versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.2 Bahía Blanca jul./dic. 2007
Introduction to Koszul Algebras
Dedicated to Héctor A. Merklen and María Inés Platzeck on their birthday.
Partially supported by a grant from PAPIIT, Universidad Nacional Autónoma de México. The author grateful thanks Dan Zacharia for helping him to improve the quality of the notes.
Abstract Las álgebras de Koszul fueron inventadas por Priddy [P] y han tenido un enorme desarrollo durante los últimos diez años, el artículo de Beilinson, Ginsburg y Soergel [BGS] ha sido muy influyente. En estas notas veremos los teoremas básicos de Álgebras de Koszul usando métodos de teoría de anillos y módulos, como se hizo en los artículos [GM1],[GM2], después nos concentraremos en el estudio de las álgebras Koszul autoinyectivas, primero las de radical cubo cero y posteriormente el caso general y por último aplicaremos los resultados obtenidos al estudio de las gavillas coherentes sobre el espacio proyectivo.
1. GRADED ALGEBRAS
In this lecture we recall the basic notions and definitions that will be used throughout this mini-course. We will always denote the base field by . We will say that an algebra is a positively graded -algebra, if
- , for all , and
- is a finite dimensional -vector space
We will say is locally finite if in addition, for each , is a finite dimensional -vector space.
The elements of are called homogeneous of degree , and is the degree component of . For example, the path algebra of any quiver is a graded algebra, whose degree component is the -vector space spanned by all the paths of length We can associate to each positively graded algebra a quiver with set of vertices , where corresponds to the primitive idempotent of having in the -th entry and zero everywhere else. The arrows of with correspond to the elements of some basis of Note that there is a homomorphism of graded -algebras
given by and
The morphism is surjective if and only if for all and In such a case we say is a graded quiver algebra.
If is a graded -algebra, we will denote by (and also by ) the radical (also called the graded Jacobson radical) of . It is the homogeneous ideal It is also easy to see that equals the intersection of all the maximal homogeneous ideals of . Let be a finite quiver. A homogeneous ideal in is called admissible if An element of such an ideal is called a uniform relation if where each coefficient is a non zero element of , and where are paths having the same lengths, all starting at a common vertex, and also ending at a common vertex. A uniform relation is minimal if no proper subset yields a uniform relation of the ideal. It is easy to show that every admissible ideal can be generated by a set of uniform relations. In view of this discussion it is easy to prove the following characterization of positively graded algebras:
Proposition 1.1. Let be a -algebra. Then is a graded quiver algebra if and only if there exists a finite quiver and a homogeneous admissible ideal of generated by a set of minimal uniform relations such that □
The algebra is called quadratic if all the minimal uniform relations are homogeneous of degree two. This is equivalent to saying that if we let where , then is generated by Whether is graded or not, we will always set .
Examples 1.2. (1) The polynomial algebra in variables can be described as the algebra whose quiver consists of a single vertex and loops at that vertex. The path algebra of is the free algebra in variables, and the ideal of relations is generated by all the differences and we have
(2) Every monomial relations algebra is positively graded.
(3)Let be the algebra of the quiver :
and let . Then is not graded by path lengths.
Given a quadratic algebra , its quadratic dual is obtained in the following way: let and be the subspaces of and respectively, spanned by the paths of length two. More precisely, let the set be a basis of consisting of the length two paths, and let the set denote a dual basis of . We have a bilinear form
defined on bases elements as follows:
Let denote the orthogonal subspace
and let be the two-sided ideal of generated by Then the quadratic dual of (or its shriek algebra) is
In this case we can choose as a basis for the set of relations where for each . Let us compute the orthogonal of First, it is obvious that for each , is in Choose now an element . If we have and if we have Therefore for all and we must also have It follows easily now that
and that the shriek algebra
is isomorphic to the exterior algebra in variables over
To each positively graded -algebra we associate two quadratic algebras. Firstly, if we write for a homogeneous admissible ideal of the path algebra, we construct the quadratic (quotient) algebra , and the other quadratic algebra is simply We will be particularly interested in the case where itself is a quadratic algebra, so that .
