Vector bundles on projective varieties and representations of quivers
We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles.
Saved in:
| Date: | 2015 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2015
|
| Series: | Algebra and Discrete Mathematics |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/155146 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Vector bundles on projective varieties and representations of quivers / M. Jardim, D.M. Prata // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 217-249. — Бібліогр.: 10 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-155146 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1551462025-02-09T20:14:58Z Vector bundles on projective varieties and representations of quivers Jardim, M. Prata, D.M. We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles. We thank Helena Soares for her help with the results in Section 5,and Rosa Maria Miró-Roig for describing to us the monads considered in Section 5.3. We also thank the referee for his help in improving the presentation of the paper. Some of the results presented in this paper were obtained in the PhD thesis of the second author. Partially supported by the CNPq grant number 400356/2015-5 and the FAPESP grant number 2014/14743-8.2 Supported by the FAPESP doctoral grant number 2007/07469-3 and the FAPESP post-doctoral grant number 2011/21398-7. 2015 Article Vector bundles on projective varieties and representations of quivers / M. Jardim, D.M. Prata // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 217-249. — Бібліогр.: 10 назв. — англ. 1726-3255 2010 MSC:14F05, 14J60. https://nasplib.isofts.kiev.ua/handle/123456789/155146 en Algebra and Discrete Mathematics application/pdf Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles. |
| format |
Article |
| author |
Jardim, M. Prata, D.M. |
| spellingShingle |
Jardim, M. Prata, D.M. Vector bundles on projective varieties and representations of quivers Algebra and Discrete Mathematics |
| author_facet |
Jardim, M. Prata, D.M. |
| author_sort |
Jardim, M. |
| title |
Vector bundles on projective varieties and representations of quivers |
| title_short |
Vector bundles on projective varieties and representations of quivers |
| title_full |
Vector bundles on projective varieties and representations of quivers |
| title_fullStr |
Vector bundles on projective varieties and representations of quivers |
| title_full_unstemmed |
Vector bundles on projective varieties and representations of quivers |
| title_sort |
vector bundles on projective varieties and representations of quivers |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2015 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155146 |
| citation_txt |
Vector bundles on projective varieties and representations of quivers / M. Jardim, D.M. Prata // Algebra and Discrete Mathematics. — 2015. — Vol. 20, № 2. — С. 217-249. — Бібліогр.: 10 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT jardimm vectorbundlesonprojectivevarietiesandrepresentationsofquivers AT pratadm vectorbundlesonprojectivevarietiesandrepresentationsofquivers |
| first_indexed |
2025-11-30T10:13:58Z |
| last_indexed |
2025-11-30T10:13:58Z |
| _version_ |
1850209876766097408 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 20 (2015). Number 2, pp. 217–249
© Journal “Algebra and Discrete Mathematics”
Vector bundles on projective varieties
and representations of quivers∗
Marcos Jardim1 and Daniela Moura Prata2
Communicated by V. M. Futorny
Abstract. We present equivalences between certain cat-
egories of vector bundles on projective varieties, namely cokernel
bundles, Steiner bundles, syzygy bundles, and monads, and full sub-
categories of representations of certain quivers. As an application,
we provide decomposability criteria for such bundles.
1. Introduction
Vector bundles over algebraic varieties play a central role in algebraic
geometry, and many interesting problems are still open. In particular,
constructing indecomposable vector bundles on a variety X with rank
smaller than the dimX is not an easy task for certain choices of X,
especially for projective spaces.
Monads are one of the most important tools for constructing such
bundles; indeed, the majority of examples of low rank bundles on projective
spaces, namely the Horrocks–Mumford bundle of rank 2 on P4, Horrocks’
∗Some of the results presented in this paper were obtained in the PhD thesis of the
second author.
1Partially supported by the CNPq grant number 400356/2015-5 and the FAPESP
grant number 2014/14743-8.
2Supported by the FAPESP doctoral grant number 2007/07469-3 and the FAPESP
post-doctoral grant number 2011/21398-7.
2010 MSC: 14F05, 14J60.
Key words and phrases: vector bundles, representations of quivers.
218 Vector bundles and representations of quivers
parent bundle of rank 3 on P5, and the rank 2k instanton bundles on
P2k+1, are obtained as cohomologies of certain monads.
The goal of this paper is to show that the theory of representations
of quivers might also be an interesting tool for the construction of use-
ful monads and cokernel bundles on projective varieties. More precisely,
we present equivalences between certain categories of vector bundles on
projective varieties and full subcategories of representations of certain
quivers. In this way, we translate the problems of constructing indecom-
posable vector bundles on Pn with low rank into a (possibly still very
hard) problem of linear algebra. As an application of these results, we
give decomposability criteria for cokernel bundles, syzygy bundles and
monads.
