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Revista de la Unión Matemática Argentina

versión 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 K . We will say that an algebra R is a positively graded K -algebra, if

  • R = R ⊕ R ⊕ ... 0 1 ,
  • RiRj ⊆ Ri+j , for all i,j , and
  • R0 = K × ...× K
  • R 1 is a finite dimensional K -vector space

We will say R is locally finite if in addition, for each i , Ri is a finite dimensional K -vector space.

The elements of Ri are called homogeneous of degree i , and Ri is the degree i component of R . For example, the path algebra of any quiver is a graded algebra, whose degree i component is the K -vector space spanned by all the paths of length i. We can associate to each positively graded algebra a quiver Q with set of vertices {vi}i , where vi corresponds to the primitive idempotent ei = (0,...,1,...,0) of R having 1 in the i -th entry and zero everywhere else. The arrows {α } of Q with α : v → v i j correspond to the elements of some basis {a } of ejR1ei. Note that there is a homomorphism of graded K -algebras

Φ : KQ - → R

given by Φ (vi) = ei and Φ(α) = a.

The morphism Φ is surjective if and only if RiRj = Ri+j for all i and j. In such a case we say R is a graded quiver algebra.

If R is a graded K -algebra, we will denote by J (and also by  + R ) the radical (also called the graded Jacobson radical) of R . It is the homogeneous ideal J = R+ = R1 ⊕ R2 ⊕ .... It is also easy to see that J equals the intersection of all the maximal homogeneous ideals of R . Let Q be a finite quiver. A homogeneous ideal I in KQ is called admissible if  + 2 I ⊆ (KQ ). An element ρ of such an ideal is called a uniform relation if  ∑n ρ = j=1cjγj where each coefficient cj is a non zero element of K , and where γj are paths having the same lengths, all starting at a common vertex, and also ending at a common vertex. A uniform relation ρ = ∑n cjγj j=1 is minimal if no proper subset S ⊂ {1,...,n} yields a uniform relation  ∑ ρ = j∈S bjγj 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 R be a K -algebra. Then R is a graded quiver algebra if and only if there exists a finite quiver Q and a homogeneous admissible ideal I of KQ generated by a set of minimal uniform relations such that R ~= KQ ∕I.

The algebra R ~= KQ ∕I is called quadratic if all the minimal uniform relations are homogeneous of degree two. This is equivalent to saying that if we let I = I2 ⊕ I3 ⊕ ... where In = (KQ )n ∩ I , then I is generated by I2. Whether R = KQ ∕I is graded or not, we will always set 𝕜 = R∕J .

Examples 1.2. (1) The polynomial algebra K [x1,...,xn] in n variables can be described as the algebra whose quiver Q consists of a single vertex v and n loops {x1, ...,xn} at that vertex. The path algebra of Q is the free algebra in n variables, and the ideal of relations is generated by all the differences xixj - xjxi, and we have

 ~ K--⟨x1,-...,xn⟩ K [x1,...,xn ]= ⟨xixj - xjxi⟩

(2) Every monomial relations algebra is positively graded.

(3)Let R be the algebra of the quiver Q :  2 α β ↗ ↘ 1 5 ↘ ↗ δ τ 3 - →ɛ 4

and let R = KQ ∕⟨βα - τ ɛδ⟩ . Then R is not graded by path lengths.

Given a quadratic algebra R , its quadratic dual R! is obtained in the following way: let V = (KQ ) 2 and V op = (KQop ) 2 be the subspaces of KQ and  op KQ respectively, spanned by the paths of length two. More precisely, let the set { γ} be a basis of V consisting of the length two paths, and let the set {γop} denote a dual basis of Vop . We have a bilinear form

<, > : V × V op → K

defined on bases elements as follows:

 { op 0 if γi ⁄= γj, < γi,γj >= 1 otherwise.

Let  ⊥ I2 denote the orthogonal subspace

 ⊥ op I2 = {v ∈ V | < u,v >= 0 ∀u ∈ V}

and let  ⊥ ⟨I2 ⟩ be the two-sided ideal of  op KQ generated by  ⊥ I2 . Then the quadratic dual of R (or its shriek algebra) is R! = KQop ∕⟨I⊥2 ⟩.

Example 1.3. Let R denote the polynomial algebra in n variables, that is

 K ⟨x1,...,xn⟩ R = -------------- ⟨xixj - xjxi⟩

In this case we can choose as a basis for I 2 the set of relations {ρ } i,j where ρi,j = xixj - xjxi for each i < j . Let us compute the orthogonal of I2. First, it is obvious that for each k ,  2 x k is in  ⊥ I2 . Choose now an element  ∑ μ = k⁄=lcklxkxl ∈ I⊥2 . If k < l we have < ρk,l,μ >= ckl - clk, and if k > l we have < ρl,k,μ >= clk - ckl. Therefore ckl ⁄= 0 for all k ⁄= l and we must also have c = c . kl lk It follows easily now that

 ⊥ 2 I2 = ⟨xk,xixj + xjxi⟩k,i<j

and that the shriek algebra

R! = ---K-⟨x1,...,xn-⟩--- ⟨x2k,xixj + xjxi⟩k,i<j

is isomorphic to the exterior algebra in n variables over K.

To each positively graded K -algebra R we associate two quadratic algebras. Firstly, if we write -- R = KQ ∕I for a homogeneous admissible ideal I of the path algebra, we construct the quadratic (quotient) algebra  ^ R = KQ ∕⟨I2⟩ , and the other quadratic algebra is simply  ! R . We will be particularly interested in the case where R itself is a quadratic algebra, so that R = ^R .

We need a few more definitions and basic facts. We denote by Mod R , and modR the categories of all the R -modules, and of the finitely generated R -modules respectively. By  Z Mod R and by  Z mod R we denote the category of graded (finitely generated graded respectively) R -modules. The category modZR is a full subcategory of the category l.f.R of locally finite graded R -modules, that is the graded modules ⊕ M . i∈Z i where each Mi is a finite dimensional K -vector space. The morphisms in the categories  Z Mod R and  Z mod R are the degree zero homomorphisms, that is homomorphisms f : M → N such that f(Mi ) ⊆ Ni for each i. Their space is denoted HomR (M, N )0 for M and N graded modules, If M and N are graded R -modules and N is finitely generated then we have

 ⊕ HomR (M, N ) = HomR (M, N )n n∈Z

where HomR (M, N )n denotes the degree n homomorphisms, that is those homomorphisms f : M → N such that f (Mi ) ⊆ Ni+n for each i.