We need a few more definitions and basic facts. We denote by , and the categories of all the -modules, and of the finitely generated -modules respectively. By and by we denote the category of graded (finitely generated graded respectively) -modules. The category is a full subcategory of the category l.f. of locally finite graded -modules, that is the graded modules where each is a finite dimensional -vector space. The morphisms in the categories and are the degree zero homomorphisms, that is homomorphisms such that for each Their space is denoted for and graded modules, If and are graded -modules and is finitely generated then we have
where denotes the degree homomorphisms, that is those homomorphisms such that for each
If is a graded module we define its graded shift as the graded module whose -th component is for all integers . Note that if we forget the grading then all the graded shifts are isomorphic as -modules, but they are all non isomorphic if in the category of graded modules. We have the obvious identifications
for each integer . We will also consider the truncation of . It is graded submodule given by:
Throughout this lecture will denote the full subcategory of consisting of the graded modules bounded from below, that is of all the graded modules of the form where for all less than some integer Similarly denotes the full subcategory of consisting of those graded modules bounded from above, that is such that for all greater than some integer and will denote the graded modules bounded from above and from below. The corresponding bounded subcategories l.f., l.f. and l.f. are defined in the obvious way too. It is well-known that the category has enough projective and injective objects, and that the global dimensions of and of are equal. There exists a duality
given by The category of finitely generated graded -modules is contained in l.f. hence its dual is the category of finitely cogenerated graded -modules. Therefore, via this duality every object of is a submodule of a finitely cogenerated injective -module. In particular we see that the finitely cogenerated graded injective -modules are in l.f.
Let us note that the graded version of Nakayama's lemma holds, that is if we let , then if and only if Consequently, the graded Jacobson radical (that is the intersection of the graded maximal submodules) of a module in is just and adapting the standard methods one can also prove that projective covers exist in
Since we come from the representation theory of finite dimensional algebras, we are particularly interested in finite dimensional algebras and in the category of finitely generated -modules. By the Jordan-Hölder theorem, every finitely generated module can be realized by a finite sequence of extensions of simple -modules. Therefore the Yoneda algebra also called the cohomology ring of ,
should contain plenty of relevant information about the representation theory of Note that is a positively graded -algebra where the multiplication is induced by the Yoneda product and the finite dimensionality of also ensures that each graded component is finite dimensional.
Example 1.4. An interesting situation in which we deal with the cohomology ring of an algebra is in this context the trivial extension , in the case the quiver is a connected bipartite graph that is not a Dynkin diagram.
The algebra has quiver where and and where is the ideal generated by relations and where
We will prove later that the Yoneda algebra of is the preprojective algebra with the same quiver as and ideal generated by all the quadratic relations of the form where runs over the vertices of the completed quiver
The preprojective algebras were introduced by Gelfand and Ponomarev , who proved:
- The preprojective algebra is finite dimensional if and only if is Dynkin.
Baer, Geigle and Lenzing proved the following result:
- If is not a Dynkin diagram, then is noetherian if and only if is a Euclidean diagram.
The theorem proves that, in our case, actually the Yoneda algebra of controls the representation type of
The notion of Koszul algebra was introduced by S. Priddy in his study of the Steenrod algebras. Koszul algebras appear in many areas of algebra, in algebraic geometry, and their study has intensified significantly in the last ten years or so.
(1) Hereditary algebras and radical square zero algebras are Koszul.
(2) The tensor product of two Koszul algebras is Koszul.
(3) If is a Koszul algebra and a finite group of automorphsims of is a Koszul -algebra, such that characteristic of does not divide the order of , then the skew group algebra is Koszul .
(4) Both the polynomial algebra and the exterior algebra
in variables are Koszul algebras.
(5) If is not a Dynkin diagram, then the preprojective algebra of and the trivial extension are Koszul.