Let us now present more precisely the results proved here, starting
with cokernel bundles, a class a vector bundles introduced by Brambilla
in [1]. Let X be a nonsingular projective variety of dimension n, and let
E and F be simple vector bundles on X such that
(i) Hom(F , E) = Ext1(F , E) = 0;
(ii) E∨ ⊗ F is globally generated;
(iii) dim Hom(E ,F) 6 3.
A cokernel bundle of type (E ,F) on X is a vector bundle C with a resolution
of the form
0 // Ea
α // Fb // C // 0 .
We prove (cf. Thm 3.5 below):
Theorem 1.1. The category of cokernel bundles of type (E ,F) is equiva-
lent to a full subcategory of the category of representation of the Kronecker
quiver with w = dim Hom(E ,F) arrows:
•
1 //
...
w
//
•
As application of this equivalence, we obtain new proofs of simplicity
and exceptionality criteria for cokernel bundles that were originally es-
tablished by Brambilla in [1, Thm. 4.3] (cf. Thm 3.8 below) and Soares
in [10, Theorem 2.2.7] (cf. Cor 3.13 below).
Next, we consider 1st-syzygy bundles on projective spaces; recall that
syzygy bundles are those given as kernel of surjective morphisms of the
form
OPn(−d1)a1 ⊕ · · · ⊕ OPn(−dm)am α
→ Oc
Pn .
M. Jardim, D. M. Prata 219
Let G := kerα; we refer to [2] as a general reference on syzygy bundles.
The case m = 1 can be regarded as a cokernel bundle; for the remainder
of the paper, we focus on the case m = 2, though it is not hard to generalize
our results for m > 2 (see Remark 4.6 below). More precisely, we prove
the following result, including a new decomposability criterion for syzygy
bundles.
Theorem 1.2. For any fixed integers d1 > d2 > 0, there is a faithful
functor from the category of representations of the quiver
•
1 //
...
w1
//
• •...
w2
oo
1oo
to the category of syzygy bundles given by sequences of the form
0 → G → OPn(−d1)a1 ⊕ OPn(−d2)a2 α
→ Oc
Pn → 0 (1)
where wj = h0(OPn(dj)), j = 1, 2. Moreover, if a2
1 + a2
2 + c2 − w1a1c −
w2a2c > 1, then G is decomposable.
Finally, we consider the relation between monads and representations
of quivers. Recall that a monad on a nonsingular projective variety X is
a complex of locally free sheaves of the form
M• : Aa // Bb // Cc (2)
whose only nontrivial cohomology is the middle one, which we assume, in
this paper, to also be a locally free sheaf. We prove:
Theorem 1.3. If A, B and C are simple vector bundles, then the category
of monads of the form (2) is equivalent to a full subcategory of the category
of representations of the quiver
•
1 //
...
m
//
• ...
n
//
1 //
•
where m = dim Hom(A,B) and n = dim Hom(B, C). In addition, if a2 +
b2 + c2 −mab−nbc > 1 then the cohomology sheaf of (2) is decomposable.
This generalizes the results of [4] (in particular, [4, Thm 1.1]) con-
cerning linear monads on Pn, i.e. when A = OPn(−1), B = OPn and
C = OPn(1).
220 Vector bundles and representations of quivers
Furthermore, if A, B and C are elements of distinct blocks of an n-block
collection generating the bounded derived category Db(X) of coherent
sheaves of OX -modules, then we also prove that the cohomology sheaf E of
(2) is decomposable, if and only if the corresponding quiver representation
is decomposable, cf. Theorem 5.5.
Notation. Throughout this paper, κ denotes an algebraically closed field
with characteristic zero, and X is always a nonsingular projective variety
over κ of dimension n.
2. Preliminary definitions and results
In this section we revise some key definitions and results on the theory
of representations of quivers and on the derived category of coherent
sheaves that will be relevant in the following sections.
2.1. Representations of quivers
We begin by revising some basic facts about representations of quivers.
Recall that a quiver Q consists on a pair (Q0, Q1) of sets where Q0 is the
set of vertices and Q1 is the set of arrows and a pair of maps t, h : Q1 → Q0
the tail and head maps. An example is the Kronecker quiver, denoted Kw,
which consists of 2 vertices and w arrows.
•
1 //
...
w
//
• (3)
A representation R = ({Vi}, {Aa}) of Q consists of a collection of
finite dimensional κ-vector spaces {Vi; i ∈ Q0} together with a collection
of linear maps {Aa : Vt(a) → Vh(a); a ∈ Q1}. A morphism f between two
representations R1 = ({Vi}, {Aa}) and R2 = ({Wi}, {Ba}) is a collection
of linear maps {fi} such that for each a ∈ Q1 the diagram bellow is
commutative
Vt(a)
Aa //
ft(a)
��
Vh(a)
fh(a)
��
Wt(a) Ba
//Wh(a)
With these definitions, representations of Q form an abelian category
hereby denoted by R(Q).
