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Revista de la Unión Matemática Argentina
Print version ISSN 0041-6932On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.48 no.3 Bahía Blanca 2007
Derived categories and their applications
María Julia Redondo and Andrea Solotar
Abstract. In this notes we start with the basic definitions of derived categories, derived functors, tilting complexes and stable equivalences of Morita type. Our aim is to show via several examples that this is the best framework to do homological algebra, We also exhibit their usefulness for getting new proofs of well known results. Finally we consider the Morita invariance of Hochschild cohomology and other derived functors.
Derived categories were invented by A. Grothendieck and his school in the early sixties. The volume [Ve77] reproduces some notes of his pupil J.L.Verdier, dating from 1963, which are the original source on derived categories (see also [Ve96]).
Let be an abelian category and
the category of complexes in
, that is, objects are sequences of maps
![n n-1 d ∘ d = 0](/img/revistas/ruma/v48n3/3a014x.png)
![n ∈ ℤ](/img/revistas/ruma/v48n3/3a015x.png)
![f : X → Y](/img/revistas/ruma/v48n3/3a016x.png)
![fn : Xn → Y n](/img/revistas/ruma/v48n3/3a017x.png)
![A](/img/revistas/ruma/v48n3/3a018x.png)
![dnY ∘ fn = f n+1 ∘ dnX](/img/revistas/ruma/v48n3/3a019x.png)
![X](/img/revistas/ruma/v48n3/3a0110x.png)
![f : X → Y](/img/revistas/ruma/v48n3/3a0112x.png)
![n n n H (f) : H (X ) → H (Y )](/img/revistas/ruma/v48n3/3a0113x.png)
![f](/img/revistas/ruma/v48n3/3a0114x.png)
![n H (f)](/img/revistas/ruma/v48n3/3a0115x.png)
![n ∈ ℤ](/img/revistas/ruma/v48n3/3a0116x.png)
The derived category of
is obtained from
by formally inverting all quasi-isomorphisms.
Recall that the definition of derived functors in homological algebra is as follows. Assume that has enough projectives and let
be a contravariant left exact functor. Then the right derived functor
is defined in the following way. Let
be an object in
and let
![M](/img/revistas/ruma/v48n3/3a0126x.png)
is a quasi-isomorphism. Then is the group homology in degree
of the complex
![M](/img/revistas/ruma/v48n3/3a0131x.png)
![f g 0 → M1 → M2 → M3 → 0](/img/revistas/ruma/v48n3/3a0132x.png)
![δ](/img/revistas/ruma/v48n3/3a0134x.png)
In fact, when we are doing homological algebra, we are not dealing with objects and cohomology groups but with complexes up to quasi-isomorphisms and their cohomology groups. Hence it is natural to work in instead of
or
.
Any object in
can be identified in
by the complex concentrated in degree zero, which will be denoted by
. Let
be a short exact sequence in
. Then
is a short exact sequence in
. Since
is a quasi-isomorphism, we can replace by the complex appearing in the first line of the previous picture, hence the short exact sequence
can be identified in
by the sequence of complexes
which is not an exact sequence.
In this new category, the concept of "short exact sequences" (which are determined by two morphisms ) will be replaced by that of "distinguished triangles"(determined by three morphisms
). If
is a short exact sequence in
, then
is a triangle in .
We will consider "cohomological functors" defined in , that applied to distinguished triangles will give long exact sequences with morphisms induced by
. That is, the extra morphism we are considering when defining triangles is a morphism of complexes that induces the corresponding connecting morphism.
So, when considering the derived category of
, we replace objects in
by complexes, and invert quasi-isomorphisms between complexes. As we shall see,
is not abelian if
is not semisimple. The abelian structure of
, and of
, has to be replaced by a triangulated structure.
Given an algebra over a commutative ring
, for simplicity, we will write
for the derived category
. Given two
-algebras
and
the natural question is: when are
and
equivalent categories? (triangle equivalent?). Of course, if
and
are Morita equivalent (that is,
-
and
-
are
-linearly equivalent) then
and
are equivalent. We will see that there other equivalences and we shall present some examples.