If M is a graded module we define its graded shift M [n ] as the graded module whose i -th component is M [n] = M i i+n for all integers i . Note that if we forget the grading then all the graded shifts are isomorphic as R -modules, but they are all non isomorphic if n ⁄= 0 in the category of graded modules. We have the obvious identifications

HomR (M, N )n = HomR (M, N [n])0 = HomR (M [- n],N )0

for each integer n . We will also consider the truncation M ≥n of M . It is graded submodule given by:

 { (M ) = 0 if t < n. ≥n t Mt if t ≥ n.

Throughout this lecture ModZ,+ R will denote the full subcategory of  Z Mod R consisting of the graded modules bounded from below, that is of all the graded modules of the form M = ⊕i ∈ZMi where Mj = 0 for all j less than some integer k0. Similarly ModZ,R - denotes the full subcategory of ModZ R consisting of those graded modules M bounded from above, that is such that Mj = 0 for all j greater than some integer t0, and  Z,b Mod R will denote the graded modules bounded from above and from below. The corresponding bounded subcategories l.f.- R , l.f.+R and l.f.bR are defined in the obvious way too. It is well-known that the category ModZ R has enough projective and injective objects, and that the global dimensions of ModR and of  Z Mod R are equal. There exists a duality

D : l.f.R -→ l.f.R

given by D (M )j = HomK (M - j,K ). The category modZR of finitely generated graded R -modules is contained in l.f.+R hence its dual D (modZ ) R is the category of finitely cogenerated graded Rop -modules. Therefore, via this duality every object of  Z D (mod R) is a submodule of a finitely cogenerated injective  op R -module. In particular we see that the finitely cogenerated graded injective R -modules are in l.f.- R.

Let us note that the graded version of Nakayama's lemma holds, that is if we let M ∈ ModZ,R+ , then JM = M if and only if M = 0. Consequently, the graded Jacobson radical (that is the intersection of the graded maximal submodules) of a module M in ModZ,+ R is just JM, and adapting the standard methods one can also prove that projective covers exist in  Z,+ Mod R .

Since we come from the representation theory of finite dimensional algebras, we are particularly interested in finite dimensional algebras R = KQ ∕I and in the category mod R of finitely generated R -modules. By the Jordan-Hölder theorem, every finitely generated module M can be realized by a finite sequence of extensions of simple R -modules. Therefore the Yoneda algebra E (R) also called the cohomology ring of R ,

E (R ) = ⊕t≥0ExttR(𝕜,𝕜 )

should contain plenty of relevant information about the representation theory of R. Note that E(R ) is a positively graded K -algebra where the multiplication is induced by the Yoneda product and the finite dimensionality of R also ensures that each graded component E (R )i 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 Λ = T(KQ ) = KQ ⋉ D (KQ ) , in the case the quiver Q is a connected bipartite graph that is not a Dynkin diagram.

The algebra T (KQ ) has quiver ˆQ = ( ˆQ0,Qˆ1 ) where Qˆ0 = Q0, and Qˆ1 = Q1 ∪ Qop1 and T (KQ ) ~= K ˆQ ∕I where I is the ideal generated by relations α αop - ββop and αβ where β ⁄= αop.

We will prove later that the Yoneda algebra E (Λ) of Λ is the preprojective algebra Γ = K Qˆ∕L with the same quiver as Λ and ideal L generated by all the quadratic relations of the form ∑ α,s(α)=v ααop where v runs over the vertices of the completed quiver Qˆ.

The preprojective algebras were introduced by Gelfand and Ponomarev [GP ] , who proved:

  • The preprojective algebra Γ is finite dimensional if and only if Q is Dynkin.

Baer, Geigle and Lenzing [BGL ] proved the following result:

  • If Q is not a Dynkin diagram, then Γ is noetherian if and only if Q is a Euclidean diagram.

The theorem proves that, in our case, actually the Yoneda algebra of Λ controls the representation type of Λ.

Definition 1.5. Let R be a graded quiver K -algebra with graded Jacobson J and let E (R ) be its cohomology ring. We say that R is a Koszul algebra, if as an algebra, E (R) 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.

Examples 1.6.

(1) Hereditary algebras and radical square zero algebras are Koszul.

(2) The tensor product of two Koszul algebras is Koszul.

(3) If R is a Koszul algebra and G a finite group of automorphsims of R is a Koszul K -algebra, such that characteristic of K does not divide the order of G , then the skew group algebra R * G is Koszul [M 1] .

(4) Both the polynomial algebra K [x1,...,xn ] and the exterior algebra

---K-⟨x1,...,xn⟩---- ⟨x2,x x + x x ⟩ k i j j i k,i<j in n variables are Koszul algebras.

(5) If Q is not a Dynkin diagram, then the preprojective algebra of KQ and the trivial extension T (KQ ) = KQ ⋉ D (KQ ) are Koszul.

If R is a positively graded K -algebra, then it is easy to show that the vector space dimension of  1 ExtR (𝕜,𝕜) is finite since  2 D (J∕J ) and E (R )1 = Ext1R(𝕜,𝕜) are isomorphic as K -vector spaces . Therefore, if R is a Koszul algebra, then E(R ) is again a (locally finite) positively graded algebra. There is a functor F : ModZ → ModZ R E(R) given by

 t F (M ) = ⊕t≥0Ext R(M, 𝕜)

and if M is finitely generated and has a finitely generated projective resolution, then F (M ) is locally finite, but not necessarily finitely generated over E (M ).

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 R = KQ ∕I be a graded K -algebra. A graded R -module M is linear or Koszul if it has a graded projective resolution:

... → Pn+1 → Pn → ...→ P1 → P0 → M → 0

such that for each j , the projective module Pj is finitely generated by a set of homogeneous elements of degree j. Then we will say that R is a Koszul algebra if each graded simple R -module is linear, that is if R∕J is a linear R -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. R = KQ ∕I be a graded K -algebra.

(1) If R is a Koszul algebra, then R is quadratic. Moreover, its Yoneda algebra E (R ) is isomorphic as a graded algebra to the quadratic dual  ! R .

(2) R is a Koszul algebra if and only if  op R is Koszul.

(3) If M is a linear R -module, then J M [1] and ΩM [1 ] are also linear, where ΩM denotes the first szyzygy.

(4) If R is Koszul, then so is its Yoneda algebra E (R), and then R and E (E (R )) are isomorphic as graded K -algebras.

(5) Let R be now a Koszul algebra, and let KR and KE (R) denote the full subcategories of  Z mod R and of  Z mod E(R) consisting of the linear R -modules (linear E (R) -modules respectively.) Then the functor F : ModZ → ModZ R E(R) induces a duality between the categories of linear modules F : KR → KE (R). Moreover, for each linear R -module M , F (J M ) = ΩF (M )[1].