If is a positively graded -algebra, then it is easy to show that the vector space dimension of is finite since and are isomorphic as -vector spaces . Therefore, if is a Koszul algebra, then is again a (locally finite) positively graded algebra. There is a functor given by
and if is finitely generated and has a finitely generated projective resolution, then is locally finite, but not necessarily finitely generated over
We end the first lecture with a more standard definition of Koszul algebras which, as will see later is equivalent to the previous one. Let be a graded -algebra. A graded -module is linear or Koszul if it has a graded projective resolution:
such that for each , the projective module is finitely generated by a set of homogeneous elements of degree Then we will say that is a Koszul algebra if each graded simple -module is linear, that is if is a linear -module. The following theorem is a summary of some of the main properties of Koszul algebras and linear modules. We will sketch the proofs of some parts in the following lectures.
(1) If is a Koszul algebra, then is quadratic. Moreover, its Yoneda algebra is isomorphic as a graded algebra to the quadratic dual .
(2) is a Koszul algebra if and only if is Koszul.
(3) If is a linear -module, then and are also linear, where denotes the first szyzygy.
(4) If is Koszul, then so is its Yoneda algebra and then and are isomorphic as graded -algebras.
(5) Let be now a Koszul algebra, and let and denote the full subcategories of and of consisting of the linear -modules (linear -modules respectively.) Then the functor induces a duality between the categories of linear modules Moreover, for each linear -module , □
We should talk now about a very nice consequence of this theorem. Assume that is a Koszul algebra and is a linear -module. Recall that the Loewy length of is finite if for some Then parts (3) and (6) of the above theorem tell us that a linear module has finite Loewy length if and only if the module has finite projective dimension over the Yoneda algebra Moreover, we get the following:
We will talk now about Koszul duality and we will sketch proofs of some parts of theorem 1.7. stated at the end of the first lecture. We start with the following:
be an exact sequence of graded -modules. Then
- If is generated in degree zero, then so is
- If and are generated in degree zero, then so is
- If is generated in degree zero, then is generated in degree zero if and only if
Proof. (1) is obvious.
(2) Let and be projective covers of and of respectively. By assumption, they are both generated in degree zero. It is then easy to show that is a projective module mapping onto , and since it is generated in degree zero, so is
(3) We always have Assume now that is generated in degree zero, so . Let be a homogeneous element of degree in the intersection so since is generated in degree zero. Thus . Conversely, assume that Then we have the following commutative diagram with exact rows: since , we have an exact commutative diagram: since , we have an exact commutative diagram:
The bottom sequence is a sequence of semisimple modules generated in degree zero, so the top of lies entirely in degree zero. The graded version of Nakayama's lemma tells us now that the projective cover of is generated in degree zero, hence is also generated by its degree zero part. □
We have the following immediate consequence:
be an exact sequence of graded -modules, all generated in degree zero, then for each nonnegative integer we have □
We will now introduce two generalizations of the notion of linear module that make sense even in the non graded case. First, recall that a ring is semiperfect if every finitely generated module has a projective cover.
- The module is quasi-Koszul if for each .
- The module is weakly Koszul if for each and
So, every weakly Koszul module is quasi-Koszul, and in the graded case we also have that every shift of a linear module is weakly and thus also quasi-Koszul. To see this, let be a linear module with projective cover Therefore and are both generated in degree one and by applying corollary 2.2. we get for each positive integer Since every syzygy of a linear module is linear, the rest follows by induction.
We have the following characterization:
- is quasi-Koszul.
- is weakly Koszul.
- is linear.