Let be an additive category with an additive automorphism
called the translation functor. We will write
and
for
and
respectively. A triangle in
is a diagram of the form
![(X, Y, Z,f,g, h)](/img/revistas/ruma/v48n3/3a0192x.png)
![T](/img/revistas/ruma/v48n3/3a0194x.png)
A morphism of triangles is said to be an isomorphism if are isomorphisms in
.
DEFINITION 2.1. A structure of triangulated category on is given by a translation functor
and a class of distinguished triangles verifying the following four axioms:
-
is a distinguished triangle, for any object
;
- Every triangle isomorphic to a distinguished one is distinguished;
- Every morphism
can be embedded in a distinguished triangle
.
- (Rotation) A triangle
is distinguished if and only if
is distinguished.
- (Morphisms) Every commutative diagram
whose rows are distinguished triangles can be completed to a morphism of triangles by a morphism
.
- (The octahedral axiom) Given
and
morphisms in
, and distinguished triangles
,
such that
is a commutative diagram.
Observe that all the rows in the previous diagram are distinguished triangles, and morphisms between rows determined the following morphisms of triangles
The name of last axiom comes from the fact that it can be viewed as a picture of a octahedron, where four faces are distinguished triangles, and all the other faces are commutative.
DEFINITION 2.2. Let be a triangulated category and
an abelian category. An additive functor
is said to be a cohomological functor if, for any distinguished triangle
in
, we get an exact sequence in
![i H](/img/revistas/ruma/v48n3/3a01125x.png)
![i H ∘ T = H (- [i])](/img/revistas/ruma/v48n3/3a01126x.png)
REMARK 2.3. By TR2, if is a distinguished triangle then
is also distinguished. Then, if
is a cohomological functor,
LEMMA 2.4. Let be a distinguished triangle in
. Then
;
- If
is such that
, then there exists
such that
;
- If
is such that
, then there exists
such that
.
Proof.
- From TR1 we know that
is a distinguished triangle, and from TR3 we know that there exists a morphism of triangles
and hence
.
- From TR2 and TR3 we know that the diagram
can be completed to a morphism of triangles. Hence, there exists
such that
.
- It follows as (ii).
![□](/img/revistas/ruma/v48n3/3a01150x.png)
REMARK 2.5. Using TR2 we also get that and
.
COROLLARY 2.6. If is an object in a triangulated category
, then
and
are cohomological functors.
COROLLARY 2.7. Any distinguished triangle is determined, up to isomorphisms, by one of its morphisms.
Proof. From TR2, it suffices to prove that the distinguished triangles and
are isomorphic. By TR3, there exists a morphism of triangles
If we apply the cohomological functors and
, by the 5-lemma we get that
![t](/img/revistas/ruma/v48n3/3a01163x.png)
![t](/img/revistas/ruma/v48n3/3a01164x.png)
![□](/img/revistas/ruma/v48n3/3a01165x.png)
PROPOSITION 2.8. For any distinguished triangle , the following conditions are equivalent:
is a monomorphism;
;
- there exists
such that
;
is an epimorphism;
- there exists
such that
.
Proof.
and
are immediate.
By
and Lemma 2.4(i) we have that
. Since
is a monomorphism,
.
If
, by Corollary 2.6,
![s : Z → Y](/img/revistas/ruma/v48n3/3a01184x.png)
![g ∘ s = idZ](/img/revistas/ruma/v48n3/3a01185x.png)
By Lemma 2.4(i) we know that
. Since
is an epimorphism we have that
, and by Corollary 2.6,
![t : Y → X](/img/revistas/ruma/v48n3/3a01191x.png)
![t ∘ f = idX](/img/revistas/ruma/v48n3/3a01192x.png)
![□](/img/revistas/ruma/v48n3/3a01193x.png)
COROLLARY 2.9. In any triangulated category any monomorphism splits and any epimorphism splits. Moreover, is an isomorphism if and only if
is a monomorphism and an epimorphism.
Let be an abelian category, and let
be the category of complexes in
. The homotopy category
is defined as follows: the objects of
are the objects in
, and morphisms in
are the homotopy equivalence classes of morphisms in
. That is,
, where
if there exists
with
. So, the morphisms homotopic to zero in
become the zero morphisms in
and the homotopic equivalences become isomorphisms.