We should talk now about a very nice consequence of this theorem. Assume that R is a Koszul algebra and M is a linear R -module. Recall that the Loewy length of M is finite if JtM = 0 for some t > 0. Then parts (3) and (6) of the above theorem tell us that a linear module M has finite Loewy length if and only if the module F (M ) has finite projective dimension over the Yoneda algebra E(R ). Moreover, we get the following:

Theorem 1.8. Let R be a Koszul algebra with Koszul dual S = E (R ). Then R is finite dimensional algebra over K if and only if S has finite graded global dimension.

2. KOSZUL DUALITY

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 R = KQ ∕I be a graded quiver algebra and let

0 → A → B → C → 0

be an exact sequence of graded R -modules. Then

  1. If B is generated in degree zero, then so is C.
  2. If A and C are generated in degree zero, then so is B.
  3. If B is generated in degree zero, then A is generated in degree zero if and only if JB ∩ A = J A.

Proof. (1) is obvious.

(2) Let PA and PC be projective covers of A and of C respectively. By assumption, they are both generated in degree zero. It is then easy to show that PA ⊕ PC is a projective module mapping onto B , and since it is generated in degree zero, so is B.

(3) We always have J A ⊆ A ∩ JB. Assume now that A is generated in degree zero, so A = R A i i 0 . Let x i be a homogeneous element of degree i in the intersection A ∩ JB, so i ≥ 1 since B is generated in degree zero. Thus  i xi ∈ Ai ⊆ J A . Conversely, assume that J A = A ∩ JB. Then we have the following commutative diagram with exact rows: since JA = A ∩ J B , we have an exact commutative diagram: since J A = A ∩ JB , we have an exact commutative diagram:

 0 0 0 | | | 0 ------ JA --------J B -------- JC ------ 0 | | | | | | ------- --------- --------- ------- 0 A| B| C| 0 | | | 0 -----A ∕JA -----B ∕JB -----C ∕JC -----0 | | | | | | 0 0 0

The bottom sequence is a sequence of semisimple modules generated in degree zero, so the top of A lies entirely in degree zero. The graded version of Nakayama's lemma tells us now that the projective cover of A is generated in degree zero, hence A is also generated by its degree zero part. □

We have the following immediate consequence:

Corollary 2.2. Let R be a graded quiver K -algebra and let

0 → A → B → C → 0

be an exact sequence of graded R -modules, all generated in degree zero, then for each nonnegative integer k we have JkB ∩ A = J kA.

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 R is semiperfect if every finitely generated module has a projective cover.

Definition 2.3. Let M be a finitely generated module over a quiver algebra R = KQ ∕I and assume that M has a minimal finitely generated projective resolution

... → Pn+1 → Pn → ...→ P1 → P0 → M → 0
  1. The module M is quasi-Koszul if  k 2 k Ω M ∩ J Pk -1 = JΩ M for each k > 0 .
  2. The module M is weakly Koszul if ΩkM ∩ Jt+1Pk -1 = JtΩkM for each t ≥ 1 and k > 0.

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 M be a linear module with projective cover P0. Therefore JP0 and ΩM are both generated in degree one and by applying corollary 2.2. we get Jt+1P0 ∩ ΩM = JtΩM for each positive integer t. Since every syzygy of a linear module is linear, the rest follows by induction.

We have the following characterization:

Theorem 2.4. Let R = KQ ∕I be a graded quiver algebra and let M be a finitely generated R -module generated in degree zero. The following statements are equivalent:

  1. M is quasi-Koszul.
  2. M is weakly Koszul.
  3. M is linear.

Proof. We showed earlier that every linear module is weakly Koszul so we only need prove that under our assumptions, if M is quasi-Kosul then it must be linear. But M being quasi-Koszul implies that we have J 2P0 ∩ ΩM = JΩM where P0 is a projective cover of M and we also have an exact sequence

0 → ΩM → J P → JM → 0 0

From the first lemma of this lecture we deduce that ΩM is generated in degree one. Induction takes care of the rest. □

We have the following properties of weakly Koszul modules:

Lemma 2.5. Let R = KQ ∕I be a semiperfect or a a graded quiver algebra and let 0 → A → B → C → 0 be a short exact sequence such that JtB ∩ A = JtA for all t ≥ 0 . Then:

  1. If A and C are weakly Koszul then so is B.
  2. If A and B are weakly Koszul, then so is C.

Proof. The proof of the lemma is an exercise in diagram chasing. First, we use the fact that J tB ∩ A = J tA to infer that we have the following commutative diagram with exact rows and columns

 0 0 0| | | | 0 ------ t -------- t -------- t ------ 0 J|A J B J C | | | -------- ---------- ---------- -------- 0 A B| C| 0 | | | 0 -----A ∕JtA ----- B∕J tB -----C ∕J tC -----0 | | | | | | 0 0 0

In particular, the sequence 0 → A∕J A → B ∕J B → C ∕JC → 0 is exact and the projective cover PB0 of B is the direct sum of the projective covers P0A of A and P0C of C , hence we have an induced commutative diagram

 0 0 0 | | | | | | ------ ------ ------ ------ 0 ΩA| ΩB| ΩC| 0 | | | 0 -----J P0A -----JP B0 -----J P0C -----0 | | | | | | ------ ------ ------ ------ 0 J A JB JC| 0 | | | 0 0 0

It is now easy to see that by taking radicals we have for each t ≥ 0 the following commutative diagram

 0 0 0 | | | | | | 0 -----ΩA ∩ J|t+1P A0 -----ΩB ∩ J|t+1P B0 ----- ΩC ∩ Jt+1P0C -----0 | | | -------- t+1 A ----------- t+1 B ----------- t+1 C -------- 0 J |P0 J |P0 J |P0 0 | | | 0 ---------J t+1A ------------J t+1B ------------J t+1C ---------0 | | | 0 0 0

and also

 0| 0 0 | | | 0 --------- t ------------- t ------------- t ---------0 J ΩA J ΩB J ΩC | | | 0 -----ΩA ∩ J t+1P A0 -----ΩB ∩ J t+1P B0 ----- ΩC ∩ Jt+1P0C -----0

where the top row is a complex acyclic in each term except possibly in J tΩA . It is easy however to verify that the equality JtΩA = ΩA ∩ Jt+1P A 0 implies that the first row is in fact exact. This further implies that  t t+1 B J ΩB = ΩB ∩ J P0 iff  t t+1 C J ΩC = ΩC ∩ J P0 . The results follow now from these observations and an easy induction. □

We have the following very useful consequence:

Corollary 2.6. Let R = KQ ∕I be a length graded quiver algebra and let 0 → A → B → C → 0 be an exact sequence of linear R -modules. Then there exist for each t ≥ 1 exact sequences 0 → ΩtA → ΩtB → ΩtC → 0 satisfying JlΩtA = JlΩtB ∩ ΩtA for each l ≥ 0.