M. Jardim, D. M. Prata 221
Given a representation R ∈ R(Q), we associate a vector v ∈ ZQ0
called dimension vector, whose entries are vi = dimVi.
The Euler form on ZQ0 is a bilinear form associated to Q, given by
< v,w >=
∑
i∈Q0
viwi −
∑
a∈Q1
vt(a)wh(a).
The Tits form is the corresponding quadratic form, given by
q(v) =< v,v > .
For instance, the Tits form of the Kronecker quiver with w arrows is given
by
qw(v) = a2 + b2 − wab, v = (a, b) ∈ Z2. (4)
Definition 2.1. A vector v ∈ ZQ0 is a root if there is an indecomposable
representation R of Q with dimension vector v. Moreover, v is a Schur
root if there is a representation R of Q with dimension vector v satisfying
Hom(R,R) = κ.
Clearly, every Schur root is a root; note also that the condition
Hom(R,R) = κ is an open condition in the affine space
⊕a∈Q1Hom(κvt(a) , κvh(a))
of all representations with fixed dimension vector v. Thus if v is a Schur
root, then Hom(R,R) = κ for a generic representation with dimension
vector v. In particular, if v is a Schur root, then generic representation
with dimension vector v is indecomposable. A reference for generic repre-
sentations and Schur roots is [8]. For more information about roots and
root systems, we refer to [5].
The following two facts will be very relevant in what follows. The first
one follows from Kac’s theory of infinite root systems [5].
Proposition 2.2. Let Q be a quiver with Tits form q. If v is a dimension
vector satisfying q(v) > 1, then every representation with dimension vector
v is decomposable.
The second fact follows from [5, Prop 1.6] and [9, Thm 4.1].
Proposition 2.3. Let Q be the Kronecker quiver with w > 3, and let
v ∈ Z2 be a dimension vector. If qw(v) 6 1, then v is a Schur root.
222 Vector bundles and representations of quivers
2.2. Derived categories
In [7], Miró-Roig and Soares gave a cohomological characterisation of
Steiner bundles and later Marques and Soares [6], gave a cohomological
characterisation of a class of bundles given as cohomology of monads.
Both results will be relevant for us, so we review them here.
Let Db(X) be the bounded derived category of the abelian category of
coherent sheaves of OX -modules. An exceptional collection is an ordered
collection (F0, · · · ,Fm) of objects of Db(X) such that
Hom0
Db(X)(Fi,Fi) ≃ κ, Extp(Fi,Fi) = 0, for all p > 1,
Extp(Fi,Fj) = 0 for all i > j, and p > 0.
In addition, if
Extp(Fi,Fj) = 0 for i 6 j and p 6= 0 ,
then (F0, . . . ,Fm) is called a strongly exceptional collection. It is a full
(strongly) exceptional collection if it generates Db(X).
An exceptional collection (F0, · · · ,Fm) is called a block if
Extp(Fj ,Fi) = 0 ∀ p > 0 and i 6= j.
An m-block collection of type (t0, . . . , tm) is an exceptional collection
B = (F0, . . . ,Fm) where each F i = (F i
1, . . . ,F
i
ti
) is a block.
Definition 2.4. Let B = (F0, . . . ,Fm) be an m-block collection of type
(t0, . . . , tm). The left dual m-block collection of B is the m-block collection
∨B of type (u0, . . . , um) with ui = um−i
∨B = (H0, . . . ,Hm) = (H0
1, . . . ,H
0
u0
, . . . ,Hm
1 , . . . ,H
m
um
)
where
Homk
Db(X)(H
i
j ,F
l
p) = 0
for all indices, with the only exception
Exti(Hi
j ,F
m−i
j ) ≃ κ.
These conditions uniquely determine ∨B.
We are now able to define Steiner bundles in the sense of [7] and state
their cohomological characterisation.
M. Jardim, D. M. Prata 223
Definition 2.5. A vector bundle S on X is a Steiner bundle of type
(F0,F1) if it is given by a short exact sequence of the form
0 // Fa
0
α // Fb
1
// S // 0
such that a, b > 1 and (F0,F1) is an ordered pair of vector bundles on X
satisfying
(i) (F0,F1) is strongly exceptional;
(ii) F∨
0 ⊗ F1 is globally generated.
The cohomological characterisation is the following, cf. [7, Thm 2.4].
Theorem 2.6. Let X be a smooth projective variety of dimension n with
an n-block collection B = (F0, . . . ,Fn), F i = (F i
1, . . . ,F
i
ti
) of locally
free sheaves which generate Db(X), and let ∨B be its left dual basis.