It can be checked that is well defined as a category, and moreover, it is an additive category, and the quotient
is an additive functor.
Observe that can also be defined as the quotient of
by the subgroup
![Ht](/img/revistas/ruma/v48n3/3a01216x.png)
![C(A )](/img/revistas/ruma/v48n3/3a01217x.png)
![g ∘ f ∈ Ht](/img/revistas/ruma/v48n3/3a01218x.png)
![f](/img/revistas/ruma/v48n3/3a01219x.png)
![g](/img/revistas/ruma/v48n3/3a01220x.png)
![Ht](/img/revistas/ruma/v48n3/3a01221x.png)
Let be the automorphism defined in the following way: for any complex
in
,
and
; for any morphism
in
,
. The functor
is additive and it is an automorphism. Since
is a morphism homotopic to zero if and only if
is a morphism homotopic to zero, it induces an additive automorphism
. We denote
and
, for any
.
We will see that is a triangulated category with translation functor
.
LEMMA 3.1. The cohomology functors induce well defined functors
.
Proof. If are homotopic, then
, so
.
DEFINITION 3.2. Let be a morphism in
.
- The mapping cone of
is the complex
such that
, and the differential
is given by
- The mapping cylinder of
is the complex
such that
, and the differential
is given by
PROPOSITION 3.3. Any short exact sequence in
fits into a commutative diagram
defined in degree by the following diagram
with exact rows, ,
homotopy inverse equivalences and
a quasi-isomorphism.
Proof. A direct computation shows that all the maps are morphisms of complexes. It is clear that . On the other hand,
![′ h(x,x ,y) = (0,x, 0)](/img/revistas/ruma/v48n3/3a01267x.png)
![α](/img/revistas/ruma/v48n3/3a01268x.png)
![β](/img/revistas/ruma/v48n3/3a01269x.png)
![β](/img/revistas/ruma/v48n3/3a01270x.png)
![γ](/img/revistas/ruma/v48n3/3a01271x.png)
![□](/img/revistas/ruma/v48n3/3a01272x.png)
- A short exact sequence
in
semi-splits if
splits, for all
.
- A triangle in
is a distinguished triangle if it is isomorphic, in
, to one of the form
.
THEOREM 3.5. [Ve96, II.1.3.2] The category is a triangulated category with translation functor
.
Let be the functor defined by
. For any morphism
homotopic to zero,
. Then
factors uniquely through
.
The following are immediate consequences of Proposition 3.3.
PROPOSITION 3.6. Any short exact sequence in is quasi-isomorphic to a semi-split short exact sequence.
PROPOSITION 3.7. Any distinguished triangle in is quasi-isomorphic to one induced by a semi-split short exact sequence in
.
PROPOSITION 3.8. The functor is a cohomological functor, for any
.
Proof. Let be a distinguished triangle in
. By definition, it is isomorphic to
![h δ 0 → Y → cone(f) → X [1] → 0](/img/revistas/ruma/v48n3/3a01296x.png)
![C (A )](/img/revistas/ruma/v48n3/3a01297x.png)
![□](/img/revistas/ruma/v48n3/3a01299x.png)
A simple computation shows that the connecting morphism associated to the exact sequence appearing in the previous proof is
: recall that the connecting morphism is defined by chasing the following diagram
so, if is such that
, then
. Finally,
REMARK 3.9. Let be a semi-split short exact sequence in
, that is,
. Then
is isomorphic to the complex
![h : Z → X [1 ]](/img/revistas/ruma/v48n3/3a01312x.png)
![K (A )](/img/revistas/ruma/v48n3/3a01314x.png)
![δ](/img/revistas/ruma/v48n3/3a01315x.png)
![Hn (h)](/img/revistas/ruma/v48n3/3a01317x.png)
As we said in the introduction, the derived category of
is obtained from
by formally inverting all quasi-isomorphisms. So
is the localization of
with respect to the class of quasi-isomorphisms in
: there exists a functor
sending quasi-isomorphisms to isomorphisms which is universal for such property, that is, for any functor
sending quasi-isomorphisms to isomorphisms, there exists a functor
making the following diagram commutative
The category can be obtained just by adding formally inverses for all quasi-isomorphisms. But in this case, morphisms in
are just formal expressions of the form
![fi](/img/revistas/ruma/v48n3/3a01331x.png)
![C (A )](/img/revistas/ruma/v48n3/3a01332x.png)
![si](/img/revistas/ruma/v48n3/3a01333x.png)
![C (A )](/img/revistas/ruma/v48n3/3a01334x.png)
![D (A )](/img/revistas/ruma/v48n3/3a01335x.png)
![K(A )](/img/revistas/ruma/v48n3/3a01337x.png)
![- 1 D (A ) = K (A)[S ]](/img/revistas/ruma/v48n3/3a01338x.png)
![K (A )](/img/revistas/ruma/v48n3/3a01339x.png)
![S](/img/revistas/ruma/v48n3/3a01340x.png)
DEFINITION 4.1. A class of maps in a category
is said to be a multiplicative system if it satisfies the following conditions:
- For any object
in
,
. If
are composable maps in
, then
.
