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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.48 n.2 Bahía Blanca jul./dic. 2007
Introduction to Koszul Algebras
Roberto Martínez-Villa
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
Example 1.3. Let denote the polynomial algebra in
variables, that 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
Definition 1.5. Let be a graded quiver
-algebra with graded Jacobson
and let
be its cohomology ring. We say that
is a Koszul algebra, if as an algebra,
is generated in degrees 0 and 1.
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.
Theorem 1.7. be a graded
-algebra.
(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:
Theorem 1.8. Let be a Koszul algebra with Koszul dual
Then
is finite dimensional algebra over
if and only if
has finite graded global dimension. □
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:
Lemma 2.1. Let be a graded quiver algebra and let
![]() |
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:
Corollary 2.2. Let be a graded quiver
-algebra and let
![]() |
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.
Definition 2.3. Let be a finitely generated module over a quiver algebra
and assume that
has a minimal finitely generated projective resolution
![]() |
- 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:
Theorem 2.4. Let be a graded quiver algebra and let
be a finitely generated
-module generated in degree zero. The following statements are equivalent:
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:
Lemma 2.5. Let be a semiperfect or a a graded quiver algebra and let
be a short exact sequence such that
for all
. Then:
- 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
![]() |
and also
![]() |
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:
Corollary 2.6. Let be a length graded quiver algebra and let
be an exact sequence of linear
-modules. Then there exist for each
exact sequences
satisfying
for each
□
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:
Theorem 2.8. Let be a graded quiver algebra. Then
is a Koszul algebra if and only if the graded semisimple part
is a linear module. □
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:
Theorem 2.9. Let be a graded quiver algebra. Then
is a Koszul algebra if and only if the opposite algebra
is also a Koszul algebra .□
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.
Theorem 2.10. If is a Koszul algebra, then the Yoneda algebra
is Koszul and the functor
given by
![]() |
induces a "duality" between the categories of linear -modules and the linear
-modules over the Yoneda algebra
; in particular
![]() |
for every linear -module
Proof. (sketch)
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.
It follows:
Since is Koszul, it follows by induction
is linear. In particular,
is a linear
-module, hence;
is Koszul.
Define .
Since has projective cover
![]() |
![]() |
It follows:
![]() |
In particular:
![]() |
where F(R) is the semisimple part of E(R).
Therefore:
![]() |
The functors
are quasi inverse in the category of linear modules. □
Using this, one can prove another characterization of weakly Koszul modules:
Proposition 2.11. Let be a Koszul algebra and let
be a finitely generated graded
-module. Then
is weakly Koszul if and only if
is a linear
-module.□
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:
Proposition 2.12. Let be a Koszul algebra and
is a weakly Koszul module, then every graded shift of
is weakly Koszul, and so are
and
□
We also have the following interpretation of weakly Koszul modules. In this context we have the following:
Proposition 2.13. Let be a Koszul algebra and
a finitely generated graded
-module. Then
is a weakly Koszul module if and only if its associated graded module is a linear
-module.□
Another result related with weakly Koszul modules is the following:
Proposition 2.14. Let be a Koszul algebra and let
. a graded weakly Koszul module with
Let
be the submodule generated by the degree zero part of
Then the following statements hold:
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.□
Definition 2.17. Given a finite dimensional -algebra
and a finitely generated
module
we define de Poincare series of
as:
We can prove the following:
Theorem 2.18. Let be a finite dimensional Koszul algebra with noetherian Yoneda algebra
and let
be a finitely generated R -module. Then the Poincare series
is rational.
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.
Proposition 2.19. Let be a finite dimensional algebra.
(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.
3. SELFINJECTIVE KOSZUL ALGEBRAS
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.
Definition 3.1. Denote by the full subcategory of
consisting of those module having a linear presentation.
The following is a reformulation of the above discussion.
Corollary 3.2. Let be an indecomposable nonprojective graded
-module having a linear presentation. Then
has a linear presentation.□
We have the following :
Proposition 3.3. There exists an equivalence between and
In particular a graded
-module
having a linear presentation is indecomposable if and only if
is indecomposable.
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:
Proposition 3.4. Let be a selfinjective Koszul algebra with
and let
be an indecomposable nonprojective weakly Koszul module. Let
![]() |
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.
Lemma 3.6. Let be a selfinjective graded quiver algebra. Then all the indecomposable projective
-modules have the same Loewy length.□
Theorem 3.7. Let
be a selfinjective quiver algebra, with radical
where
but
. Then
is a Koszul algebra if and only if
is of infinite representation type.
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:
3) for
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.
Theorem 4.1. Let be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume
. Then the following statements hold:
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
□
Corollary 4.2. Let be as in the theorem,
an indecomposable non projective
module with almost split sequence:
![]() |
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.□
Corollary 4.6. Let and
as in the lemma and
an irreducible epimorphism. Then the map:
is an irreducible epimorphism.□
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 :
Theorem 4.8. Let be a finite dimensional indecomposable selfinjective Koszul algebra and assume
Let
be an Auslander Reiten graded component. Then the stable part of
is of the form
□
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:
Theorem 4.9. has Auslander Reiten triangles and the Auslander Reiten quivers are of type
□
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:
Theorem 4.10. (Serre) Let
be the polynomial algebra in
variables. Then there exists an equivalence of categories
.□
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:
![]() |
in
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
□
Corollary 4.13. has Auslander Reiten sequences.□
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.
Proposition 4.14. The preprojective component of the A-R quiver of consists entirely of locally free sheaves.
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:
Proposition 4.15. Let be the locally free sheaves lying in the preprojective component of some subcategory
of
. Denote by
![]() |
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.