Using passage to the Yoneda algebra, we have the following characterization of quasi-Koszul modules:

Theorem 2.7. Let R = KQ ∕I be a semiperfect quiver algebra and let M be a finitely generated R -module. Then M is quasi-Koszul if and only if it has a finitely generated minimal projective resolution, and for each t ≥ 1 we have

 t 1 1 Ext R(M, 𝕜) = Ext R(𝕜,𝕜) ⋅ ...⋅ ExtR (𝕜,𝕜) ⋅ HomR (M, 𝕜)

Proof. Suppose that the module M is quasi-Koszul. Then the short exact sequence

 α π 0 → ΩM → P0 → M → 0

where P 0 is projective cover of M, induces a short exact sequence

 ¯α π¯ 0 → ΩM → J P0 → JM → 0

and therefore we also have the folowing exact sequence of semisimple modules:

 ΩM JP0 J M 0 → ------→ --2-- → -2---→ 0 JΩM J P0 J M

Applying Hom (- ,𝕜) 𝕜 to the above sequence, we have that the induced sequence

 J M JP ΩM 0 → Hom 𝕜(-----,𝕜) → Hom 𝕜(---0-,𝕜) → Hom 𝕜(------,𝕜) → 0 J2M J 2P0 J ΩM

is also exact, and this implies that we have a short exact sequence

0 → HomR (JM, 𝕜 ) → HomR (JP0, 𝕜) → HomR (ΩM, 𝕜 ) → 0

or equivalently every R -homomorphism g : ΩM → 𝕜 extends to J P0. We now use induction on t, so we will prove first that  1 1 ExtR (M, 𝕜) = ExtR (𝕜,𝕜) ⋅ HomR (M, 𝕜). The inclusion  1 1 Ext R(M, 𝕜) ⊇ ExtR (𝕜,𝕜) ⋅ HomR (M, 𝕜) holds always, so we must prove the reverse inclusion. Pick a nonzero element  1 x ∈ ExtR (M, 𝕜) and write it as a nonsplit exact sequence

x : 0 → 𝕜 →i E →p M → 0

We have the following commutative diagram with exact rows:

 ----- --¯α-- --¯π-- ----- 0 ΩM|| J P0 JM| 0 || |¯j |j || α π | 0 -----ΩM| ------P0 -------M| ------0 | | | |g |h | 0 ------ 𝕜 ---i---E ---p---M ------0

We have seen that g extends to J P 0 so there exists s: J P → 𝕜 0 such that s¯α = g. Then we have  ¯ is ¯α = ig = hα = hj¯α. Continuing, we obtain that  ¯ (is - hj)¯α = 0 so there exists a map t: JM → E such that t¯π = - is + h¯j. It follows that pt = j. We have now the following commutative diagram with exact rows and columns:

 0 0| | | J M -------JM | | |t j ----- ---i-- | ---p----- |------- 0 𝕜| E| M | 0 || | q || | | 0 -----𝕜 ------Z| ------M ∕J M -----0 | | | | 0 0

and denote by y the bottom exact sequence. Now M ∕J M decomposes into a direct sum of simple modules  ⊕n M ∕J M = l=1 Si. Therefore we have

 ⊕n ⊕m y ∈ Ext1R( Si,𝕜) ⊂ Ext1R(𝕜, 𝕜) l=1 l=1

for some positive integer m. We can then write  ∑ y = mi=1yi where

y : 0 → 𝕜 → Z → 𝕜 → 0 i i

and letting q′ be the induced map q′: M → ⊕n Si `→ ⊕m 𝕜 l=1 l=1 we have q′ = (q ,q ,...,q ) 1 2 m whre the maps q : M → 𝕜 i are defined in the obvious way. Let us consider the pullbacks

0 -----𝕜|-----Ei|----- M| -----0 | | | | |qi 0 ----- ----- Z ------ ------0 𝕜 i 𝕜

and denote by x i the top exact sequences. We have x = ∑ x = ∑ y q i i i which proves the inclusion  1 1 Ext R(M, 𝕜) ⊆ Ext R(𝕜,𝕜) ⋅ HomR (M, 𝕜). By induction and dimension shift it follows that  t 1 1 ExtR(M, 𝕜 ) = Ext R(𝕜,𝕜) ⋅ ...⋅ Ext R(𝕜,𝕜) ⋅ HomR (M, 𝕜) . For the reverse implication we will go backwards, so assume now that we have the equality Ext1R(M, 𝕜) = Ext1R(𝕜,𝕜 ) ⋅ HomR (M, 𝕜 ). We want to prove that every map g: ΩM → 𝕜 extends to JP 0 where again P 0 denotes the projective cover of M. The map g induces the pushout x :

 ----- -i--- --p-- ----- 0 ΩM| P0 M| 0 |g | | | | | 0 ------𝕜 -------E -----M -----0

By hypothesis we can write  ∑ ∑ x = yihi = xi where yi are exact sequences

 u v yi: 0 → 𝕜 →i Zi→ i𝕜 → 0

and xi are the pullbacks

 ----- ----- ----- ----- 0 𝕜| Ei| M| 0 | | |hi | | 0 -----𝕜 ----- Zi------𝕜 ------0

Each xi induces a map gi: ΩM → 𝕜 such that  ∑ g = gi :

 ----- -i--- --p-- ----- 0 ΩM| P0 M| 0 |gi |¯gi | | | 0 ------𝕜 ------ Ei -----M -----0

It is enough to prove that each gi extends to JP0 . We have the following commutative diagram:

 ¯α ¯π 0 -----ΩM|| -----J P0 -----JM| -----0 || |¯j |j || | 0 -----ΩM --α---P0 ---π---M ------0 | | | |gi |¯hi |hi ------ | ---ui-- --vi--- |------ 0 𝕜 Zi 𝕜 0

The composition hij = 0 yields the existence of a map si: JP0 → 𝕜 such that uisi = ¯hi¯j. It is an immediate exercise now to show that gi = siα¯ as claimed and that this implies that J2P0 ∩ ΩM = JΩM. Again an induction proves that M 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 R = KQ ∕I be a graded quiver algebra. Then R is a Koszul algebra if and only if the graded semisimple part 𝕜 = R ∕J is a linear module.

We consider the situation where our algebra is a Koszul algebra. Using the usual duality and the fact: ExtkR(M, N ) ~= ExtkRop(D(N ),D (M ) for locally finite graded modules M, N it is easy to prove:

Theorem 2.9.  Let R = KQ ∕I be a graded quiver algebra. Then R is a Koszul algebra if and only if the opposite algebra  op R is also a Koszul algebra .