- Given
,
and
with
, there exist morphisms in
completing the following diagrams in a commutative way
with
, that is,
and
.
- Given
, there exists
with
if and only if there exists
with
,
.
The multiplicative system is said to be saturated if it satisfies:
- A morphism
belongs to
if and only if there exist morphisms
and
such that
and
.
If is a triangulated category with translation functor
, the multiplicative system
is said to be compatible with the triangulation if it satisfies:
- A morphism
belongs to
if and only if
belongs to S.
- Given a morphism of triangles
, if
and
belong to
, then
belongs to
.
If is a multiplicative system, the morphisms of the localization of
with respect to
can be described by a "calculus of fractions".
If is saturated and
is the localization functor, then
belongs to
if and only if
is an isomorphism.
Finally, if is compatible with the triangulation,
is a triangulated category, with distinguished triangles isomorphic to images of distinguished triangles in
.
We refer to [Ve96, II.2] for more details.
PROPOSITION 4.2. Let be the class of quasi-isomorphisms in
. Then
is a saturated multiplicative system in
, compatible with the triangulation.
Proof. See for instance [Ha66, I.4].
- The class
of quasi-isomorphism in
is not a multiplicative system.
- The localization of
with respect to
is isomorphic to the localization of
with respect to
.
Now we can describe . The objects are those of
. The morphisms
in
are equivalence classes of pairs
![s ∈ S](/img/revistas/ruma/v48n3/3a01411x.png)
with , that is,
if and only if
. The composition of morphisms in
can be visualized by the diagram
that is,
![u](/img/revistas/ruma/v48n3/3a01420x.png)
![h](/img/revistas/ruma/v48n3/3a01421x.png)
Finally, the functor is defined as the identity in objects and sends a morphism
to the equivalence class of the pair
.
Since is saturated, a morphism
in
is a quasi-isomorphism if and only if
is an isomorphism.
- A complex
in
is quasi-isomorphic to zero if and only if
in
.
- Let
be a morphism in
. Then
in
if and only if there exists a quasi-isomorphism
such that
is homotopic to zero in
. Moreover, if
is a monomorphism (epimorphism) in
, then
is so.
- The cohomological functor
sends quasi-isomorphisms to isomorphisms, so it factors through
, inducing a cohomological functor
, for any
.
The automorphism sends quasi-isomorphisms to quasi-isomorphisms, so it induces an automorphism
.
THEOREM 4.5. The category is a triangulated category with translation functor
and
is an additive functor of triangulated categories.
Proof. The class of quasi-isomorphisms in is a multiplicative system compatible with the triangulation, so the localization is triangulated and the distinguished triangles are those isomorphic to images by
of distinguished triangles in
. Clearly
commutes with the translation functor
and sends distinguished triangles to distinguished triangles.
Since is fully faithful, we identify objects and morphisms in
with objects and morphisms in
concentrated in degree zero.
PROPOSITION 4.6. The composition is a fully faithful functor.
Proof. Denote the composition functor. For any object
in
,
is the complex concentrated in degree zero, and for any morphism
,
is the equivalence class of the pair
. Observe that the composition of
with the cohomological functor
is the identity of
.
The functor is faithful: if
, there exists a quasi-isomorphism
such that
is homotopic to zero; then
, but
is an isomorphism, and hence
.
