The translation functor $ T $ is defined by $ T ( X ^ { \bullet } ) ^ {i} = X ^ {i+ 1} $, $ d _ {T( X)} = - d _ {X} $, and one often writes $ X ^ { \bullet } $ instead of $ T ( X ^ { \bullet } ) $[a1]. Denoted by $ \mathsf{ K } ( {\mathcal A} ) $ the additive category whose objects are the objects of $ \mathsf{ C } ( {\mathcal A} ) $ and whose morphisms are homotopy equivalence classes of morphisms in $ \mathsf{ C } ( {\mathcal A} ) $. Call a triangle distinguished if it is isomorphic to a triangle of the form $ X ^ { \bullet } \rightarrow ^ {u} Y ^ { \bullet } \rightarrow C _ {u} \rightarrow T ( X ^ { \bullet } ) $. Here $ C _ {u} = T ( X ^ { \bullet } ) \oplus Y ^ { \bullet } $ denotes the maping cone (cf. Mapping-cone construction) of $ u $. Similarly one defines $ \mathsf{ K } ^ {+} ( {\mathcal A} ) $ (respectively, $ \mathsf{ K } ^ {-} ( {\mathcal A} ) $, respectively, $ \mathsf{ K } ^ {\mathsf{ b } } ( {\mathcal A} ) $), the category of bounded below (respectively, bounded above, respectively, bounded) complexes of $ {\mathcal A} $. A complex $ X ^ { \bullet } $ is bounded above if $ X ^ {n} = 0 $ for $ n $ large enough, etc.

Let $ X ^ { \bullet } , Y ^ { \bullet } \in \mathsf{ K } ( {\mathcal A} ) $. A morphism $ f : X ^ { \bullet } \rightarrow Y ^ { \bullet } $ is called a quasi-isomorphism if it induces an isomorphism on cohomology. Let $ \mathop{\rm Qis} $ be the collection of all quasi-isomorphisms. The localized category (cf. Localization in categories) $ \mathsf{ D } ( {\mathcal A} ) = \mathsf{ K } ( {\mathcal A} ) _ { \mathop{\rm Qis} } $ is called the derived category of $ {\mathcal A} $. Similarly one defines $ \mathsf{ D } ^ {+} ( {\mathcal A} ) $ (respectively, $ \mathsf{ D } ^ {-} ( {\mathcal A} ) $, respectively, $ \mathsf{ D } ^ {b } ( {\mathcal A} ) $). Every short exact sequence gives rise to a distinguished triangle in $ \mathsf{ D } ( {\mathcal A} ) $.

Assume that $ {\mathcal A} $ has enough injectives (cf. Injective object). Denote by $ {\mathcal I} \subset {\mathcal A} $ the collection of injective objects in $ {\mathcal A} $ and let $ \mathsf{ K } ^ {+} ( {\mathcal I} ) $ be the triangulated subcategory of $ \mathsf{ K } ^ {+} ( {\mathcal A} ) $ consisting of bounded below complexes of injective objects in $ {\mathcal A} $. The canonical functor $ Q : \mathsf{ K } ^ {+} ( {\mathcal A} ) \rightarrow \mathsf{ D } ^ {+} ( {\mathcal A} ) $ induces an equivalence of categories $ \mathsf{ K } ^ {+} ( {\mathcal I} ) \rightarrow \mathsf{ D } ^ {+} ( {\mathcal A} ) $. A similar discussion applies to $ \mathsf{ D } ^ {-} ( {\mathcal A} ) $ in case $ {\mathcal A} $ has enough projectives (cf. Projective object of a category).

Finally, let $ {\mathcal A} $ be an Abelian category and let $ {\mathcal A} _ {1} \subset {\mathcal A} $ be a thick Abelian subcategory. Define $ \mathsf{ K } _ { {\mathcal A} _ {1} } ( {\mathcal A} ) $ as the full triangulated subcategory of $ \mathsf{ K } ( {\mathcal A} ) $ consisting of the complexes whose cohomology objects are in $ {\mathcal A} _ {1} $, and put $ \mathsf{ D } _ { {\mathcal A} _ {1} } ( {\mathcal A} ) = \mathsf{ K } _ { {\mathcal A} _ {1} } ( {\mathcal A} ) _ { \mathop{\rm Qis} } $. This is the full subcategory of $ \mathsf{ D } ( {\mathcal A} ) $ consisting of those complexes whose cohomology objects are in $ {\mathcal A} _ {1} $.

The derived functor.

Let $ {\mathcal A} $ and $ {\mathcal B} $ be Abelian categories. Let $ F : \mathsf{ K } ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ K } ( {\mathcal B} ) $ be a $ \delta $-functor (where $ * $ is $ \emptyset $, $ + $, $ - $, or b). One says that the right derived functor $ \mathsf{ R } ^ {*} F $ (respectively, left derived functor $ \mathsf{ L } ^ {*} F $) of $ F $ exists if the functor $ G \mapsto \Hom ( Q F , G Q ) $ (respectively, $ G \mapsto \Hom ( G Q , Q F ) $) from the category of $ \delta $-functors $ G : \mathsf{ D } ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ D } ( {\mathcal B} ) $ to the category of sets is representable (cf. Representable functor). In that case $ \mathsf{ R } ^ {*} F : \mathsf{ D } ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ D } ( {\mathcal B} ) $ (respectively, $ L ^ {*} F $) is, by definition, a representative. For every $ i \in \mathbf Z $ one puts $ \mathsf{ R } ^ {i} F = H ^ {i} \circ \mathsf{ R } ^ {*} F $ (respectively, $ \mathsf{ L } ^ {i} F = H ^ {i} \circ \mathsf{ L } ^ {*} F $).