Proof. We showed earlier that every linear module is weakly Koszul so we only need prove that under our assumptions, if is quasi-Kosul then it must be linear. But being quasi-Koszul implies that we have where is a projective cover of and we also have an exact sequence
From the first lemma of this lecture we deduce that is generated in degree one. Induction takes care of the rest. □
We have the following properties of weakly Koszul modules:
- If and are weakly Koszul then so is
- If and are weakly Koszul, then so is
Proof. The proof of the lemma is an exercise in diagram chasing. First, we use the fact that to infer that we have the following commutative diagram with exact rows and columns
In particular, the sequence is exact and the projective cover of is the direct sum of the projective covers of and of , hence we have an induced commutative diagram
It is now easy to see that by taking radicals we have for each the following commutative diagram
where the top row is a complex acyclic in each term except possibly in . It is easy however to verify that the equality implies that the first row is in fact exact. This further implies that iff The results follow now from these observations and an easy induction. □
We have the following very useful consequence:
Using passage to the Yoneda algebra, we have the following characterization of quasi-Koszul modules:
Theorem 2.7. Let be a semiperfect quiver algebra and let be a finitely generated -module. Then is quasi-Koszul if and only if it has a finitely generated minimal projective resolution, and for each we have
Proof. Suppose that the module is quasi-Koszul. Then the short exact sequence
where is projective cover of induces a short exact sequence
and therefore we also have the folowing exact sequence of semisimple modules:
Applying to the above sequence, we have that the induced sequence
is also exact, and this implies that we have a short exact sequence
or equivalently every -homomorphism extends to We now use induction on so we will prove first that The inclusion holds always, so we must prove the reverse inclusion. Pick a nonzero element and write it as a nonsplit exact sequence
We have the following commutative diagram with exact rows:
We have seen that extends to so there exists such that Then we have Continuing, we obtain that so there exists a map such that It follows that We have now the following commutative diagram with exact rows and columns:
and denote by the bottom exact sequence. Now decomposes into a direct sum of simple modules Therefore we have
for some positive integer We can then write where
and letting be the induced map we have whre the maps are defined in the obvious way. Let us consider the pullbacks
and denote by the top exact sequences. We have which proves the inclusion By induction and dimension shift it follows that . For the reverse implication we will go backwards, so assume now that we have the equality We want to prove that every map extends to where again denotes the projective cover of The map induces the pushout
By hypothesis we can write where are exact sequences
and are the pullbacks
Each induces a map such that :
It is enough to prove that each extends to . We have the following commutative diagram:
The composition yields the existence of a map such that It is an immediate exercise now to show that as claimed and that this implies that Again an induction proves that is quasi-Koszul □
We return now to the situation where our algebra is a Koszul algebra. We have the following characterization:
We consider the situation where our algebra is a Koszul algebra. Using the usual duality and the fact: for locally finite graded modules it is easy to prove:
First note that if is a module over a K-algebra , then its associated graded module is Gr is a graded module over the associated graded algebra Gr Assume now that is a graded quiver algebra so we immediately have that as graded algebras. If the module is graded and generated in degree zero, then is isomorphic to its associated graded module.
¿From the previous lemmas it is easy to prove the main theorem on Koszul duality stated at the end of last lecture.
induces a "duality" between the categories of linear -modules and the linear -modules over the Yoneda algebra ; in particular
for every linear -module
Assume is a Koszul module, since and have the same projective cover and is Koszul, the exact sequence:
induces an exact sequence:
where, and are Koszul.
It follows by previous lemmas that is Koszul and for each we get exact sequences:
Hence; by above lemma, exact sequences:
It follows that the sequences:
Adding all this sequences we get an exact sequence:
where is a projective generated in degree zero.
Since is Koszul, it follows by induction is linear. In particular, is a linear -module, hence; is Koszul.
Since has projective cover
where F(R) is the semisimple part of E(R).
The functors are quasi inverse in the category of linear modules. □
Using this, one can prove another characterization of weakly Koszul modules:
Note also that if is a Koszul algebra and is an exact sequence of linear -modules, then is an exact sequence of linear -modules. We can use some of the results of this lecture to construct new weakly Koszul modules from existing ones:
We also have the following interpretation of weakly Koszul modules. In this context we have the following:
Another result related with weakly Koszul modules is the following:
1) is linear.