The functor is full: let
be a representative of the equivalence class of a morphism in
from
to
. Since
is a quasi-isomorphism, the complex
has cohomology
in degree zero, and zero otherwise. Then, the morphism of complexes
is a quasi-isomorphism and is the kernel of
. Now,
, so there exists a unique morphism
in
such that
. Finally the commutative diagram
says that and
are equivalent, so
is full.
From now on, we will identify with the full subcategory
. We already know that
for any pair of objects
in
. Now we want to describe
for any
.
It is clear that , so we only have to study
for
.
Following Yoneda, let be the set of isomorphism classes of exact sequences
![A](/img/revistas/ruma/v48n3/3a01511x.png)
![Extn (X,Y ) → HomD (A)(X,Y [n]) A](/img/revistas/ruma/v48n3/3a01512x.png)
![ψ](/img/revistas/ruma/v48n3/3a01513x.png)
![f s (X → Z ← Y [n])](/img/revistas/ruma/v48n3/3a01514x.png)
with a quasi-isomorphism.
On the other hand, let be a representative of a map from
to
. Since
is a quasi-isomorphism, the complex
has cohomology zero except in degree
. Consider the following quasi-isomorphism
Observe that if , then
, so
. Hence
for any
.
If , consider the quasi-isomorphism
given by
and observe that there exists a commutative diagram
where are given by
Finally the morphism from
to
can be associated with the exact sequence appearing in the first row of the diagram
This shows that there is a close connection between and
. In fact, the following theorem holds.
THEOREM 4.7. Let be objects in
. Then
for all
;
;
for all
;
for all
.
An abelian category is said to be semisimple if any short exact sequence in
splits.
THEOREM 4.8. The derived category is abelian if and only if
is semisimple.
Proof. If is semisimple then
is equivalent to the abelian semisimple category
, as we shall see in the second example of the following section.
Assume that is abelian. We know from Proposition 2.8 that any monomorphism splits, and that any epimorphism splits. Let
be a morphism in
. Then
is equal to the composition
. Since
is an epimorphism and
is a monomorphism, they split. Hence there exist
and
such that if
then
. Assume that
is not semisimple and let
be a non-split monomorphism. Then there exists a morphism
in
such that
. Now,
is fully faithfull, so there exists a morphism
in
such that
. But
is a monomorphism, so
, a contradiction.
5.1. Hereditary categories. Let be an hereditary category, that is,
. For instance, the category of abelian groups is hereditary. In this case we can easily describe objects, morphisms and triangles in
.
We start with the description of the objects of . Let
be a complex in
. The vanishing of
implies that
is right exact for any
in
. Let
and consider the short exact sequences
which induces the following quasi-isomorphisms
where is the composition
. So
is quasi-isomorphic to
.
Let be objects in
. Then
![m - n ∈ {0,1 }](/img/revistas/ruma/v48n3/3a01604x.png)
![HomD (A)(X [n ],Y [n]) ≃ HomA (X, Y )](/img/revistas/ruma/v48n3/3a01605x.png)
![HomD (A)(X [n],Y [n + 1]) ≃ Ext1A(X, Y)](/img/revistas/ruma/v48n3/3a01606x.png)
Concerning triangles, let be a morphism in
. The previous computation applied to the complex
![f g 0 → X → Y → Z → 0](/img/revistas/ruma/v48n3/3a01612x.png)
![A](/img/revistas/ruma/v48n3/3a01613x.png)
![D(A )](/img/revistas/ruma/v48n3/3a01615x.png)
![δ ∈ HomD (A)(Z,X [1])](/img/revistas/ruma/v48n3/3a01616x.png)
5.2. Semisimple categories. Let be a semisimple category. In this case,
is hereditary, so the conclusions in the previous example hold. But now
, so
and
for all
. Hence
is equivalent to
.
5.3. . Let
be the full subcategory of finitely generated left modules over the path algebra associated to the quiver
. In this case we only have three non-isomorphic indecomposable modules: the simple projective module
, the projective module of length two
, and the simple module
. Moreover, if
, then
if
or
and it is zero otherwise, and
if
and it is zero otherwise. Then
has the following picture
where the composition of any two consecutive arrows is zero. Observe that this category has neither monomorphisms nor epimorphisms. All distinguished triangles can be visualized in the picture as the diagrams of three consecutive arrows.