Concerning existence one has the following. Suppose $ L \subset \mathsf{ K } ^ {*} ( {\mathcal A} ) $ is a triangulated subcategory such that: 1) every object of $ \mathsf{ K } ^ {*} ( {\mathcal A} ) $ admits a quasi-isomorphism into (respectively, from) an object of $ L $; and 2) for every acyclic object $ I ^ { \bullet } \in L $, $ F ( I ^ { \bullet } ) $ is acyclic. (An acyclic complex $ X ^ { \bullet } $ is one whose cohomology is zero.) Then the right derived functor $ \mathsf{ R } ^ {*} F $ (respectively, left derived functor $ \mathsf{ L } ^ {*} F $) exists and for every object $ I ^ { \bullet } \in L $ one has $ Q F ( I ^ { \bullet } ) \cong \mathsf{ R } ^ {*} F ( Q ( I ^ { \bullet } ) ) $ (respectively, $ Q F ( I ^ { \bullet } ) \cong \mathsf{ L } ^ {*} F ( Q ( I ^ { \bullet } ) ) $).

Let $ {\mathcal A} $ and $ {\mathcal B} $ be Abelian categories and let $ F : {\mathcal A} \rightarrow {\mathcal B} $ be an additive left exact (respectively, right exact) functor (cf. Exact functor). Suppose that $ {\mathcal A} $ has enough injective (respectively, projective) objects. Then $ \mathsf{ R } ^ {+} F $ (respectively, $ \mathsf{ L } ^ {-} F $) exists. The functor $ \mathsf{ R } ^ {i} F $ (respectively, $ \mathsf{ L } ^ {i} F $) coincides with the usual $ i $-th right (respectively, left) derived functor of $ F $.

The most important property is the following. Let $ F : {\mathcal A} \rightarrow {\mathcal B} $, $ G : {\mathcal B} \rightarrow {\mathcal C} $ be additive left exact functors between Abelian categories. Assume that $ {\mathcal A} $ and $ {\mathcal B} $ have enough injective objects. Assume $ F $ sends injective objects into $ G $-acyclic objects. Then $ \mathsf{ R } ^ {+} ( G \circ F ) \cong \mathsf{ R } ^ {+} G \circ \mathsf{ R } ^ {+} F $. A similar statement holds for left derived functors. See also Derived functor.

Verdier duality.

The concept of derived categories is very well suited to state and prove a result on duality by Verdier (cf. [a8]). For related topics such as Alexander duality and Poincaré duality see also [a6]. Let $ X $ and $ Y $ be topological spaces and let $ R $ be a Noetherian ring. Suppose that $ X $ and $ Y $ are locally compact and of finite dimension. Let $ \mathop{\rm Sh} ( X , R ) $ be the Abelian category of sheaves of $ R $-modules. This category has enough injective objects. Denote by $ \mathsf{ D } ^ {+} ( X , R ) = \mathsf{ D } ^ {+} ( \mathop{\rm Sh} ( X , R ) ) $ the derived category. Consider a continuous mapping $ f : X \rightarrow Y $ and let $ f _ {!} $ be the functor direct image with proper support. This is an additive left exact functor.

Verdier duality. There exists an additive functor $ f ^ { ! } : \mathsf{ D } ^ {+} ( Y , R ) \rightarrow \mathsf{ D } ^ {+} ( X , R ) $ and a natural isomorphism $ \mathsf{ R } \Hom ( \mathsf{ R } f _ {!} F , G ) \cong \mathsf{ R } f _ {*} \mathsf{ R} \Hom ( F , f ^ { ! } G ) $, for all $ F \in \mathsf{ D } ^ {-} ( X , R ) $, $ G \in \mathsf{ D } ^ {+} ( Y , R) $.

Suppose that $ Y = \{ \mathop{\rm pt} \} $ and put $ D _ {X} = f ^ { ! } R _ { \mathop{\rm pt} } $. This is called the dualizing sheaf on $ X $. For any object $ F \in \mathsf{ D } ^ {b } ( X , R ) $ the Verdier dual of $ F $ is $ \mathsf{ R } \Hom ( F , D _ {X} ) $.

References

[a1]	A.A. Beilinson, J. Bernstein, P. Deligne, "Faisceaux pervers" Astérisque. Analyse et topologie sur les espaces singuliers (I) , 100 (1982)
[a2]	A. Borel, et al., "Intersection cohomology" , Birkhäuser (1984)
[a3]	P. Deligne, "Cohomology à supports propres" , Sem. Geom. Alg. 4. Exp. 17 , Lect. notes in math. , 305 , Springer (1973) pp. 82–115
[a4]	P.-P. Grivel, "Catégories derivées et foncteurs derivés" A. Borel (ed.) et al. (ed.) , Algebraic $D$-modules , Acad. Press (1987) pp. 1–108
[a5]	R. Hartshorne, "Residues and duality" , Springer (1966)
[a6]	B. Iversen, "Cohomology of sheaves" , Springer (1986)
[a7]	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
[a8]	J.-L. Verdier, "Dualité dans la cohomologie des espaces localement compacts" , Sem. Bourbaki. Exp 300 (1965–1966)