2) For each
3) is weakly Koszul.□
Not all modules over a Koszul algebra are Koszul, however we have the following approximation:
Lemma 2.15. Let be a Koszul algebra and a graded module with minimal projective resolution consisting of finitely generated projective and of finite projective dimension. Then there exists an integer such that is Koszul.□
The lemma has the following partial dual:
Proposition 2.16. Let be a finite dimensional Koszul algebra with Yoneda algebra noetherian and let be a finitely generated module. Then there exists a non negative integer such that is weakly Koszul.□
We can prove the following:
Proof. The proof consists in reducing to a module which is weakly Koszul and then to use Wilson's result . □
We know that every Koszul algebra is quadratic. In certain cases the converse also holds. We end this lecture with two more examples of Koszul algebras.
(1) If has global dimension 2, then is a Koszul algebra if and only if it is quadratic.
(2) If is a monomial algebra, then is Koszul if and only if it is quadratic . □
Note that there are examples of quadratic algebras that are not Koszul. For instance, let be the hereditary -algebra of Loewy length two
and let denote the usual duality. Then the trivial extension algebra is quadratic but not Koszul.
We will apply now the results of the previous two lectures to the study of selfinjective Koszul algebras. Recall first that if is a graded quiver algebra, and if is a finitely presented -module, then its transpose is a finitely presented module and can prove that we have an Auslander-Reiten sequence
in the category of locally finite graded -modules. At the beginning of this section we will study an important class of modules over a graded algebra. Let be an indecomposable finitely presented graded -module, and assume also that has a linear presentation, that is the graded projective presentation of has the property that is generated in degree 0, and is generated in degree 1. Then the transpose is linear; it has a presentation where is generated in degree 0 and is generated in degree . Then the truncation and we will prove in a minute that is indecomposable. In this way we obtain a non-split exact sequence
with which is an Auslander-Reiten sequence in the category of finitely generated graded modules generated in degree zero. The proof that is indecomposable follows from more general considerations.
The following is a reformulation of the above discussion.
We have the following :
Proof. We only sketch a proof of the fact that the functor
is full and faithful.
(1) "Fullness": Let be a nonzero morphism and let
be a projective cover of in . We have the following commutative diagram:
It is easy to see that the composition lifts to using the projectivity of , hence there is a homomorphism such that Therefore factors through But is generated in degree 1 and is generated in degree 2, hence and we can use the universal property of the cokernel to get a morphism with From here it is immediate to see that
(2) "Faithfulness": A nonzero graded homomorphism is given by a family of maps where for each , . Since is generated in degree zero, and this implies that the induced map is nonzero. □
We can show now, as promised that the module is indecomposable if has a linear presentation: indeed, is indecomposable by the preceding arguments and its dual is isomorphic to .
Let be now a selfinjective Koszul algebra, and let us assume from now on that is indecomposable as an algebra. Recall that the Nakayama functor is an autoequivalence of that restricts to an autoequivalence of taking projective modules into projective modules. It is also known that if is a nonprojective indecomposable module over any finite dimensional graded algebra, then there exists an Auslander-Reiten sequence ending at in the category of graded modules. Moreover, if we ignore the grading, this sequence is in fact an Auslander-Reiten sequence in the (ungraded) module category . It turns out that all the predecessors of a weakly Koszul module in the graded Auslander-Reiten quiver of are weakly Koszul:
be an Auslander-Reiten sequence in . Then both and are weakly Koszul. Consequently, the category of weakly Koszul -modules has left Auslander-Reiten sequences.
Proof. We only sketch the idea of the proof. First one proves that we apply the Nakayama equivalence functor to a weakly Koszul module we obtain a weakly Koszul module and we use this fact to infer that if is weakly Koszul then is also weakly Koszul.. Then one shows that if an indecomposable module is weakly Koszul then its second syzygy is not simple. Finally, we use these facts to show that the Auslander-Reiten sequence Let
satisfies the conditions of Lemma 2.6. (1) of the previous lecture to conclude that the middle term is also weakly Koszul. □
We will now give a characterization of the selfinjective Koszul algebra of radical cube zero. We start with the following general results:
Lemma 3.5. Let be a quiver algebra, with radical where is the two-sided ideal of generated by the arrows of Assume that Then the ideal is a length-homogeneous ideal of the path algebra, hence is a graded algebra with the grading induced by the path lengths.