Let F i
i0
∈ F i and F j
j0
∈ F j, where 0 6 i < j 6 n and 1 6 i0 6 ai,
1 6 j0 6 aj, and let S be a locally free sheaf on X. Then S is a Steiner
bundle of type (F i
i0
,F j
j0
) given by the short exact sequence
0 // (F i
i0
)a // (F j
j0
)b // S // 0
if and only if (F i
i0
)∨ ⊗F j
j0
is globally generated and all Extl(Hm
p ,S) vanish,
with the only exceptions of
dim Extn−i−1(Hn−i
i0
,S) = a and dim Extn−j(Hn−j
j0
,S) = b. (5)
Now we turn our attention to the cohomological characterisation for
the bundles obtained as cohomology of monads, due to Marques and
Soares in [6].
Definition 2.7. A monad M• on a smooth projective variety X is a
complex of locally free coherent sheaves on X
M• : A
α // B
β // C
such that α is injective, β is surjective; the coherent sheaf E = kerβ/imα
is called the cohomology of M•.
The following two definitions are important for the main result we
would like to present.
224 Vector bundles and representations of quivers
Definition 2.8. Let B = (F0, · · · ,Fm), F i = (F i
1, · · · ,F i
ti
), be an m-
block collection. A coherent sheaf E on X has natural cohomology with
respect to B if for each 0 6 p 6 m and 1 6 j 6 tp there is at most one
q > 0 such that Extq(Fp
j , E) 6= 0.
Definition 2.9. Let X be a smooth projective variety with an m-block
collection B = (F0, · · · ,Fm), F i = (F i
1, · · · ,F i
ti
) of coherent sheaves on
X. A Beilinson monad for E is a bounded complex G• in Db(X) whose
terms are finite direct sums of elements of B and whose cohomology is E ,
that is,
⊕
i∈Z
H i(G•) = H0(G•) = E .
The next result tell us when a coherent sheaf E on X is isomorphic to
a Beilinson monad G•, see [6, Cor 1.7].
Lemma 2.10. Let X be a smooth projective variety of dimension n
with an n-block collection B = (F0, · · · ,Fn) generating Db(X). Let
∨B = (H0, · · · ,Hn) with Hi = (Hi
1, · · · ,Hi
ui
), be its left dual n-block
collection. Then each coherent sheaf E on X is isomorphic to a Beilinson
monad G• with each Gr given by
Gr =
⊕
p,q
Extn−q+r(Hn−q
p , E) ⊗ Fq
p .
The cohomological characterisation for monads is the following, cf.
[6, Thm 2.2].
Theorem 2.11. Let X be a nonsingular projective variety of dimension
n, and let B = (F0, · · · ,Fn), where F i = (F i
1, · · · ,F i
ti
), be an n-block
collection of coherent sheaves on X generating Db(X). Let ∨B be its left
dual n-block collection, and let F i
i0
, F j
j0
, and Fk
k0
be elements of the blocks
F i,F j and Fk, respectively, with 0 6 i < j < k 6 n.
A torsion-free sheaf E on X is the cohomology sheaf of a monad of
the form
M• : (F i
i0
)a // (F j
j0
)b // (Fk
k0
)c (6)
for some b > 1 and a, c > 0 if and only if E has:
(1) rank b · rk(F j
j0
) − a · rk(F i
i0
) − c · rk(Fk
k0
);
(2) Chern polynomial ct(E) = ct(F
j
j0
)bct(F
i
i0
)−act(F
k
k0
)−c;
(3) natural cohomology with respect to ∨B.
M. Jardim, D. M. Prata 225
Remark 2.12. The original statement of [6, Thm 2.2] requires a, b, c > 1.
However, following the same steps of the proof of [6, Thm 2.2], one can
prove that the result also holds for a, c > 0; in other words, one can allow
for degenerate monads.
This result will be very useful in the last section of this paper, in
which we study the decomposability of sheaves given by the cohomology
of monads of the above form.
3. Cokernel and Steiner bundles
In this section we explain the relation between cokernel and Steiner
bundles and representations of the Kronecker quiver.
3.1. Cokernel bundles
Let E and F be vector bundles on a nonsingular projective variety X
of dimension n > 2, satisfying the following conditions:
(1) E and F are simple, that is, Hom(E , E) = Hom(F ,F) = κ;
(2) Hom(F , E) = 0;
(3) Ext1(F , E) = 0;
(4) the sheaf E∨ ⊗ F is globally generated;
(5) W = Hom(E ,F) has dimension w > 3.
The next definition is due to Brambilla [1].
Definition 3.1. A cokernel bundle of type (E ,F) on Pn is a vector bundle
C with resolution of the form
0 // Ea
α // Fb // C // 0 (7)
where E ,F satisfy the conditions (1) through (5) above, a > 0 and
b · rk(F) − a · rk(E) > n.
Cokernel bundles of type (E ,F) form a full subcategory of the category
of coherent sheaves on X; this category will be denoted by CX(E ,F).
Let us now see how cokernel bundles are related to quivers. Fix a
basis σ = {σ1, · · · , σw} of Hom(E ,F).