First note that if M is a module over a K-algebra R , then its associated graded module is Gr 2 M = M ∕JM ⊕ J M ∕J M ⊕ ... is a graded module over the associated graded algebra GrR = R ∕J ⊕ J∕J 2 ⊕ .... Assume now that R is a graded quiver algebra so we immediately have that R ~= GrR as graded algebras. If the module M is graded and generated in degree zero, then M 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 R is a Koszul algebra, then the Yoneda algebra E (R ) is Koszul and the functor  Z Z F : Mod R → Mod E (R ) given by

F (M ) = ⊕t≥0ExttR(M, 𝕜 )

induces a "duality" between the categories of linear R -modules and the linear E (R )op -modules over the Yoneda algebra E(R ) ; in particular

 * * ~ Ext E(R)(ExtR(M, 𝕜),𝕜)= M

for every linear R -module M.

Proof. (sketch)

Assume M is a Koszul R - module, since M and M ∕J M have the same projective cover and M ∕J M is Koszul, the exact sequence:

0 → J M → M → M ∕JM → 0 induces an exact sequence:

0 → Ω (M ) [1] → Ω (M ∕J M )[1] → J M [1] → 0

where,Ω(M ) [1] and Ω(M ∕J M ) [1] are Koszul.

It follows by previous lemmas that J M [1] is Koszul and for each k ≥ 1 we get exact sequences:

 k k k- 1 0 → Ω (M ) → Ω (M ∕J M ) → Ω (J M ) → 0

Hence; by above lemma, exact sequences:

0 → HomR (Ωk- 1(J M ),𝕜) → HomR (Ωk (M ∕JM ),𝕜) → HomR (Ωk(M ),𝕜 ) → 0

It follows that the sequences:

0 → ExtkR-1(J M, 𝕜) → ExtkR (M ∕JM, 𝕜) → ExtkR (M, 𝕜) → 0

Adding all this sequences we get an exact sequence:

0 → F (JM )[- 1 ] → F (M ∕JM ) → F (M ) → 0

where F (M ∕JM ) is a projective generated in degree zero.

It follows: ΩF (M ) = F (J M [1])[- 1].

Since J M [1] is Koszul, it follows by induction F (M ) is linear. In particular, F (R ) = 𝕜 is a linear E (R ) -module, hence; E (R ) is Koszul.

Define G (N ) = ⊕t≥0ExttE(R)(N,𝕜 ) .

Since Ωk (F (M )) has projective cover F (JkM ∕J k+1M ),

HomE (R )(Ωk (F (M )),𝕜) = HomE (R)(F (J kM ∕Jk+1M ),𝕜)
= Hom (Hom (J kM ∕J k+1M, 𝕜),𝕜) ~= J kM ∕J k+1M. E (R) R

It follows:

 ~ k k+1 ~ GF (M ) = k⊕≥0J M ∕J M = M

In particular:

 ~ GF (R) = R

where F(R) is the semisimple part of E(R).

Therefore:

GF (R) = E (E (R)) ~= R

The functors F, G are quasi inverse in the category of linear modules. □

Using this, one can prove another characterization of weakly Koszul modules:

Proposition 2.11. Let R be a Koszul algebra and let M be a finitely generated graded R -module. Then M is weakly Koszul if and only if F (M ) is a linear E (R ) -module.

Note also that if R is a Koszul algebra and 0 → A → B → C → 0 is an exact sequence of linear R -modules, then 0 → F (C) → F (B ) → F(A ) → 0 is an exact sequence of linear E(R ) -modules. We can use some of the results of this lecture to construct new weakly Koszul modules from existing ones:

Proposition 2.12. Let R be a Koszul algebra and M is a weakly Koszul module, then every graded shift of M is weakly Koszul, and so are JM and ΩM.

We also have the following interpretation of weakly Koszul modules. In this context we have the following:

Proposition 2.13. Let R be a Koszul algebra and M a finitely generated graded R -module. Then M is a weakly Koszul module if and only if its associated graded module is a linear R -module.

Another result related with weakly Koszul modules is the following:

Proposition 2.14. Let R be a Koszul algebra and let M = M0 ⊕ M1 ⊕ ...Mn ⊕ .. . a graded weakly Koszul module with M0 ⁄= 0. Let KM = ⟨M0 ⟩ be the submodule generated by the degree zero part of M. Then the following statements hold:

1) KM is linear.

2) For each k ≥ 0,  k k J M ∩ KM = J KM .

3) M ∕KM is weakly Koszul.

Not all modules over a Koszul algebra are Koszul, however we have the following approximation:

Lemma 2.15. [AE ] Let R be a Koszul algebra and M a graded R - module with minimal projective resolution consisting of finitely generated projective and M of finite projective dimension. Then there exists an integer n such that M [n] ≥n is Koszul.

The lemma has the following partial dual:

Proposition 2.16. Let R be a finite dimensional Koszul algebra with Yoneda algebra E (R) noetherian and let M be a finitely generated R - module. Then there exists a non negative integer k such that Ωk (M ) is weakly Koszul.

Definition 2.17. Given a finite dimensional K -algebra R and a finitely generated R - module M, we define de Poincare series of M as:

 M ∑ m m PR (t) = dimK Ext R (M, 𝕜)t . m≥0

We can prove the following:

Theorem 2.18.  Let R be a finite dimensional Koszul algebra with noetherian Yoneda algebra E (R) and let M be a finitely generated R -module. Then the Poincare series  M PR (t) is rational.

Proof. The proof consists in reducing to a module ΩnM which is weakly Koszul and then to use Wilson's result [W ] .[GM RSZ ]

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 R be a finite dimensional algebra.

(1) If R has global dimension 2, then R is a Koszul algebra if and only if it is quadratic.

(2) If R is a monomial algebra, then R is Koszul if and only if it is quadratic [GZ ] .

Note that there are examples of quadratic algebras that are not Koszul. For instance, let R be the hereditary K -algebra of Loewy length two

 ⌊ ⌋ K 0 0 R = ⌈K K 0⌉ K 0 K

and let D = HomK (- ,K ) denote the usual duality. Then the trivial extension algebra A = R ⋉ D (R ) 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 R = KQ ∕I is a graded quiver algebra, and if M is a finitely presented R -module, then its transpose TrM is a finitely presented  op R module and can prove that we have an Auslander-Reiten sequence

0 → DTrM → E → M → 0

in the category of locally finite graded R -modules. At the beginning of this section we will study an important class of modules over a graded algebra. Let M be an indecomposable finitely presented graded R -module, and assume also that M has a linear presentation, that is the graded projective presentation P → P → M → 0 1 0 of M has the property that P0 is generated in degree 0, and P1 is generated in degree 1. Then the transpose TrM is linear; it has a presentation P *0 → P *1 → TrM → 0 where P0* is generated in degree 0 and P1* is generated in degree - 1 . Then the truncation (DTrM )≥0 = soc2DTrM and we will prove in a minute that soc2DTrM is indecomposable. In this way we obtain a non-split exact sequence

0 → soc2DTrM → F → M → 0

with F = E ≥0 which is an Auslander-Reiten sequence in the category  Z mod 0 R of finitely generated graded modules generated in degree zero. The proof that soc2DTrM is indecomposable follows from more general considerations.