Proof. We have and so and since is an admissible ideal we get For each , let . To show that is a homogeneous ideal of the path algebra we must show that Note that for each we have so for all Let and write it as a linear combination of paths in where denotes the length of the path . Therefore we have and so This proves the homogeneity of □
Using the indecomposability of as an algebra, the following result is not hard to prove.
Proof. "" Assume that is Koszul of finite representation type and let be a simple -module. Since is selfinjective every syzygy of is indecomposable and since there are only finitely many nonisomorphic indecomposable modules, must be isomorphic to the graded shift of some syzygy. It cannot be the first syzygy since the Loewy length of is 3. But then after shifting we may assume that there exists a linear module whose syzygy is , again contradicting our assumption on the Loewy length. Therefore is of infinite representation type.
"" For this direction, we will only sketch the proof. We observe first that if is an indecomposable non projective -module then having Loewy length two is generated in a single degree. Let be that degree. Then it is easy to see that its first syzygy is generated in degree or is simple and then it is generated in degree Therefore, in order to prove that is a Koszul algebra it is enough to show that for every simple modules and and for every nonzero integer , and then use the preceding argument. To prove then that no simple module can occur as some syzygy of another simple module one uses the fact that every radical square zero algebra is stably equivalent to a hereditary algebra. In our case, the indecomposable non projective -modules are all -modules and is stably equivalent to a hereditary algebra also of infinite representation type. Then, using this stable equivalence we translate our problem over to this where we prove the equivalent statement that no simple -module can occur as some power of of some other simple -module. □
Using some general considerations about the quiver and relations of trivial extension algebras, the following is an immediate consequence of Theorems 3.7. and 1.7.
Corollary 3.8. Let be a connected bipartite graph. Then the trivial ex- tension algebra is a selfinjective Koszul algebra if and only if the underlying graph is not a Dynkin diagram. In that case, the Koszul dual of R is the preprojective algebra of .□
We want to look now at the Auslander-Reiten sequences, and also at the graded Auslander-Reiten quiver of a selfinjective Koszul algebra of radical cube zero. Let be such an algebra. Then as we mentioned above is stably equivalent to a hereditary algebra that is we have an equivalence of categories [Re]
We use this equivalence and the structure of the Auslander-Reiten quiver of to determine the shape of the connected components of the Auslander-Reiten quiver of . From the observations above one can prove that up to shifts, the graded components of the Auslander-Reiten quiver are of the following types: connecting components that are obtained by taking preprojective and preinjective components over and pasting together their images in , and regular components that are of type or tubes.
Over a Koszul algebra we also have the dual notion of colinear modules. These are the duals of the linear -modules, or equivalently those graded modules having a finitely generated colinear injective resolution. Then the "preinjective" part of the connecting components consists of linear modules and the preprojective part of these components consists of colinear modules. Finally one can show that the regular components consist of modules that are both linear and colinear. Note that this means that over a radical cube zero selfinjective Koszul algebra every indecomposable regular module is up to shift, both linear and colinear.
We have seen that the exterior algebra is Koszul. It is of interest to characterize the selfinjective Koszul algebras, we can do this in terms of their Yoneda algebras.
Let be a finite dimensional indecomposable Koszul algebra with Yoneda algebra Then is selfinjective if and only if the following conditions hold:
1) The algebra has global dimension
2) If is a graded simple then we have:
4) is a simple module and gives a bijection between the simple graded -modules and the simple graded modules.□
We will call Artin Schelter regular algebras to the algebras satisfying conditions 1-4 of the thoerem
Artin-Schelter algebras play an important role in non commutative algebraic geometry. Using our preceding results, it follows that the global dimension two quadratic Artin-Schelter regular algebras are the Koszul dual of selfinjective algebras on infinite representation type with radical cube zero.
