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is a distinguished triangle.
 
is a distinguished triangle.
  
An additive functor between two triangulated categories is called a  $  \delta $-
+
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.
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 ) $
 
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 ) $
Line 81: Line 80:
  
 
where  $  w $
 
where  $  w $
is thought of as a  "morphism of degree 1"  from  $  Z \rightarrow X $(
+
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) $).  
which, by definition, is the same thing as a morphism  $  Z \rightarrow T ( X) $).  
+
Whence the terminology  "triangulated category" . Writing  $  \Hom  ^ {i} ( X , Y ) $
Whence the terminology  "triangulated category" . Writing  $  \mathop{\rm Hom} ^ {i} ( X , Y ) $
 
 
for the group of morphisms  $  {\mathcal C} ( X , T  ^ {i} 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 $
 
one finds straightforwardly from TR1)–TR3) for each distinguished triangle and object  $  M $
Line 90: Line 88:
  
 
$$  
 
$$  
\dots \rightarrow   \mathop{\rm Hom} ^ {i} ( M , X )  \rightarrow   \mathop{\rm Hom^ {i}
+
\dots \rightarrow \Hom  ^ {i} ( M , X )  \rightarrow \Hom   ^ {i}
 
( M , Y ) \rightarrow
 
( M , Y ) \rightarrow
 
$$
 
$$
Line 96: Line 94:
 
$$  
 
$$  
 
\rightarrow \  
 
\rightarrow \  
  \mathop{\rm Hom} ^ {i} ( M , Z )  \rightarrow   \mathop{\rm Hom} ^ {i+} 1 ( M , X )  \rightarrow \dots ,
+
  \Hom  ^ {i} ( M , Z )  \rightarrow \Hom  ^ {i+ 1} ( M , X )  \rightarrow \dots ,
 
$$
 
$$
  
 
$$  
 
$$  
\dots \rightarrow   \mathop{\rm Hom} ^ {i} ( Z , M )  \rightarrow   \mathop{\rm Hom} ^ {i} ( Y , M ) \rightarrow
+
\dots \rightarrow \Hom  ^ {i} ( Z , M )  \rightarrow \Hom  ^ {i} ( Y , M ) \rightarrow
 
$$
 
$$
  
 
$$  
 
$$  
 
\rightarrow \  
 
\rightarrow \  
\mathop{\rm Hom} ^ {i} ( X , M )  \rightarrow   \mathop{\rm Hom^ {i+} 1 ( Z , M )  \rightarrow \dots .
+
\Hom  ^ {i} ( X , M )  \rightarrow \Hom   ^ {i+ 1} ( Z , M )  \rightarrow \dots .
 
$$
 
$$
  
Line 178: Line 176:
  
 
Let  $  {\mathcal A} $
 
Let  $  {\mathcal A} $
be an [[Abelian category|Abelian category]]. Denote by  $  \textrm{ C } ( {\mathcal A} ) $
+
be an [[Abelian category|Abelian category]]. Denote by  $  \mathsf{ C } ( {\mathcal A} ) $
 
the additive category of complexes of  $  {\mathcal A} $.  
 
the additive category of complexes of  $  {\mathcal A} $.  
 
The translation functor  $  T $
 
The translation functor  $  T $
is defined by  $  T ( X ^ { \bullet } )  ^ {i} = X  ^ {i+} 1 $,  
+
is defined by  $  T ( X ^ { \bullet } )  ^ {i} = X  ^ {i+ 1} $,  
$  d _ {T(} X) = - d _ {X} $,  
+
$  d _ {T( X)} = - d _ {X} $,  
 
and one often writes  $  X ^ { \bullet } $
 
and one often writes  $  X ^ { \bullet } $
instead of  $  T ( X ^ { \bullet } ) $[[#References|[a1]]]. Denoted by  $  \textrm{ K } ( {\mathcal A} ) $
+
instead of  $  T ( X ^ { \bullet } ) $[[#References|[a1]]]. Denoted by  $  \mathsf{ K } ( {\mathcal A} ) $
the additive category whose objects are the objects of  $  \textrm{ C } ( {\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  $  \textrm{ 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 } ) $.  
 
