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.

 

1. Introduction

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

with for all , and a morphism between complexes is a family of morphisms in such that . The cohomology groups of the complex are by definition

A morphism induces a group morphism in each degree. We say that is a quasi-isomorphism if is an isomorphism for all .

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

be a projective resolution of , that is,

is a quasi-isomorphism. Then is the group homology in degree of the complex

These are well defined functors since the projective resolution of an object is unique up to quasi-isomorphisms. Moreover, for any short exact sequence there is a long exact sequence

where is the so called connecting morphism.

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.

2. Triangulated categories

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

that will be denoted as , and a morphism of triangles is a triple

forming a commutative diagram in

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

  

  there exist morphisms , such that

  

  is a distinguished triangle and

  

  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

where denotes .

REMARK 2.3. By TR2, if is a distinguished triangle then is also distinguished. Then, if is a cohomological functor,

is also exact. So we get a long exact sequence

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).

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

are isomorphisms. Then, has left and right inverse, and therefore, is an isomorphism.

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,

is exact, so there exists such that .

By Lemma 2.4(i) we know that . Since is an epimorphism we have that , and by Corollary 2.6,

is exact, so there exists such that .

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.

3. Distinguished triangles in

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

since is an ideal in , that is, if or belongs to .

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,

where . Then and are homotopy equivalences, so they are quasi-isomorphisms. Now, since is a quasi-isomorphism, the 5-lemma applied to the long exact sequences obtained from the second and the third row, implies that is a quasi-isomorphism.

DEFINITION 3.4.

• 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

Since is a short exact sequence in , it induces an exact sequence

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

where is a morphism of complexes and

is a distinguished triangle in . Moreover, the connecting morphism in the long exact sequence

is just .

4. Derived categories

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

where all are morphism in and are quasi-isomorphisms en . To work with these kind of expressions, we would like to find a "common denominator". It was Verdier's observation that we can obtain a better description of the morphisms by a "calculus of fractions" if we construct in several steps:

that is, we consider complexes and their morphisms, then complexes and homotopy classes of morphisms of complexes, and finally, we invert all quasi-isomorphisms in . Then is the localization of with respect to the class of all quasi-isomorphisms.

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].

REMARK 4.3.

• 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

with . The equivalence relation is defined as follows:

if there exists a commutative diagram

with , that is, if and only if . The composition of morphisms in can be visualized by the diagram

that is,

where the existence of and comes from S2.

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.

REMARK 4.4.

• 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

in . It is clear that we can define a map by sending to the equivalence class of the morphism given by

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. Examples

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

Since

is exact, there exists a morphism between short exact sequences

which induces the following quasi-isomorphisms

where is the composition . So is quasi-isomorphic to .

Let be objects in . Then

which is zero except for . Finally, and .

Concerning triangles, let be a morphism in . The previous computation applied to the complex

says that it is quasi-isomorphic to the complex

Hence any distinguished triangle is isomorphic to one of the form

In particular, if is a short exact sequence in , then

is a distinguished triangle in , and is the morphism associated to the given short exact sequence.

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.

6. Morita theory in

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].

DEFINITION 6.1.

1. 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 .

2. 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

1. .
2. (that is, there are no self-extensions of ).
3. 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:

1. The unbounded derived categories of and are equivalent as triangulated categories.
2. There is a tilting complex in whose endomorphism ring is isomorphic to .
3. 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.

EXAMPLE 7.1.

1. Let be a -algebra, and .
2. 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.

EXAMPLE 7.3.

1. The DG modules for the DG algebra of the first example of the previous subsection are the same as the cochain complexes of -modules.
2. 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

for all . We shall denote by the full subcategory of formed by closed objects.

EXAMPLE 7.8.

1. Free DG -modules of finte type are closed.
2. Complexes of f.g. projective -modules are closed.
3. 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 ?

PROPOSITION 7.10.

1. For all there is a quasi-isomorphism with closed.
2. The map may be completed to a triangulated functor commuting with infinite direct sums and such that it gives a triangulated equivalence .
3. 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.

EXAMPLE 7.12.

1. If is a right bounded complex, is a "projective resolution" of the complex (see [Ha66]).
2. For , and arbitrary the description of has been obtained by Spaltenstein ([Sp88]).
3. 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:

1. The functor induces bijections

   

2. The functor commutes with (infinite) direct sums.
3. 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

1. The unbounded derived categories of and are equivalent as triangulated categories.
2. 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.

9. Appendix

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:

The are the projective covers of the simple modules , and . In particular, is projective and for .

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