226 Vector bundles and representations of quivers
Definition 3.2. A representation R = ({κa, κb}, {Ai}
w
i=1) of Kw is
(E ,F ,σ)-globally injective when the map
α(P ) :=
w
∑
i=1
Ai ⊗ σi(P ) : κa ⊗ EP → κb ⊗ FP
is injective for every P ∈ X; here, EP and FP denote the fibers of E and
F over the point P , respectively.
(E ,F ,σ)-globally injective representations of Kw form a full subcate-
gory of the category of representations of Kw; we denote it by R(Kw)gi.
From now on, since (E ,F ,σ) are fixed, we will just refer to globally in-
jective representations. It is a simple exercise to establish the following
properties of R(Kw)gi.
Lemma 3.3. The category R(Kw)gi is closed under sub-objects, i.e. every
subrepresentation R′ of a representation R in R(Kw)gi is also in R(Kw)gi.
Lemma 3.4. The category R(Kw)gi is closed under extensions and under
direct summands, that is, respectively:
(i) if R1, R2 ∈ R(Kw)gi and
0 // R1
// R // R2
// 0
is a short exact sequence in R(Kw)gi, then R ∈ R(Kw)gi;
(ii) if R ∈ R(Kw)gi with R ≃ R1 ⊕R2, then Ri ∈ R(Kw)gi, i = 1, 2.
Our next result relates the category of globally injective representations
of Kw to the category of cokernel bundles.
Theorem 3.5. For every choice of basis σ of Hom(E ,F), there is an
equivalence between R(Kw)gi, the category of (E ,F ,σ)-globally injective
representations of Kw, and CX(E ,F), the category of cokernel bundles of
type (E ,F).
Proof. Given a basis σ of Hom(E ,F), we construct a functor
Lσ : R(Kw)gi → CX(E ,F)
and show that it is essentially surjective and fully faithful.
Let R = ({κa, κb}, {Ai}
w
i=1) be a globally injective representation of
Kw. Define a map α : Ea → Fb given by
α = A1 ⊗ σ1 + · · · +Aw ⊗ σw.
M. Jardim, D. M. Prata 227
Since R is globally injective, we have that dim cokerα(P ) = b · rk(F) −
a · rk(E) for each P ∈ X. Therefore α is injective as a map of sheaves,
and C := cokerα is a cokernel bundle.
Now given two globally injective representations
R1 = ({κa, κb}, {Ai}
w
i=1) and R2 = ({κc, κd}, {Bi}
w
i=1),
and a morphism f = (f1, f2) between them, let Lσ(R1) = C1,Lσ(R2) = C2
be the cokernel bundles and α1, α2 the maps associated to R1 and R2,
respectively. We want to define a morphism Lσ(f) : C1 → C2.
Since we have f1 : κa → κc, f2 : κb → κd, we have maps f
′
1 = f1 ⊗1E ∈
Hom(Ea, Ec) and f
′
2 = f2 ⊗ 1F ∈ Hom(Fb,Fd). Consider the diagram
0 // Ea
f
′
1
��
α1 // Fb
f
′
2 ��
π1 // C1
//
��
0
0 // Ec
α2 // Fd π2 // C2
// 0
(8)
where π1, π2 are the projections. Applying the left exact contravariant
functor Hom(−, C2) to the upper sequence on (8) we find a map φ ∈
Hom(C1, C2) and we define Lσ(f) := φ.
Now given C an object of CX(E ,F) we take α =
∑w
i=1Ai ⊗ σi, with
Ai ∈ Hom(κa, κb), i = 1, · · · , w. Hence R = ({κa, κb}, {Ai}
w
i=1) is a glob-
ally injective representation of R(Kw) such that Lσ(R) = C. Therefore
Lσ is essentially surjective.
Finally, we need to prove that Lσ is fully faithful. To check that it is full,
given φ ∈ Hom(Lσ(R1),Lσ(R2)) we want f = (f1, f2) ∈ Hom(R1, R2)
such that Lσ(f) = φ. Let φ̃ = φπ1 ∈ Hom(Fb, C2). Let us apply the left
exact covariant functor Hom(Fb,−) to the lower sequence in diagram (9)
below:
0 // Ea
f
′
1
��
α1 // Fb
f
′
2 ��
π1 // C1
//
φ
��
0
0 // Ec
α2 // Fd π2 // C2
// 0
(9)
we conclude that
ρ2 : Hom(Fb,Fd) → Hom(Fb, C2) (10)
is an isomorphism since Hom(Fb, Ec) = Ext1(Fb, Ec) = 0. It follows that
there is a morphism f
′
2 ∈ Hom(Fb,Fd) such that
ρ2(f
′
2) = π2f
′
2 = φπ1
228 Vector bundles and representations of quivers
with f
′
2 = f2 ⊗ 1F and f2 ∈ Hom(κb, κd).