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 } $
 
Here  $  C _ {u} = T ( X ^ { \bullet } ) \oplus Y ^ { \bullet } $
 
denotes the maping cone (cf. [[Mapping-cone construction|Mapping-cone construction]]) of  $  u $.  
 
denotes the maping cone (cf. [[Mapping-cone construction|Mapping-cone construction]]) of  $  u $.  
Similarly one defines  $  \textrm{ K }  ^ {+} ( {\mathcal A} ) $(
+
Similarly one defines  $  \mathsf{ K }  ^ {+} ( {\mathcal A} ) $ (respectively,  $  \mathsf{ K }  ^ {-} ( {\mathcal A} ) $,  
respectively,  $  \textrm{ K }  ^ {-} ( {\mathcal A} ) $,  
+
respectively,  $  \mathsf{ K } ^ {\mathsf{ b } } ( {\mathcal A} ) $),  
respectively,  $  \textrm{ K } ^ {\textrm{ b } } ( {\mathcal A} ) $),  
 
 
the category of bounded below (respectively, bounded above, respectively, bounded) complexes of  $  {\mathcal A} $.  
 
the category of bounded below (respectively, bounded above, respectively, bounded) complexes of  $  {\mathcal A} $.  
 
A complex  $  X ^ { \bullet } $
 
A complex  $  X ^ { \bullet } $
Line 199: Line 196:
 
large enough, etc.
 
large enough, etc.
  
Let  $  X ^ { \bullet } , Y ^ { \bullet } \in \textrm{ K } ( {\mathcal A} ) $.  
+
Let  $  X ^ { \bullet } , Y ^ { \bullet } \in \mathsf{ K } ( {\mathcal A} ) $.  
 
A morphism  $  f :  X ^ { \bullet } \rightarrow Y ^ { \bullet } $
 
A morphism  $  f :  X ^ { \bullet } \rightarrow Y ^ { \bullet } $
 
is called a quasi-isomorphism if it induces an isomorphism on cohomology. Let  $  \mathop{\rm Qis} $
 
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|Localization in categories]])  $  \textrm{ D } ( {\mathcal A} ) = \textrm{ K } ( {\mathcal A} ) _ { \mathop{\rm Qis}  } $
+
be the collection of all quasi-isomorphisms. The localized category (cf. [[Localization in categories|Localization in categories]])  $  \mathsf{ D } ( {\mathcal A} ) = \mathsf{ K } ( {\mathcal A} ) _ { \mathop{\rm Qis}  } $
 
is called the derived category of  $  {\mathcal A} $.  
 
is called the derived category of  $  {\mathcal A} $.  
Similarly one defines  $  \textrm{ D }  ^ {+} ( {\mathcal A} ) $(
+
Similarly one defines  $  \mathsf{ D }  ^ {+} ( {\mathcal A} ) $ (respectively,  $  \mathsf{ D }  ^ {-} ( {\mathcal A} ) $,  
respectively,  $  \textrm{ D }  ^ {-} ( {\mathcal A} ) $,  
+
respectively,  $  \mathsf{ D } ^ {b } ( {\mathcal A} ) $).  
respectively,  $  \textrm{ D } ^ {\textrm{ b } } ( {\mathcal A} ) $).  
+
Every short [[Exact sequence|exact sequence]] gives rise to a distinguished triangle in  $  \mathsf{ D } ( {\mathcal A} ) $.
Every short [[Exact sequence|exact sequence]] gives rise to a distinguished triangle in  $  \textrm{ D } ( {\mathcal A} ) $.
 
  
 
Assume that  $  {\mathcal A} $
 
Assume that  $  {\mathcal A} $
 
has enough injectives (cf. [[Injective object|Injective object]]). Denote by  $  {\mathcal I} \subset  {\mathcal A} $
 
has enough injectives (cf. [[Injective object|Injective object]]). Denote by  $  {\mathcal I} \subset  {\mathcal A} $
 
the collection of injective objects in  $  {\mathcal A} $
 
the collection of injective objects in  $  {\mathcal A} $
and let  $  \textrm{ K }  ^ {+} ( {\mathcal I} ) $
+
and let  $  \mathsf{ K }  ^ {+} ( {\mathcal I} ) $
be the triangulated subcategory of  $  \textrm{ K }  ^ {+} ( {\mathcal A} ) $
+
be the triangulated subcategory of  $  \mathsf{ K }  ^ {+} ( {\mathcal A} ) $
 
consisting of bounded below complexes of injective objects in  $  {\mathcal A} $.  
 