Given an algebra over a commutative ring
, we shall denote
for the derived category
. Given two
-algebras
and
the natural question is: when are
and
equivalent categories? (triangle equivalent?). Of course, if
and
are Morita equivalent (that is,
-
and
-
are
-linearly equivalent) then
and
are equivalent. Are there other equivalences ? The answer is yes. In fact, we shall present some examples later.
Rickard developed [Rick89], [Rick91] a Morita theory for derived categories based on the notion of tilting complex. As we shall see this is a generalization of the notion of tilting module. A summary of the history of the subject is developed for example in [KZ98]. Keller's approach [Ke94] is a little different and we will follow it. This is also the point of view of [DG02].
- A functor
between triangulated categories
and
is said exact if it commutes whith shifts and preserves distinguished triangles, that is,
is equipped with a natural isomorphism
,
such that for every distinguished triangle
in
,
is a distinguished triangle in
.
- An equivalence between two triangulated categories is an equivalence of categories which is exact and whose inverse functor is also exact.
REMARK 6.2. If is exact then it is automatically additive.
We know from classical Morita theory that two -algebras
and
are Morita equivalent if and only if there exists a bimodule
such that it is finitely generated projective, balanced and generator
-module and
as a
-bimodule. We have that
is a generator for
-
, that is for every
-
, there exists a set
and an epimorphism
. For example,
is a generator
-module but this progenerator gives the trivial equivalence. The fact that
is f.g. projective implies that
is a direct summand of
, for some
, in particular
commutes with direct sums. The notion of tilting complex appears naturally if we look at the properties verified by
in
.
The free rank one -module
considered as a cochain complex concentrated in degree
verifies that,
,
DEFINITION 6.3. A complex in
is a generator of this category if and only if the smallest full triangulated subcategory of
containing
and closed by infinite direct sums is
.
EXAMPLE 6.4. is a generator of
.
DEFINITION 6.5. A tilting complex for a
-algebra
is a bounded cochain complex of f.g. projective
-modules which generates the derived category
and such that the graded ring of endomorphisms
is concentrated in degree
.
As a special case of tilting complexes we have the tilting modules over a finite dimensional -algebra (
is a field) (when thinking them as their projective resolutions).
DEFINITION 6.6. Let be a finite dimensional
-algebra and
a finitely generated
-module. We say that
is a tilting module if
.
(that is, there are no self-extensions of
).
- There exists a short exact sequence of
-modules
such that
are direct summands of finite direct sums of
(that is,
) for
.
REMARK 6.7. If is a projective
-module the first and second conditions are automatic. In that case the exact sequence in 3 splits, so
is a direct summand of
, for some
. The third condition is verified if and only if
is a generator in
-
. As a consequence
.
When is self-injective, every
-module of finite projective dimension is projective. Then for these algebras, tilting complexes are the same as Morita equivalences. The following theorem is due to Rickard [Rick89]
THEOREM 6.8. Given two -algebras
and
such that
or
is
-flat, then the following are equivalent:
- The unbounded derived categories of
and
are equivalent as triangulated categories.
- There is a tilting complex
in
whose endomorphism ring
is isomorphic to
.
- There exists a cochain complex of
-
-bimodules
such that the derived tensor product
is an equivalence of categories.
We shall give a proof of this theorem following Keller. For this we must recall some preliminaries first.
7. Derived category of a differential graded algebra
7.1. DG algebras. Let be a differential graded algebra (DG algebra for short), that is,
is
-graded algebra with a morphism
of degree one such that
, if
,
. The map
is called differential.
- Let
be a
-algebra, and
.
- Given a
-algebra
and a complex
of
-modules, let us take
, that is,
with
and differential defined by
, for
.
7.2. DG modules. Let be a DG algebra.
DEFINITION 7.2. A differential graded -module (DG
-module for short) is a
-graded right
-module
together with a
-linear differential
(of degree one) such that
A morphism of DG modules and
is an
-linear map
of degree zero wich commutes with the differentials.