Consider ˜̃φ = f
′
2α1 ∈ Hom(Ea,Fd). Applying the left exact covariant
functor Hom(Ea,−) to the lower sequence on (9) we get
0 // Hom(Ea, Ec)
γ1 // Hom(Ea,Fd)
γ2 // Hom(Ea, C2) // · · ·
Once we have an exact sequence,
γ2(f
′
2α1) = π2f
′
2α1 = φπ1α1 = 0
then f
′
2α1 ∈ ker γ2 = im γ1, and there is a map f
′
1 ∈ Hom(Ea, Ec) such
that γ1(f
′
1) = α2f
′
1 = f
′
2α1 and f
′
1 = f1 ⊗ 1E with f1 ∈ Hom(κa, κc).
Since α1 =
∑w
i=1Ai ⊗ σi, α2 =
∑w
i=1Bi ⊗ σi, α2f
′
1 = f
′
2α1, and σ is a
basis then f2Ai = Bif1, i = 1, · · · , w, thus
f = (f1, f2) ∈ HomR(Kw)gi(R1, R2).
Now we need to prove that Lσ(f) = φ. Suppose Lσ(f) = φ such that
φπ1 = π2f
′
2 = φπ1. Then (φ − φ)π1 = 0 and C1 = im π1 ⊂ ker(φ − φ)
therefore φ = φ.
Finally, we show that Lσ : Hom(R1, R2) → Hom(Lσ(R1),Lσ(R2)) is
injective. Let f = (f1, f2), g = (g1, g2) ∈ Hom(R1, R2) be morphisms such
that Lσ(f) = φ1 = φ2 = Lσ(g), that is, φ1 − φ2 = 0.
0 // Ea
f
′
1−g
′
1
��
α1 // Fb
f
′
2−g
′
2 ��
π1 // C1
//
0
��
0
0 // Ec
α2 // Fd π2 // C2
// 0
(11)
Given φ1 − φ2 = 0 ∈ Hom(C1, C2), doing the same construction as
before,
0π1 = 0 ∈ Hom(Fb, C2) ≃ Hom(Fb,Fd)
with isomorphism given by ρ2 in (10). Since
ρ2(f
′
2 − g
′
2) = π2 ◦ (f
′
2 − g
′
2) = 0
then f
′
2 − g
′
2 = 0 and so f
′
2 = g
′
2. Similarly, 0α1 = 0 ∈ Hom(Ea,Fd) and
γ1(f
′
1 − g
′
1) = α2(f
′
1 − g
′
1) = 0α1 = 0.
Since γ1 injective, f
′
1 − g
′
1 = 0, then f
′
1 = g
′
1. Therefore Lσ is faithful.
M. Jardim, D. M. Prata 229
Remark 3.6. Note that the functor Lσ depends on the choice of the basis
σ. However let σ′ be another basis for Hom(E ,F). Let Lσ′ be the equiva-
lence between the category of (E ,F ,σ′)-globally injective representations
of Kw and the cokernel bundles on Pn. Then if G is the inverse functor
of Lσ′ we have that the functor G ◦ Lσ′ gives an equivalence between the
categories (E ,F ,σ)- and (E ,F ,σ′)-globally injective representations of
Kw.
Lemma 3.7. For any choice of basis σ, the functor Lσ : R(Kw)gi →
CX(E ,F) defined above is additive and exact. In particular, if R ≃ R1⊕R2
is a globally injective representation, then Lσ(R) ≃ Lσ(R1) ⊕ Lσ(R2).
Proof. Checking the additivity of Lσ is a simple exercise. We show its
exactness in detail.
Let us prove that Lσ preserves exact sequences. Let R1 = ({κa1 , κb1},
{Ai}
w
i=1), R2 = ({κa2 , κb2}, {Bi}
w
i=1) and R3 = ({κa3 , κb3}, {Ci}
w
i=1) be
globally injective representations of Kw and let f : R1 → R2 and g :
R2 → R3 be morphisms such that the sequence
0 // R1
f // R2
g // R3
// 0
is exact. We want to prove that
0 // C1
ϕ // C2
ψ // C3
// 0
is also exact, where Ci = Lσ(Ri), i = 1, 2, 3 and ϕ = Lσ(f), ψ = Lσ(g).
From the exact sequence of representations we get
0
��
0
��
0
��
0 // Ea1
α1 //
1E ⊗f1
��
Fb1
π1 //
1F ⊗f2
��
C1
ϕ
��
// 0
0 // Ea2
α2 //
1E ⊗g1
��
Fb2
π2 //
1F ⊗g2
��
C2
//
ψ
��
0
0 // Ea3
α3 //
��
Fb3
π3 //
��
C3
//
��
0
0 0 0
We need to show that ϕ is injective and ψ is surjective.
230 Vector bundles and representations of quivers
• ψ is surjective:
It follows from the fact that π3(1F ⊗ g2) is surjective.
• ϕ is injective.