We will look now at the Gelfand-Kirillov dimension of these algebras. Therefore let be a selfinjective algebra of infinite representation type with radical such that and We know that is stably equivalent to a hereditary algebra and the quiver of called the separated quiver of is obtained in a prescribed way from the original quiver It also follows from our previous discussions that the quiver is not a union of Dynkin diagrams. Keeping in mind that the quiver of a Koszul algebra is the opposite quiver of its Koszul dual, we have the following:
Let be a quadratic Artin-Schelter regular algebra of global dimension two. Let be an indecomposable linear -module. Then:
(a) If a connected component of the separated quiver of is a Euclidean diagram, then GK or GK
(b) If there is a non Euclidean component of the separated quiver of , then GK □
We obtain the following corollary:
Corollary 3.11. Let be a quadratic Artin-Schelter algebra of global dimension two with associated bipartite graph If there exists a connected component of of Euclidean type, then GK, and if there exists a component that is neither Euclidean nor Dynkin then GK In particular is noetherian if and only if it has finite GK-dimension. □
We start recalling the main results we have so far proved:
In the first lecture we saw various definitions concerning graded algebras and modules, the definition of Koszul algebras and Koszul modules, we stated the main theorem on Koszul algebras and we looked to some examples.
We dedicated the second lecture to the study of quasi Koszul, weakly Koszul and Koszul modules and their relations. We also sketched a proof of the main theorem on Koszul algebras.
In the third lecture we initiated the study of selfinjective Koszul algebras, looking in detail to selfinjective algebras of radical cube zero. For such algebras we studied the existence of almost split sequences in the category of Koszul modules and we described the shape of the Auslander Reiten components.
The aim of the last lecture is: to continue the study of selfinjective Koszul algebras, to find the shape of the Auslander Reiten components of graded modules, to study the existence of left almost split sequences in the category of Koszul modules and to describe the shape of the A-R sequences. In the particular case of the exterior algebra, we will apply these results to the study of the category of coherent sheaves on projective space and we will investigate the existence of A-R components in the subcategory of locally free sheaves. We will end the lectures giving some theorems concerning the growth of the ranks of the locally free sheaves.
We will start with a series of propositions which will culminate in a theorem about the graded stable A-R components of a selfinjective Koszul algebra.
1. For any indecomposable non projective -module there exists an integer such that is weakly Koszul, where denotes the Auslander Reiten translation.
2. If is an indecomposable weakly Koszul module generated in degrees and
is the almost split sequence, then is generated in degrees with where is the Loewy length of and is a subset of □
Assume is an epimorphism. Then is not an epimorphism.□
Lemma 4.3. Let be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume Let is an indecomposable non projective weakly Koszul module generated in degrees and
the almost split sequence. If is a monomorphism, then is not a monomorphism.□
Lemma 4.4. Let be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume Let is an indecomposable non projective weakly Koszul module with almost split sequence
Then exactly one map is an epimorphism and exactly one map is a monomorphism.□
Lemma 4.5. Let be a finite dimensional indecomposable selfinjective Koszul algebra and assume Let be an indecomposable non projective weakly Koszul module and an irreducible epimorphism. Then is not simple.□
Putting together the previous lemmas we can prove as in the following:
Theorem 4.7. Let be a finite dimensional indecomposable selfinjective Koszul algebra and assume Let be an Auslander Reiten component containing an indecomposable non projective graded weakly Koszul module . Let be the cone consisting of the predecessor of Then each module in has the property that the middle term of the almost split sequence has at most two terms.□
From this result we get our first important theorem, which generalizes a result by Ringel :
We will see now how this result applies to the coherent sheaves on projective space:
If we consider the exterior algebra in variables, then the Yoneda algebra is noetherian and our results will apply, so all graded Auslander Reiten components of are of type
By a theorem of Bernstein-Gelfand-Gelfand [BGG]there exists an equivalence of triangulated categories: where denotes the derived category of bounded complexes of coherent sheaves on projective space. As a consequence we have:
We will recall some well known results on quotient categories:
We will consider graded quiver algebras and define the torsion part of a graded module as such that is a submodule of of finite length, for any module the torsion part satisfies: We say is torsion if and torsion free if The category is a Serre subcategory of In this situation there exists a quotient category with the same objects as , and is a direct limit of where the limit is taken over all the pairs such that and are torsion. In particular, if and are two finitely generated graded -modules, it is easy to see that for any integers , we have
The category is abelian and there is an exact functor: such that a map goes to an isomorphism under if and only if and are in
We have the following:
We have a sequence of functors:
Since is noetherian any finitely generated graded module has a truncation with Koszul, but implies is a union of categories:
. If we denote by the sheaf corresponding to then
We have seen before that there is an equivalence of categories: , where denotes the category of finitely related graded modules with linear presentations.