- The DG modules for the DG algebra
of the first example of the previous subsection are the same as the cochain complexes of
-modules.
- Let
be the DG algebra of the second example of the previous subsection. Each complex
of
-modules gives rise to a DG
-module
with
-action
Also,
is a DG
-module by means of the action:
7.3. The homotopy category . We recall that the homotopy category
is defined as follows.
Its class of objects is given by -
and the set of morphisms
is defined as
where
is the equivalence relation given by identifying homotopic maps.
There is a shift operator
defined by
and ,
.
We recall that endowed with and all the triangles isomorphic to the standard triangles,
becomes a triangulated category.
EXAMPLE 7.4. For the first example of the previous subsection we have that is the standard homotopy category of cochain complexes of
-modules.
7.4. The derived category. We also recall that , where
denotes the class of all the homotopy classes of quasi-isomorphisms.
EXAMPLE 7.5. For the first example of the previous subsection we have that may be identified with the standard derived category of cochain complexes of
-modules.
REMARK 7.6. We notice that has infinite direct sums (ordinary sums of DG
-modules).
Consider now the free DG -module
and let
be a DG
-module. Then
![]() |
is bijective. In particular, each quasi-isomorphism induces a bijection
![]() | (1) |
Since then
.
DEFINITION 7.7. A DG -module
is said to be closed if
![N](/img/revistas/ruma/v48n3/3a01832x.png)
![Hp(A )](/img/revistas/ruma/v48n3/3a01833x.png)
![H (A)](/img/revistas/ruma/v48n3/3a01834x.png)
- Free DG
-modules of finte type are closed.
- Complexes of f.g. projective
-modules are closed.
- Suppose that
and
are closed and let
be a morphism of DG
-modules. Consider the mapping cone
. Then
is also closed.
REMARK 7.9. As a consequence is a triangulated subcategory of
.
Why are we interested in considering the subcategory ?
- For all
there is a quasi-isomorphism
with
closed.
- The map
may be completed to a triangulated functor commuting with infinite direct sums and such that it gives a triangulated equivalence
.
is the smallest full subcategory of
containing
and closed under infinite direct sums.
We will say that is the projective resolution of
.
EXAMPLE 7.11. For and
concentrated in degree zero, choose
to be the homotopy class of any projective resolution of
. Then
is closed since all epimorphisms split.
- If
is a right bounded complex,
is a "projective resolution" of the complex
(see [Ha66]).
- For
,
and arbitrary
the description of
has been obtained by Spaltenstein ([Sp88]).
- For
and
arbitrary, see [Ke94].
REMARK 7.13. The two last items of the previous proposition imply that coincides with the smallest full subcategory containing
and closed by infinite direct sums.
7.5. Left derived tensor functors. Let and
be two DG algebras and let
be a DG
-
-bimodule, that is,
with a
-linear map
of degree one such that
for ,
and
.
Define the DG algebra by
and the map
is given by
dn : (Bop ⊗kA)n | → (Bop ⊗kA)n+1 |
d(b ⊗ a) | = dB(b) ⊗ a + (-1)pb ⊗ dA(a), |
for and
. The product is given by
for ,
,
and
.
We notice then that is a right DG
-module since
Let be a right DG
-module. We define
as the DG
-module with action of
in
as before and with the DG structure given by
![]() |
for and
.
The -submodule generated by all differences
is stable under
and under multiplication by elements of
. So
, the quotient module of
by this submodule, is a well defined DG
-module. Moreover, this construction is functorial in
and
.
The functor -
-
-
-
yields a triangulated functor from
to
, denoted by the same symbol.
We define the left derived tensor product by
Notice that commutes with direct sums since
and
do.
LEMMA 7.14. The functor is an equivalence if and only if the following conditions hold:
- The functor
induces bijections
- The functor
commutes with (infinite) direct sums.
- The smallest full triangulated subcategory of
containing
and closed under (infinite) direct sums coincides with
.
Proof. The conditions are necessary since they hold in for
and they must be preserved by equivalences.