Let us suppose ϕ(s) = 0, s ∈ C1. Then s = π1(v), v ∈ Fb1 and
0 = ϕπ1(v) = π2(1F ⊗ f2)(v).
Since kerπ2 = imα2, there is u ∈ Ea2 such that
(1F ⊗ f2)(v) = α2(u) (12)
Note that
α3(1E ⊗ g1)(u) = (1F ⊗ g2)(α2)(u) = (1F ⊗ g2)(1F ⊗ f2)(v) = 0
and since α3 is injective, (1E ⊗ g1)(u) = 0 so u = (1E ⊗ f1)(u′) with
u′ ∈ Ea1 . We have
α2(u) = α2(1E ⊗ f1)(u′) = (1F ⊗ f2)α1(u′).
From (12) we have (1F ⊗ f2)(v) = (1F ⊗ f2)(α1(u′)). Since (1F ⊗ f2) is
injective, it follows that v = α1(u′) therefore
s = π1(v) = π1α1(u′) = 0.
Now suppose R ≃ R1⊕R2. Let us prove that Lσ(R1⊕R2) ≃ Lσ(R1)⊕
Lσ(R2). We have the short exact sequence
0 // R1
iR1 // R1 ⊕R2
πR2 // R2
iR2
oo
// 0
where iRj
is the inclusion and πRj
the projection, j = 1, 2. Since the
sequence above is split, πR2 ◦ iR2 = 1R2 . Now since Lσ is an exact functor,
we have
0 // Lσ(R1)
Lσ(iR1
)
// Lσ(R1 ⊕R2)
Lσ(πR2
)
// Lσ(R2) //
Lσ(iR2
)
oo
0 (13)
Then
Lσ(πR2 ◦ iR2) = Lσ(πR2) ◦ Lσ(iR2) = Lσ(1R2) = 1Lσ(R2)
therefore the sequence (13) is split. Hence Lσ(R1 ⊕ R2) ≃ Lσ(R1) ⊕
Lσ(R2).
M. Jardim, D. M. Prata 231
As an application of the previous results, we give a new, functorial
proof for a result due to Brambilla, cf. [1, Thm 4.3].
Theorem 3.8. Let C be a cokernel bundle of type (E ,F), given by the
resolution
0 // Ea
α // Fb // C // 0 , (14)
and let w = dim Hom(E ,F).
(i) If C is simple, then a2 + b2 − wab 6 1.
(ii) If a2 + b2 − wab 6 1, then there exists a non-empty open subset
U ⊂ Hom(Ea,Fb) such that for every α ∈ U the corresponding
cokernel bundle is simple.
Proof. To prove (i), let C be a cokernel bundle given by resolution (14)
and suppose C is simple. By Theorem 3.5 there is a globally injective
representation R of Kw such that C = Lσ(R). Since Lσ is full, we have
that κ = Hom(C, C) ≃ Hom(R,R), thus R is simple and therefore, by
Proposition 2.2, qw(a, b) = a2 + b2 − wab 6 1.
For the second claim, note that if qw(a, b) 6 1, there is a generic
representation R with dimension vector (a, b) such that R is Schur, by
Proposition 2.2. Then there is a non-empty open subset
U ⊂ Hom(κa, κb) ⊗ κw ≃ Hom(Ea,Fb)
such that every R ∈ U is simple. Since Hom(C, C) ≃ Hom(R,R) = κ, it
follows that C is simple.
The previous Theorem implies that if a2 + b2 − wab > 1 then C is
not simple. However, more is true, and it is not difficult to establish the
following stronger statement.
Proposition 3.9. Under the same conditions as in Theorem 3.8, if
a2 + b2 − wab > 1, then C is decomposable.
Under more restrictive conditions, Brambilla proved in [1, Thm 6.3]
that if C is a generic cokernel bundle such that a2 + b2 − wab > 1, then
C ≃ Cnk ⊕ Cmk+1, where Ck and Ck+1 are Fibonacci bundles, n,m ∈ N (we
refer to [1] for the definition of Fibonacci bundles).
232 Vector bundles and representations of quivers
Proof. Let C be any cokernel bundle given by the exact sequence (14), such
that a2 + b2 − wab > 1. Then there is a globally injective representation
R of Kw, such that C = Lσ(R) with dimension vector (a, b) satisfying and
qw(a, b) = a2 + b2 − wab > 1. By Lemma 2.2, R is decomposable. Then
by Lemma 3.7, C is also decomposable.
Next, recall that a vector bundle E on X is exceptional if it is simple
and Extp(E , E) = 0 for p > 1.
Proposition 3.10. Under the same conditions as in Theorem 3.8, if C
is exceptional, then a2 + b2 − wab = 1.
Proof. Since the functor Lσ is exact, we have an isomorphism
Ext1(R,R) ≃ Ext1(Lσ(R),Lσ(R)).