It is clear and for any indecomposable non projective Koszul module we have an almost split sequence:
in , where The sequence induces an almost split sequence:
Assume decomposes in sum of indecomposables: Then decomposes in sum of indecomposables: and if and only if
We know , that is stably equivalent to the Kronecker algebra with arrows:
The module belongs either a preprojective or preinjective component in or to a component of type
The preprojective components are of the form:
The preinjective components are of the form:
is the unique indecomposable injective -module and
It is clear that is not in the preprojective component of , otherwise there exists a Koszul module with simple.
Proposition 4.11. Let be the exterior algebra in variables, Then the Auslander Reiten quiver of has a connected component thatmardelplata2.pdf coincides with the preinjective component of and all other components are subquivers of □
Theorem 4.12. Let be the polynomial algebra in variables, . For any integer the subcategory of has left Auslander Reiten sequences and the Auslander Reiten quiver of has one component that coincides with preprojective component of and all other components are full subquivers of a quiver of type □
We look for the location of locally free sheaves on the A-R quiver and prove that their ranks are given by Chebysheff polynomials of the second kind.
Observe that if is a Koszul module corresponding to a locally free sheaf, so is and since the sequence:
with projective, splits when we localize at any maximal graded relevant ideal, then also corresponds with a locally free sheaf. Hence if is a Koszul -module such that is locally free also and correspond to locally free sheaves, it follows also is locally free. The category of locally free sheaves is closed under extensions, therefore it has right almost split sequences.
Proof. The sheaves corresponding to , and are locally free since by applying the Koszul duality, corresponds to the -module , and to a syzygy of the trivial -module .
Since the preprojective component of consists only of the orbits of and , the result follows immediately. □
It is possible to compute the ranks of the locally free sheaves in by doing an easy computation in the category of linear -modules.
Let be a free resolution of a finitely generated -module Then the Euler number is (-1)rank It follows that the sheafication is locally free, then rank =
If is a linear -module such that then we define the sheaf rank of as the alternating sum . It is clear from Koszul duality, that sheaf rank of is equal to the rank of
If is a sheaf in the preprojective component, then we may assume is obtained by the sheafication of a Koszul -module of projective dimension 1 which in turn corresponds under Koszul duality to a Koszul module of Loewy length .
It follows rank= .
It is rather easy to compute dimension vectors for the module lying in preinjective component of , since we can reduce to the and using stable equivalence, to the Kronecker algebra and then compute the growth of the dimensions using the Coxeter transformation .
We have the following:
the Chebysheff polynomials of the second kind. Then for each , we have
In addition, if , then for each , .□
By specializing to the projective plane, we get the following corollary.
We get our main theorem:
1. Every successor of in the A-R component is locally free.
2. Let be the Auslander-Reiten sequence in starting at . Then
Consequently, the ranks increase exponentially in each Auslander-Reiten quiver.□
Hartshorne [H1] asked about the existence of locally free sheaves of small rank on projective space, in this direction we have:
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Instituto de Matemáticas de la UNAM,
C. P. 61-3
58089, Morelia Michoacan, Mexico
Recibido: 23 de noviembre de 2006
Aceptado: 3 de junio de 2007