Definition 3.1. Denote by LR the full subcategory of  Z mod R consisting of those module having a linear presentation.

The following is a reformulation of the above discussion.

Corollary 3.2. Let M be an indecomposable nonprojective graded R -module having a linear presentation. Then TrM [1] has a linear presentation.

We have the following [GM RSZ ] :

Proposition 3.3. There exists an equivalence between LR and modZ0 R∕J 2. In particular a graded R -module M having a linear presentation is indecomposable if and only if M ∕J 2M is indecomposable.

Proof. We only sketch a proof of the fact that the functor

 2 Z 2 R ∕J ⊗R - : LR -→ mod 0 R ∕J

is full and faithful.

(1) "Fullness": Let  ¯ 2 2 f : M ∕J M → N ∕J N be a nonzero morphism and let

0 → K →j P g→ M → 0

be a projective cover of M in  Z mod R . We have the following commutative diagram:

 ----- --j-- ---πP- 2 ------ 0 K P| P ∕J P 0 |g |¯g | h M --πM-M ∕J2M -----0 | |¯f πN 2 N ------N∕J N -----0

It is easy to see that the composition ¯f¯gπP lifts to N using the projectivity of P , hence there is a homomorphism h : P → N such that π hj = f¯¯gπ j = ¯fπ gj = 0. N P M Therefore hj factors through J2N. But K is generated in degree 1 and  2 J N is generated in degree 2, hence hj = 0 and we can use the universal property of the cokernel to get a morphism f: M → N with fg = h. From here it is immediate to see that R ∕J2 ⊗ f = f¯.

(2) "Faithfulness": A nonzero graded homomorphism f : M → N is given by a family of maps {f } i where for each i , f : M → N i i i . Since M is generated in degree zero, f0 ⁄= 0 and this implies that the induced map  2 2 M ∕J → N∕J is nonzero. □

We can show now, as promised that the module  2 soc DTrM is indecomposable if M has a linear presentation: indeed,  2 TrM ∕JRopTrM is indecomposable by the preceding arguments and its dual is isomorphic to soc2DTrM .

Let R be now a selfinjective Koszul algebra, and let us assume from now on that R is indecomposable as an algebra. Recall that the Nakayama functor ν = DHomR (- ,R) is an autoequivalence of mod R that restricts to an autoequivalence of modZ R taking projective modules into projective modules. It is also known that if M is a nonprojective indecomposable module over any finite dimensional graded algebra, then there exists an Auslander-Reiten sequence ending at M in the category  Z mod R of graded modules. Moreover, if we ignore the grading, this sequence is in fact an Auslander-Reiten sequence in the (ungraded) module category mod R . It turns out that all the predecessors of a weakly Koszul module in the graded Auslander-Reiten quiver of R are weakly Koszul:

Proposition 3.4. Let R be a selfinjective Koszul algebra with J 2 ⁄= 0 and let M be an indecomposable nonprojective weakly Koszul module. Let

0 → DTrM → E → M → 0

be an Auslander-Reiten sequence in modZ R . Then both DTrM and E are weakly Koszul. Consequently, the category of weakly Koszul R -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 ν to a weakly Koszul module we obtain a weakly Koszul module and we use this fact to infer that if M is weakly Koszul then DTrM = ν Ω2M 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

0 → DTrM → E → M → 0

satisfies the conditions of Lemma 2.6. (1) of the previous lecture to conclude that the middle term E 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 R = KQ ∕I be a quiver algebra, with radical J = L∕I where L is the two-sided ideal of KQ generated by the arrows of Q Assume that J 3 = 0. Then the ideal I is a length-homogeneous ideal of the path algebra, hence R is a graded algebra with the grading induced by the path lengths.

Proof. We have 0 = J3 = L3 + I∕I and so L3 ⊆ I, and since I is an admissible ideal we get  3 2 L ⊆ I ⊆ L . For each n , let In = (KQ )n ∩ I . To show that I is a homogeneous ideal of the path algebra K we must show that  ∑ I = n In. Note that for each n ≥ 3 we have (KQ )n ⊆ L3 ⊆ I so In = (KQ )n for all n ≥ 3. Let ρ ∈ I and write it as a linear combination of paths γ in : ρ = ∑ cγ + ∑ c γ l(γi)=2 ii l(γj)≥3 j j where l(γ) denotes the length of the path γ . Therefore we have ∑ ∑ l(γi)=2 ciγi = ρ - l(γj)≥3cjγj ∈ I and so  ∑ I = nIn. This proves the homogeneity of I.

Using the indecomposability of R as an algebra, the following result is not hard to prove.

Lemma 3.6. Let R be a selfinjective graded quiver algebra. Then all the indecomposable projective R -modules have the same Loewy length.

Theorem 3.7. [M 2] Let R = KQ ∕I be a selfinjective quiver algebra, with radical J where J2 ⁄= 0 but J3 = 0 . Then R is a Koszul algebra if and only if R is of infinite representation type.

Proof. "⇒ " Assume that R is Koszul of finite representation type and let S be a simple R -module. Since R is selfinjective every syzygy of S is indecomposable and since there are only finitely many nonisomorphic indecomposable modules, S must be isomorphic to the graded shift of some syzygy. It cannot be the first syzygy since the Loewy length of R is 3. But then after shifting we may assume that there exists a linear module whose syzygy is S , again contradicting our assumption on the Loewy length. Therefore R is of infinite representation type.

"⇐ " For this direction, we will only sketch the proof. We observe first that if M is an indecomposable non projective R -module then M, having Loewy length two is generated in a single degree. Let i be that degree. Then it is easy to see that its first syzygy ΩM is generated in degree i + 1 or is simple and then it is generated in degree i + 2. Therefore, in order to prove that R is a Koszul algebra it is enough to show that for every simple modules S and T and for every nonzero integer n ,  n T ≇ Ω S, 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 R -modules are all R ∕J 2 -modules and R∕J 2 is stably equivalent to a hereditary algebra H also of infinite representation type. Then, using this stable equivalence we translate our problem over to this H where we prove the equivalent statement that no simple H -module can occur as some power of DTr of some other simple H -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 Q be a connected bipartite graph. Then the trivial ex- tension algebra R = KQ ⋉ D (KQ ) is a selfinjective Koszul algebra if and only if the underlying graph Q is not a Dynkin diagram. In that case, the Koszul dual  ! E (R) ~= R of R is the preprojective algebra of Q .