In order to prove that they are sufficient let us consider the full subcategory of objects
in
for which the maps
are bijective for all . This category is clearly closed under the shift and its inverse. Using the
-lemma we check that it is a triangulated subcategory. Using
we get that it is closed under (infinite) direct sums. Also, using
we see that
contains
. Thus we must have that
. So, the full subcategory
of objects
in
such that
is bijective for all in
contains
. Again it is closed under the shift and its inverse, and also closed under (infinite) direct sums. It is a triangulated category by the
-lemma. Thus
and as a consequence
is fully faithful.
Condition (c) shows that is surjective.
EXAMPLE 7.15. Suppose that is a quasi-isomorphism, that is, a morphism of DG algebras inducing an isomorphism
. Then
is an equivalence. In fact,
is isomorphic to
in
, so the conditions
and
of the lemma before hold. In order to prove
consider the commutative diagram
Similarly, is an equivalence.
We may perform compositions of left derived functors of this kind.
LEMMA 7.16. If ,
and
are DG
-algebras,
is a flat
-module,
is a DG
-
-bimodule, we have
for some DG -
-bimodule
.
Proof. Take , considering
as a DG right
-module. The morphism of functors
is clearly invertible since the composition
is an isomorphism and, using a result of [Ke94], a morphism of functor between triangulated categories is invertible if and only if it is an isomorphism when one applies the functors to a generator of the category.
Also the morphism
is invertible in for each
, using the same result and the facts that
is closed over
and that the functor
preserves quasi-isomorphisms by the
-flatness of
. Thus
and we take .
8. Applications to tilting theory
Let be a commutative ring,
a
-algebra,
a flat
-algebra. Recall Rickard's theorem:
THEOREM 8.1. Given two -algebras
and
such that
or
is
-flat, then the following are equivalent
- The unbounded derived categories of
and
are equivalent as triangulated categories.
- There exists a cochain complex of
-
-bimodules
such that the derived tensor product
is an equivalence of categories.
Proof.
Let be the given triangulated equivalence. Take
and
. There are canonical isomorphisms
Since is closed in
, we also have
Thus if
and
may be identified with
. If we view
as a DG algebra concentrated in degree zero we may view
as a DG
-A-bimodule. We claim that
is an equivalence. Since
is an equivalence, conditions
and
of lemma 7.14 clearly hold. For
use the commutative diagram
To establish a connection between and
, let us introduce the DG subalgebra
with
for
,
and
for
. Note that there are canonical morphisms
which are both quasi-isomorphisms. So, by example 7.15, we have a chain of equivalences
Applying the previous lemma twice we get the required complex of
-
-bimodules.
In this section we show an example of tilting equivalence which is not a Morita equivalence. The example is due to Schwede ([Sch04]).
Consider a field and take the
-algebra
defined by:
Up to isomorphism, there are three indecomposable -projective modules:
![P i](/img/revistas/ruma/v48n3/3a011057x.png)
![S1 = P1∕P 2](/img/revistas/ruma/v48n3/3a011058x.png)
![2 2 3 S = P ∕P](/img/revistas/ruma/v48n3/3a011059x.png)
![3 3 S = P](/img/revistas/ruma/v48n3/3a011060x.png)
![3 S](/img/revistas/ruma/v48n3/3a011061x.png)
![i pdimA (S ) = 1](/img/revistas/ruma/v48n3/3a011062x.png)
![i = 1, 2](/img/revistas/ruma/v48n3/3a011063x.png)
Let us take , which is clearly not projective. The following projective resolution of
:
may be used to compute . The module
is
-free of rank one, then
is a tilting
-module.
The -algebra
may be identified, by a direct computation, to the subalgebra of
consisting of upper triangular matrices
such that
.
Now and
are NOT Morita equivalent since their lattices of projective modules differ.
Acknowledgement: we want to thank Estanislao Herscovich for his help in the preparation of this notes.
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María Julia Redondo
Instituto de Matemática,
Universidad Nacional del Sur,
Av. Alem 1253,
(8000) Bahía Blanca, Argentina.
mredondo@criba.edu.ar
Andrea Solotar
Departamento de Matemática,
FCEyN, Universidad Nacional de Buenos Aires,
Ciudad Universitaria, Pabellón I,
(1428) Ciudad Autónoma de Buenos Aires, Argentina.
asolotar@dm.uba.ar
Recibido: 26 de octubre de 2006
Aceptado: 10 de abril de 2007