consisting of bounded below complexes of injective objects in  $  {\mathcal A} $.  
The canonical functor  $  Q :  \textrm{ K }  ^ {+} ( {\mathcal A} ) \rightarrow \textrm{ D }  ^ {+} ( {\mathcal A} ) $
+
The canonical functor  $  Q :  \mathsf{ K }  ^ {+} ( {\mathcal A} ) \rightarrow \mathsf{ D }  ^ {+} ( {\mathcal A} ) $
induces an equivalence of categories  $  \textrm{ K }  ^ {+} ( {\mathcal I} ) \rightarrow \textrm{ D }  ^ {+} ( {\mathcal A} ) $.  
+
induces an equivalence of categories  $  \mathsf{ K }  ^ {+} ( {\mathcal I} ) \rightarrow \mathsf{ D }  ^ {+} ( {\mathcal A} ) $.  
A similar discussion applies to  $  \textrm{ D }  ^ {-} ( {\mathcal A} ) $
+
A similar discussion applies to  $  \mathsf{ D }  ^ {-} ( {\mathcal A} ) $
 
in case  $  {\mathcal A} $
 
in case  $  {\mathcal A} $
 
has enough projectives (cf. [[Projective object of a category|Projective object of a category]]).
 
has enough projectives (cf. [[Projective object of a category|Projective object of a category]]).
Line 223: Line 219:
 
Finally, let  $  {\mathcal A} $
 
Finally, let  $  {\mathcal A} $
 
be an Abelian category and let  $  {\mathcal A} _ {1} \subset  {\mathcal A} $
 
be an Abelian category and let  $  {\mathcal A} _ {1} \subset  {\mathcal A} $
be a thick Abelian subcategory. Define  $  \textrm{ K } _ { {\mathcal A} _ {1}  } ( {\mathcal A} ) $
+
be a thick Abelian subcategory. Define  $  \mathsf{ K } _ { {\mathcal A} _ {1}  } ( {\mathcal A} ) $
as the full triangulated subcategory of  $  \textrm{ K } ( {\mathcal A} ) $
+
as the full triangulated subcategory of  $  \mathsf{ K } ( {\mathcal A} ) $
 
consisting of the complexes whose cohomology objects are in  $  {\mathcal A} _ {1} $,  
 
consisting of the complexes whose cohomology objects are in  $  {\mathcal A} _ {1} $,  
and put  $  \textrm{ D } _ { {\mathcal A} _ {1}  } ( {\mathcal A} ) = \textrm{ K } _ { {\mathcal A} _ {1}  } ( {\mathcal A} ) _ { \mathop{\rm Qis}  } $.  
+
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  $  \textrm{ D } ( {\mathcal A} ) $
+
This is the full subcategory of  $  \mathsf{ D } ( {\mathcal A} ) $
 
consisting of those complexes whose cohomology objects are in  $  {\mathcal A} _ {1} $.
 
consisting of those complexes whose cohomology objects are in  $  {\mathcal A} _ {1} $.
  
Line 233: Line 229:
 
Let  $  {\mathcal A} $
 
Let  $  {\mathcal A} $
 
and  $  {\mathcal B} $
 
and  $  {\mathcal B} $
be Abelian categories. Let  $  F :  \textrm{ K }  ^ {*} ( {\mathcal A} ) \rightarrow \textrm{ K } ( {\mathcal B} ) $
+
be Abelian categories. Let  $  F :  \mathsf{ K }  ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ K } ( {\mathcal B} ) $
be a  $  \delta $-
+
be a  $  \delta $-functor (where  $  * $
functor (where  $  * $
 
 
is  $  \emptyset $,  
 
is  $  \emptyset $,  
 
$  + $,  
 
$  + $,  
 
$  - $,  
 
$  - $,  
or b). One says that the right derived functor  $  \textrm{ R }  ^ {*} F $(
+
or b). One says that the right derived functor  $  \mathsf{ R }  ^ {*} F $ (respectively, left derived functor  $  \mathsf{ L }  ^ {*} F  $)  
respectively, left derived functor  $  \textrm{ L }  ^ {*} F  $)  
 