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 R be such an algebra. Then as we mentioned above R ∕J2 is stably equivalent to a hereditary algebra H that is we have an equivalence of categories [Re]

G : modR ∕J 2 → modH-

We use this equivalence and the structure of the Auslander-Reiten quiver of H to determine the shape of the connected components of the Auslander-Reiten quiver of R . 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 H and pasting together their images in mod R , and regular components that are of type ZA ∞ or tubes.

Over a Koszul algebra we also have the dual notion of colinear modules. These are the duals of the linear Rop -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.

Theorem 3.9. [S ],[M 1]

Let R = KQ ∕I be a finite dimensional indecomposable Koszul algebra with Yoneda algebra E (R). Then R is selfinjective if and only if the following conditions hold:

1) The algebra E(R ) has global dimension n.

2) If S is a graded simple then we have:

3)Extk (S,R ) = 0 R for 0 ≤ k < n.

4)  n Ext R(S, R) is a simple module and  n Ext R(S, R) gives a bijection between the simple graded R -modules and the simple graded Rop - 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 R = KQ ∕I be a selfinjective algebra of infinite representation type with radical J such that J3 = 0 and J 2 ⁄= 0. We know that R ∕J2 is stably equivalent to a hereditary algebra H = K Q¯ and the quiver Q¯ of H, called the separated quiver of Q is obtained in a prescribed way from the original quiver Q. It also follows from our previous discussions that the quiver  ¯ Q 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:

Theorem 3.10. [GM T]

Let S = KQ ∕L be a quadratic Artin-Schelter regular algebra of global dimension two. Let M be an indecomposable linear S -module. Then:

(a) If a connected component of the separated quiver of Q is a Euclidean diagram, then GKdimM = 1 or GKdimM = 2.

(b) If there is a non Euclidean component of the separated quiver of Q , then GKdimM = ∞.

We obtain the following corollary:

Corollary 3.11. Let R be a quadratic Artin-Schelter algebra of global dimension two with associated bipartite graph Q¯. If there exists a connected component of ¯ Q of Euclidean type, then GKdimR = 2 , and if there exists a component that is neither Euclidean nor Dynkin then GKdimR = ∞. In particular R is noetherian if and only if it has finite GK-dimension.

4. APPLICATIONS

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 [AR1 ],[AR2 ],[ARS ] 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. [M V Z ],[M V Z2]

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 R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume J 3 ⁄= 0 . Then the following statements hold:

1. For any indecomposable non projective R -module M there exists an integer n such that τnM is weakly Koszul, where τ denotes the Auslander Reiten translation.

2. If M is an indecomposable weakly Koszul module generated in degrees m < m < ...m 0 1 k and

0 → τM → E → M → 0

is the almost split sequence, then τM is generated in degrees ℓ0 < ℓ1 < ...ℓt with ℓi = - n + 2 + mi , j where n is the Loewy length of R and {mi } j is a subset of {m0, m1, ...mk }

Corollary 4.2. Let R be as in the theorem, M an indecomposable non projective R- module with almost split sequence:

 t 0 → τM (g1,→g2) E1 ⊕ E2 (f1→,f2)M → 0

Assume f1 is an epimorphism. Then g1 is not an epimorphism.

Lemma 4.3. Let R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume  3 J ⁄= 0. Let M is an indecomposable non projective weakly Koszul module generated in degrees m0 < m1 < ...mk and

 (g,g)t (f ,f ) 0 → τM 1→2 E1 ⊕ E2 1→ 2 M → 0

 the almost split sequence. If f1 is a monomorphism, then g1 is not a monomorphism.

Lemma 4.4. Let R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra with noetherian Yoneda algebra and assume  3 J ⁄= 0. Let M is an indecomposable non projective weakly Koszul module with almost split sequence

0 → τ M (g1,g→2,..gk)tE ⊕ E ⊕ ...E (f1,f2→,...fk)M → 0 1 2 k

Then exactly one map f 1 is an epimorphism and exactly one map g 1 is a monomorphism.

Lemma 4.5. Let R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra and assume  3 J ⁄= 0. Let M be an indecomposable non projective weakly Koszul module and f : E ′ → M an irreducible epimorphism. Then Ker f is not simple.

Corollary 4.6. Let R and M as in the lemma and f : E ′ → M an irreducible epimorphism. Then the map: τ(f ) : τ (E′) → τ (M ) is an irreducible epimorphism.

Putting together the previous lemmas we can prove as in [ABRP S] the following:

Theorem 4.7. Let R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra and assume J3 ⁄= 0. Let C be an Auslander Reiten component containing an indecomposable non projective graded weakly Koszul module M . Let C M be the cone consisting of the predecessor of M. Then each module in CM 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 [R1] :

Theorem 4.8. Let R = KQ ∕I be a finite dimensional indecomposable selfinjective Koszul algebra and assume J 3 ⁄= 0. Let C  be an Auslander Reiten graded component. Then the stable part of C is of the form ZA ∞.

We will see now how this result applies to the coherent sheaves on projective space:

If we consider the exterior algebra R in n + 1 variables, then the Yoneda algebra K [x0,x1,x2,...xn] is noetherian and our results will apply, so all graded Auslander Reiten components of R are of type ZA ∞.

By a theorem of Bernstein-Gelfand-Gelfand [BGG]there exists an equivalence of triangulated categories: mod Z b n R ~= D (CohP ), where Db (CohP n) denotes the derived category of bounded complexes of coherent sheaves on projective space. As a consequence we have:

Theorem 4.9. Db (CohP n) has Auslander Reiten triangles and the Auslander Reiten quivers are of type ZA ∞.

We will recall some well known results on quotient categories:

We will consider graded quiver algebras R = KQ ∕I and define the torsion part of a graded module M as  ∑ t(M ) = N α such that N α is a submodule of M of finite length, for any module M the torsion part satisfies: t(M ∕tM )) = 0. We say M is torsion if t(M ) = M and torsion free if t(M ) = 0. The category T = {M | M is torsion} is a Serre subcategory of M odZR. In this situation there exists a quotient category QM odZ R with the same objects as M odZ R , and HomQModZR (M, N ) is a direct limit of  ′ ′ HomModZR (M ,N ∕N ) where the limit is taken over all the pairs  ′ ′ (M ,N ) such that  ′ M ∕M and  ′ N are torsion. In particular, if X and Y are two finitely generated graded S -modules, it is easy to see that for any integers k,l , we have

Hom Z (π(Y )[l],π (X ) [k ]) QModR =

⋃ t≥0 HomModZR (Y≥t,X [k - l])

The category QM odZ R is abelian and there is an exact functor:  Z Z π : M odR → QM odR such that a map f : M → N goes to an isomorphism under π if and only if Kerf and Cokerf are in T.

