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
be an
additive category
equipped with an additive automorphism
,
called the
translation functor.
A
triangle
in
is a sextuple
of objects
,
,
in
and morphism
,
,
.
One often uses
to denote such a triangle. It is obvious what
it meant by a morphism of triangles. The category
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
for a triangle
,
these axioms are as follows.
TR1)
Each triangle isomorphic to a distinguished
triangle is distinguished. For each morphism
there is a distinguished triangle
;
is distinguished.
TR2)
is distinguished if and only if
is distinguished.
TR3)
If
,
are distinguished and
is a morphism, then there is an
such that
is a morphism of triangles.
TR4)
Let
,
,
be three distinguished triangles with
,
,
.
Then there exists two morphisms
,
such that
,
are morphisms of triangles and such that
is a distinguished triangle.
An additive functor between two triangulated categories is called a
-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
as
where
is thought of as a
"morphism of degree 1"
from
(which, by definition, is the same thing as a morphism
).
Whence the terminology
"triangulated category" .
Writing
for the group of morphisms
one finds straightforwardly from TR1)–TR3) for each distinguished triangle and object
of
long exact sequences of groups
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
be a triangulated category. A collection
of morphism s in
is called a
multiplicative system
if it satisfies properties (FR1)–(FR5) (given in
[a7]).
(FR1) If
and
are in
,
then so is
.
All identity morphisms are in
.
(FR2) If
is in
and
,
then there are an
in
and a
such that
,
and (symmetrically) if
is in
and
,
then there are an
in
and a
such that
.
(FR3) For all
there are
such that
,
.
(FR4) If
,
then also
.
(FR5) If
and
are two distinguished triangles and
is a morphism from
to
with
,
then there is an
such that
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
with respect to
is a category
together with the canonical functor
such that the pair
has the universal property: Any functor
such that
is an isomorphism for all
factors uniquely through
.
Such a pair exists and, moreover,
carries a unique structure of a triangulated category such that
is exact. Note that the objects of
are the objects of
and that a morphism from
to
in
may be represented by a diagram
of morphisms in
such that
.
Let
be an
Abelian category.
Denote by
the additive category of complexes of
.
The translation functor
is defined by
,
,
and one often writes
instead of
[a1].
Denoted by
the additive category whose objects are the objects of
and whose morphisms are homotopy equivalence classes of morphisms in
.
Call a triangle distinguished if it is isomorphic to a triangle of the form
.
Here
denotes the maping cone (cf.
Mapping-cone construction)
of
.
Similarly one defines
(respectively,
,
respectively,
),
the category of bounded below (respectively,
bounded above, respectively, bounded) complexes of
.
A complex
is
bounded above
if
for
large enough, etc.
Let
.
A morphism
is called a
quasi-isomorphism
if it induces an isomorphism on cohomology. Let
be the collection of all quasi-isomorphisms. The localized category (cf.
Localization in categories)
is called the
derived category
of
.
Similarly one defines
(respectively,
,
respectively,
).
Every short
exact sequence
gives rise to a distinguished triangle in
.
Assume that
has enough injectives (cf.
Injective object).
Denote by
the collection of injective objects in
and let
be the triangulated subcategory of
consisting of bounded below complexes of injective objects in
.
The canonical functor
induces an equivalence of categories
.
A similar discussion applies to
in case
has enough projectives (cf.
Projective object of a category).
Finally, let
be an Abelian category and let
be a thick Abelian subcategory. Define
as the full triangulated subcategory of
consisting of the complexes whose cohomology objects are in
,
and put
.
This is the full subcategory of
consisting of those complexes whose cohomology objects are in
.
The derived functor.
Let
and
be Abelian categories. Let
be a
-functor
(where
is
,
,
,
or b). One says that the right
derived functor
(respectively, left derived functor
)
of
exists if the functor
(respectively,
)
from the category of
-functors
to the category of sets is representable (cf.
Representable functor).
In that case
(respectively,
)
is, by definition, a representative. For every
one puts
(respectively,
).
Concerning existence one has the following. Suppose
is a triangulated subcategory such that: 1) every object of
admits a quasi-isomorphism into (respectively, from) an object of
;
and 2) for every acyclic object
,
is acyclic. (An
acyclic complex
is one whose cohomology is zero.) Then the right derived functor
(respectively, left derived functor
)
exists and for every object
one has
(respectively,
).
Let
and
be Abelian categories and let
be an additive left exact (respectively, right exact) functor (cf.
Exact functor).
Suppose that
has enough injective (respectively, projective) objects. Then
(respectively,
)
exists. The functor
(respectively,
)
coincides with the usual
-th
right (respectively, left) derived functor of
.
The most important property is the following. Let
,
be additive left exact functors between Abelian categories. Assume that
and
have enough injective objects. Assume
sends injective objects into
-acyclic
objects. Then
.
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
and
be topological spaces and let
be a
Noetherian ring.
Suppose that
and
are locally compact and of finite dimension. Let
be the Abelian category of sheaves of
-modules.
This category has enough injective objects. Denote by
the derived category. Consider a continuous mapping
and let
be the functor direct image with proper support. This is an additive left exact functor.
Verdier duality.
There exists an additive functor
and a natural isomorphism
,
for all
,
.
Suppose that
and put
.
This is called the
dualizing sheaf
on
.
For any object
the Verdier dual of
is
.