 
of  $  F $
 
of  $  F $
exists if the functor  $  G \mapsto \mathop{\rm Hom} ( Q F , G Q ) $(
+
exists if the functor  $  G \mapsto \Hom ( Q F , G Q ) $ (respectively,  $  G \mapsto  \Hom ( G Q , Q F  ) $)  
respectively,  $  G \mapsto  \mathop{\rm Hom} ( G Q , Q F  ) $)  
+
from the category of  $  \delta $-functors  $  G :  \mathsf{ D }  ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ D } ( {\mathcal B} ) $
from the category of  $  \delta $-
+
to the category of sets is representable (cf. [[Representable functor|Representable functor]]). In that case  $  \mathsf{ R }  ^ {*} F :  \mathsf{ D }  ^ {*} ( {\mathcal A} ) \rightarrow \mathsf{ D } ( {\mathcal B} ) $ (respectively,  $  L  ^ {*} F  $)  
functors  $  G :  \textrm{ D }  ^ {*} ( {\mathcal A} ) \rightarrow \textrm{ D } ( {\mathcal B} ) $
 
to the category of sets is representable (cf. [[Representable functor|Representable functor]]). In that case  $  \textrm{ R }  ^ {*} F :  \textrm{ D }  ^ {*} ( {\mathcal A} ) \rightarrow \textrm{ D } ( {\mathcal B} ) $(
 
respectively,  $  L  ^ {*} F  $)  
 
 
is, by definition, a representative. For every  $  i \in \mathbf Z $
 
is, by definition, a representative. For every  $  i \in \mathbf Z $
one puts  $  \textrm{ R }  ^ {i} F = H  ^ {i} \circ \textrm{ R }  ^ {*} F $(
+
one puts  $  \mathsf{ R }  ^ {i} F = H  ^ {i} \circ \mathsf{ R }  ^ {*} F $ (respectively,  $  \mathsf{ L }  ^ {i} F = H  ^ {i} \circ \mathsf{ L }  ^ {*} F  $).
respectively,  $  \textrm{ L }  ^ {i} F = H  ^ {i} \circ \textrm{ L }  ^ {*} F  $).
 
  
Concerning existence one has the following. Suppose  $  L \subset  \textrm{ K }  ^ {*} ( {\mathcal A} ) $
+
Concerning existence one has the following. Suppose  $  L \subset  \mathsf{ K }  ^ {*} ( {\mathcal A} ) $
is a triangulated subcategory such that: 1) every object of  $  \textrm{ 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 $;  
 
admits a quasi-isomorphism into (respectively, from) an object of  $  L $;  
 
and 2) for every acyclic object  $  I ^ { \bullet } \in L $,  
 
and 2) for every acyclic object  $  I ^ { \bullet } \in L $,  
 
$  F ( I ^ { \bullet } ) $
 
$  F ( I ^ { \bullet } ) $
 
is acyclic. (An acyclic complex  $  X ^ { \bullet } $
 
is acyclic. (An acyclic complex  $  X ^ { \bullet } $
is one whose cohomology is zero.) Then the right derived functor  $  \textrm{ R }  ^ {*} F $(
+
is one whose cohomology is zero.) Then the right derived functor  $  \mathsf{ R }  ^ {*} F $ (respectively, left derived functor  $  \mathsf{ L }  ^ {*} F  $)  
respectively, left derived functor  $  \textrm{ L }  ^ {*} F  $)  
 
 
exists and for every object  $  I ^ { \bullet } \in L $
 
exists and for every object  $  I ^ { \bullet } \in L $
one has  $  Q F ( I ^ { \bullet } ) \cong \textrm{ R }  ^ {*} F ( Q ( I ^ { \bullet } ) ) $(
+
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 } ) ) $).
respectively,  $  Q F ( I ^ { \bullet } ) \cong \textrm{ L }  ^ {*} F ( Q ( I ^ { \bullet } ) ) $).
 
  
 
Let  $  {\mathcal A} $
 
Let  $  {\mathcal A} $
Line 268: Line 256:
 
be Abelian categories and let  $  F :  {\mathcal A} \rightarrow {\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|Exact functor]]). Suppose that  $  {\mathcal A} $
 
be an additive left exact (respectively, right exact) functor (cf. [[Exact functor|Exact functor]]). Suppose that  $  {\mathcal A} $
has enough injective (respectively, projective) objects. Then  $  \textrm{ R }  ^ {+} F $(
+
has enough injective (respectively, projective) objects. Then  $  \mathsf{ R }  ^ {+} F $ (respectively,  $  \mathsf{ L }  ^ {-} F  $)  
respectively,  $  \textrm{ L }  ^ {-} F  $)  
+
exists. The functor  $  \mathsf{ R }  ^ {i} F $ (respectively,  $  \mathsf{ L }  ^ {i} F  $)  
exists. The functor  $  \textrm{ R }  ^ {i} F $(
+
coincides with the usual  $  i $-th right (respectively, left) derived functor of  $  F $.
respectively,  $  \textrm{ 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} $,  
 