We have the following:

Theorem 4.10. (Serre) [H2 ] Let S be the polynomial algebra in n + 1 variables. Then there exists an equivalence of categories  Z ~ n Φ : Qmod S→ cohP .

We have a sequence of functors:

 Z F Z π Z Φ mod R -→ mod S -→ Qmod S → coh(P n)

Since S is noetherian any finitely generated graded module M has a truncation M ≥k with M ≥k [k ] Koszul, but π ( M ≥k) = π(M ) implies QmodZ S is a union of categories:

 Z Qmod S = ∪ i∈Z π(KS )[i] . If we denote by  ⌢ M the sheaf corresponding to Φ π(M ), then coh(P n) = ∪ i∈Z ⌢ KS [i].

We have seen before that there is an equivalence of categories:  2 Z R ∕J ⊗R - : LR → mod R∕J2 , where LR denotes the category of finitely related graded R - modules with linear presentations.

It is clear KR ⊆ LR and for any indecomposable non projective Koszul module M we have an almost split sequence:

0 → σM → E → M → 0

in KR , where σM = soc2τM. The sequence induces an almost split sequence:

 2 2 0 → σM → E ∕J E → M ∕J M → 0

in  Z mod R∕J2.

Assume E decomposes in sum of indecomposables: E = E1 ⊕ E2 ⊕ ...Ek. Then E decomposes in sum of indecomposables: E ∕J 2E = E1 ∕J2E1 ⊕ E2∕J 2E2 ⊕ ...Ek ∕J2Ek and Ei ~= Ej if and only if E ∕J2E ~= E i i j ∕J 2E . j

We know [Re ] , that  2 R ∕J is stably equivalent to the Kronecker algebra with n + 1 arrows:  ⇉. . . . ⇉

The module  2 M ∕J M belongs either a preprojective or preinjective component in R ∕J 2 or to a component of type ZA ∞.

The preprojective components are of the form:

 ∙ ∙ Y 1 Y 3 ↗ ↘. ↗ ↘. ↗ .. .. .. .. .. Y∙0 ↗ ↘ Y∙2 ↗ ↘ Y∙4 ↗ ...

The preinjective components are of the form:

 ∙ ∙ ∙ Z4 ↘ Z2 ↘ Z0 ↗ . ↗ . ↗ .. .. Z3 .. .. Z1 .. ... ↗ ↘ ∙ ↗ ↘ ∙ ↗

Z1 is the unique indecomposable injective  2 R ∕J -module and Z0 = K.

It is clear that M ∕J2M is not in the preprojective component of modZR∕J2 , otherwise there exists a Koszul module M with σM simple.

Proposition 4.11. Let R be the exterior algebra in n + 1 variables, n > 1. Then the Auslander Reiten quiver of KR has a connected component thatmardelplata2.pdf coincides with the preinjective component of . ⇉. . ⇉. and all other components are subquivers of ZA ∞.

Theorem 4.12. Let S be the polynomial algebra in n + 1, variables, n > 1 . For any integer i the subcategory ⌢ KS [i] of coh(P n) has left Auslander Reiten sequences and the Auslander Reiten quiver of  ⌢ KS [i] has one component that coincides with preprojective component of  ⇉ . .. . ⇉ and all other components are full subquivers of a quiver of type ZA ∞.

Corollary 4.13. coh(P 1) 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 M is a Koszul S module corresponding to a locally free sheaf, so is  k M ≥k = J M and since the sequence:

0 → ΩM → P → M → 0

with P projective, splits when we localize at any maximal graded relevant ideal, then ΩM also corresponds with a locally free sheaf. Hence if N is a Koszul R -module such that  ⌢ F (N ) is locally free also J N and ΩN correspond to locally free sheaves, it follows also σN 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 coh(P n) consists entirely of locally free sheaves.

Proof. The sheaves corresponding to K , and  2 soc R [2 ] are locally free since by applying the Koszul duality, K corresponds to the S -module S , and soc2R [2] to a syzygy of the trivial S -module K .

Since the preprojective component of coh(P n) consists only of the orbits of K and soc2R [2 ] , the result follows immediately. □

It is possible to compute the ranks of the locally free sheaves in  n coh(P ) by doing an easy computation in the category of linear R -modules.

Let 0 → Fn → Fn -1...F → M → 0 be a free resolution of a finitely generated S -module M. Then the Euler number is  ∑n χ(M ) = i=0 (-1)i rank Fi. It follows that the sheafication ⌢ M is locally free, then rank⌢ M =χ(M ).

If N = N0 ⊕ N1 ⊕ ...⊕ Np is a linear R -module such that F (N ) = M, then we define the sheaf rank of N as the alternating sum ∑n i i=0(- 1 )dimNi . It is clear from Koszul duality, that sheaf rank of N is equal to the rank of ⌢M .

If F is a sheaf in the preprojective component, then we may assume F is obtained by the sheafication of a Koszul S -module M of projective dimension 1 which in turn corresponds under Koszul duality to a Koszul R - module N of Loewy length 2 .

It follows rankF = dimN0 - dimN1 .

It is rather easy to compute dimension vectors for the module lying in preinjective component of KR , since we can reduce to the  2 R ∕J and using stable equivalence, to the Kronecker algebra and then compute the growth of the dimensions using the Coxeter transformation [R2 ] .

We have the following:

Proposition 4.15. Let F0, F1,F2 ... be the locally free sheaves lying in the preprojective component of some subcategory ⌢K [i] S of coh(Pn) . Denote by

 [k∑∕2] (m - k )! Tk(x) = (- 1)m ------------(2x)k-2m m=0 m!(k - 2m )!

the Chebysheff polynomials of the second kind. Then for each k ≥ 1 , we have

 n-+-1- n-+-1- rankFk = Tk( 2 ) - Tk- 1( 2 ).

In addition, if n > 1 , then for each k , rankFk+1 > rankFk .

 By specializing to the projective plane, we get the following corollary.

Corollary 4.16. Let F ,F ,F ... 0 1 2 be the locally free sheaves lying in the preprojective component of some subcategory  ⌢ KS [i] of coh(P 2) . Then, for each k ≥ 0 , rankFk = f2k , where f0,f1,f2... is the Fibonacci numbers sequence.

We get our main theorem:

Theorem 4.17. Let n > 1 and F  n ∈ coh (P ) is an indecomposable locally free sheaf in some ⌢ KS [i]. Then

1. Every successor of F in the A-R component F is locally free.

2. Let 0 → F → B → δ- 1F → 0 be the Auslander-Reiten sequence in ⌢ KS [i] starting at F . Then

rankδ-2F > n δ-1F

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 n > 1. For each integer i each Auslander Reiten component of ⌢ KS [i] contains at most one locally free sheaf of rank less than n.

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 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