Corollary 4.16. Let be the locally free sheaves lying in the preprojective component of some subcategory
of
. Then, for each
,
, where
is the Fibonacci numbers sequence.□
We get our main theorem:
Theorem 4.17. Let and
is an indecomposable locally free sheaf in some
Then
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:
Theorem 4.18. Let For each integer
each Auslander Reiten component of
contains at most one locally free sheaf of rank less than
□
[AE] Avramov, Luchezar. Eisenbud, David. Regularity of modules over a Koszul algebra, Journal of Algebra 153 (1992), 85-90. [ Links ]
[ABPRS] Auslander, M. Bautista,R. Platzeck, M.-I. Reiten, I. Smalo, S.O. Almost split sequences whose middle term has at most two summands, Canad. J. Math. 31 (5) (1979) 942-960. [ Links ]
[AR1] Auslander, Maurice. Reiten, Idun. Representation theory of artin algebras III. Almost split sequences, Communications in Algebra 3 (1975), 239-294. [ Links ]
[AR2] Auslander, Maurice. Reiten, Idun. Almost split sequences in dimension two, Advances in Mathematics 66 (1987), 88-118. [ Links ]
[ARS] Auslander, Maurice. Reiten, Idun. Smalø, Sverre O. Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, (1995). [ Links ]
[BGL] Baer, D. Geigle, W. Lenzing, H. The preprojective algebra of a tame hereditary Artin algebra, Comm. Algebra, 15, 425-457, (1987) [ Links ]
[BGS] Beilinson, Alexander. Ginzburg, Victor. Soergel, Wolfgang. Koszul duality patterns in representation theory, Journal of the American Mathematical Society, 9, no. 2., (1996), 473-527. [ Links ]
[BGG] Bernstein, I.N. Gelfand, I.M. Gelfand, S.I. Algebraic bundles over and problems in linear algebra, Funct. Anal. Appl., 12, (1979), 212-214. [ Links ]
[GMRSZ] Green, E. L. Martínez-Villa, R. Reiten, I. Solberg, Ø. Zacharia, D. On modules with linear presentations, J. Algebra 205, (1998), no. 2, 578-604. [ Links ]
[GM1] Green, Edward L. Martínez-Villa, Roberto. Koszul and Yoneda algebras, Representation theory of algebras (Cocoyoc, 1994), 247-297, CMS Conf. Proc., 18, Amer. Math. Society., Providence, RI, (1996). [ Links ]
[GM2] Green, Edward L. Martínez-Villa, Roberto. Koszul and Yoneda algebras II, Algebras and modules, II (Geiranger, 1996), 227-244, CMS Conf. Proc., 24, Amer. Math. Society., Providence, RI, (1998). [ Links ]
[GZ] Green, Edward L. Zacharia, Dan. The cohomology ring of a monomial algebra. Manuscripta Math. 85 (1994), no. 1, 11-23. [ Links ]
[GMT] Guo, Jin Yun. Martínez-Villa, Roberto. Takane, Martha. Koszul generalized Auslander regular algebras. Proc. of ICRA VIII, CMS Conference Proc. CMS-AMS, Vol. 24, (1998), 263-283. [ Links ]
[GP] Gelfand I. M. Ponomarev V.A. Model algebras and representations of graphs, Funct. Anal. Appl., 13, 157-166, (1979). [ Links ]
[H1] Hartshorne, Robin. Algebraic vector bundles on projective spaces: a problem list, Topology, 18, 117-128, 1979. [ Links ]
[H2] Hartshorne, Robin. Algebraic Geometry, Springer-Verlag, 1987. [ Links ]
[M1] Martínez-Villa, Roberto. Graded, selfinjective, and Koszul algebras, J. Algebra 215 (1999), no. 1, 34-72. [ Links ]
[M2] Martínez-Villa, Roberto. Applications of Koszul Algebras: the Preprojective Algebra. Proc. of ICRA VII, CMS Conference Proc. CMS-AMS. Vol. 19, (1996) pag. 487-504. [ Links ]
[M3] Martínez-Villa, Roberto. Skew group algebras and their Yoneda algebras, Math. J. Okayama Univ. 43 , 1-16 (2001) [ Links ]
[MVZ] Martínez-Villa, R. Zacharia, D. Approximations with modules having linear resolutions. J. Algebra 266 (2003) 671-697. [ Links ]
[MVZ2] Martínez-Villa, R. Zacharia, D. Auslander-Reiten sequences, Locally Free sheaves and Chebysheff Polynomials. Compositio Math. 142 (2006) 397-408. [ Links ]
[P] Priddy, Stewart B. Koszul Resolutions, Transactions AMS 152 (1970), 39-60. [ Links ]
[Re] Reiten, Idun. Stable equivalence for some categories with radical square zero. Trans. Amer. Math. Soc. 212 (1975), 333-345. [ Links ]
[R1] Ringel, C. M. Cones, Representation theory of algebras (Cocoyoc, 1994), 583-586, CMS Conf. Proc.,18, Amer. Math. Soc., Providence, RI, 1996. [ Links ]
[R2] Ringel, C. M. Tame Algebras and Integral Quadratic Forms. Lecture Notes in Mathematics 1099, Springer, (1984). [ Links ]
[S] Smith, P. S. Some finite dimensional algebras related to elliptic curves, "Rep. Theory of Algebras and Related topics", CMS Conference Proceedings, Vol. 19, 315-348, AMS (1996). [ Links ]
[W] Wilson, G. The Cartan map on categories of graded modules, J. Algebra 85 (1983) 390-398. [ Links ]
Roberto Martínez-Villa
Instituto de Matemáticas de la UNAM,
Unidad Morelia,
C. P. 61-3
58089, Morelia Michoacan, Mexico
mvilla@matmor.unam.mx
Recibido: 23 de noviembre de 2006
Aceptado: 3 de junio de 2007