The most important property is the following. Let  $  F :  {\mathcal A} \rightarrow {\mathcal B} $,  
Line 280: Line 265:
 
and  $  {\mathcal B} $
 
and  $  {\mathcal B} $
 
have enough injective objects. Assume  $  F $
 
have enough injective objects. Assume  $  F $
sends injective objects into  $  G $-
+
sends injective objects into  $  G $-acyclic objects. Then  $  \mathsf{ R }  ^ {+} ( G \circ F  ) \cong \mathsf{ R }  ^ {+} G \circ \mathsf{ R }  ^ {+} F $.  
acyclic objects. Then  $  \textrm{ R }  ^ {+} ( G \circ F  ) \cong \textrm{ R }  ^ {+} G \circ \textrm{ R }  ^ {+} F $.  
 
 
A similar statement holds for left derived functors. See also [[Derived functor|Derived functor]].
 
A similar statement holds for left derived functors. See also [[Derived functor|Derived functor]].
  
Line 291: Line 275:
 
and  $  Y $
 
and  $  Y $
 
are locally compact and of finite dimension. Let  $  \mathop{\rm Sh} ( X , R ) $
 
are locally compact and of finite dimension. Let  $  \mathop{\rm Sh} ( X , R ) $
be the Abelian category of sheaves of  $  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 ) ) $
modules. This category has enough injective objects. Denote by  $  \textrm{ D }  ^ {+} ( X , R ) = \textrm{ D }  ^ {+} (  \mathop{\rm Sh} ( X , R ) ) $
 
 
the derived category. Consider a continuous mapping  $  f :  X \rightarrow Y $
 
the derived category. Consider a continuous mapping  $  f :  X \rightarrow Y $
 
and let  $  f _ {!} $
 
and let  $  f _ {!} $
 
be the functor direct image with proper support. This is an additive left exact functor.
 
be the functor direct image with proper support. This is an additive left exact functor.
  
Verdier duality. There exists an additive functor  $  f ^ { ! } :  \textrm{ D }  ^ {+} ( Y , R ) \rightarrow \textrm{ D }  ^ {+} ( X , R ) $
+
Verdier duality. There exists an additive functor  $  f ^ { ! } :  \mathsf{ D }  ^ {+} ( Y , R ) \rightarrow \mathsf{ D }  ^ {+} ( X , R ) $
and a natural isomorphism  $  \textrm{ R }   \mathop{\rm Hom} ( \textrm{ R } f _ {!} F , G ) \cong \textrm{ R } f _ {*} \textrm{ R Hom  } ( F , f ^ { ! } G ) $,  
+
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 \textrm{ D }  ^ {-} ( X , R ) $,  
+
for all  $  F \in \mathsf{ D }  ^ {-} ( X , R ) $,  
$  G \in \textrm{ D }  ^ {+} ( Y , R) $.
+
$  G \in \mathsf{ D }  ^ {+} ( Y , R) $.
  
 
Suppose that  $  Y = \{  \mathop{\rm pt} \} $
 
Suppose that  $  Y = \{  \mathop{\rm pt} \} $
 
and put  $  D _ {X} = f ^ { ! } R _ { \mathop{\rm pt}  } $.  
 
and put  $  D _ {X} = f ^ { ! } R _ { \mathop{\rm pt}  } $.  
 
This is called the dualizing sheaf on  $  X $.  
 
This is called the dualizing sheaf on  $  X $.  
For any object  $  F \in \textrm{ D } ^ {\textrm{ b } } ( X , R ) $
+
For any object  $  F \in \mathsf{ D } ^ {b } ( X , R ) $
 
the Verdier dual of  $  F $
 
the Verdier dual of  $  F $
is  $  \textrm{ R Hom } ( F , D _ {X} ) $.
+
is  $  \mathsf{ R } \Hom ( F , D _ {X} ) $.
  
 
====References====
 
====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 $D$-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>
 
<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 $D$-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>

Latest revision as of 06:56, 10 May 2022


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)
How to Cite This Entry:
Derived category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_category&oldid=51294
This article was adapted from an original article by M.G.M. van Doorn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article