Now we know from [8] that
qw(a, b) = dim Hom(R,R) − dim Ext1(R,R) (15)
hence if C is an exceptional cokernel bundle, then qw(a, b) =
a2 + b2 − wab = 1.
However, the converse of the Proposition 3.10 is not true. For instance,
consider the generic cokernel bundle given by the exact sequence
0 // OP3 // OP3(4)35 // C // 0 .
We have q35(1, 35) = 1, but from the long exact sequence of cohomologies,
Ext2(C, C) ≃ κ35 hence C is not exceptional. In order to establish the
converse statement, we need stronger assumption, provided by Steiner
bundles.
3.2. Steiner bundles
Note that the Steiner bundles of type (E ,F) satisfying
dim Hom(E ,F) > 3, are a particular case of cokernel bundles,
therefore all results in the previous section also hold for such Steiner
bundles. Furthermore, the additional hypotheses satisfied by the sheaves E
and F allow to establish the converse of Lemma 3.7 and Proposition 3.10.
Let us first consider the converse of Lemma 3.7; more precisely, we
prove the following statement.
M. Jardim, D. M. Prata 233
Theorem 3.11. Let X be a nonsingular projective variety of dimension
n, and let B = (F0, · · · ,Fn) be an n-block collection generating Db(X).
A Steiner bundle of type (F i
i0
,F j
j0
) such that w = dim Hom(F i
i0
,F j
j0
) > 3
is decomposable if and only if, for any choice of basis γ for Hom(F i
i0
,F j
j0
),
the corresponding (F i
i0
,F j
j0
,γ)-globally injective representation of Kw is
also decomposable.
The theorem follows easily from Lemma 3.7 and the following claim.
Let S
F i
i0
,F
j
j0
(X) denote the category of Steiner bundles of type (F i
i0
,F j
j0
)
over X.
Proposition 3.12. The category S
F i
i0
,F
j
j0
(X) is closed under direct sum-
mands.
Proof. Let ∨B = (H0, · · · ,Hn) where Hi = (Hi
1, · · · ,Hi
ui
), be the n-
block collection which is left dual to B, and let S be a Steiner bundle of
type (F i
i0
,F j
j0
) given by the short exact sequence
0 // (F i
i0
)a // (F j
j0
)b // S // 0 ,
where F i
i0
and F j
j0
are elements of blocks F i and F j respectively, 0 6
i < j 6 n.
If S ≃ S1 ⊕ S2, 0 6= Si ( S, i = 1, 2, then we have that
Extp(Hm
q ,S) ≃ Extp(Hm
q ,S1) ⊕ (Hm
q ,S2).
It follows that Extp(Hm
q ,Sl), l = 1, 2, vanish except for
Extn−i−1(Hn−i
i0
,Sl) = al, al > 0, l = 1, 2
and
Extn−j(Hn−j
j0
,Sl) = bl, bl > 0, l = 1, 2
with a1 + a2 = a and b1 + b2 = b. Then from the cohomological character-
isation, Theorem 2.11, one of the following possibilities must hold.
1) For al 6= 0 and bl 6= 0, l = 1, 2, the bundles Sl are Steiner bundles
given by
0 // (F i
i0
)al // (F j
j0
)bl // Sl // 0 .
234 Vector bundles and representations of quivers
2) For a1, b1, b2 6= 0 and a2 = 0, we have
0 // (F i
i0
)a1 // (F j
j0
)b1 // S1
// 0 and S2 ≃ (F j
j0
)b2 .
3) For a1 = 0 and b1, a2, b2 6= 0, we have
S1 ≃ (F i
i0
)a1 and 0 // (F i
i0
)a2 // (F j
j0
)b2 // S2
// 0 .
To complete this section, we consider the converse of Proposition 3.10.
Proposition 3.13. Let S be a Steiner bundle of type (E ,F) with w =
dim Hom(E ,F) > 3, given by the short exact sequence:
0 // Ea
α // Fb // S // 0 . (16)
(i) If S is exceptional then a2 + b2 − wab = 1.
(ii) If a2 + b2 − wab = 1 then there is a non-empty open subset U ⊂
Hom(Ea,Fb) such that for every α ∈ U the corresponding bundle S
is exceptional.
Proof. The first claim is just Proposition 3.10. For the second state-
ment, we first show that if S1,S2 are Steiner bundles of type (E ,F), then
Extp(S1,S2) = 0 for p > 2.
Indeed, suppose Si, i = 1, 2, are given by short exact sequences
0 // Eai // Fbi // Si // 0 (17)
Applying the functor Hom(−,F) to the sequence (17) for i = 1, we have
Extp(S1,F) = 0, p > 2. Applying Hom(−, E) to the same sequence, we
obtain Extq(S1, E) = 0, q > 0. Finally applying the functor Hom(S1,−)
to the sequence (17), i = 2, we conclude that Extj(S1,S2) = 0 for j > 2.
Now to prove the second claim, start by supposing that qw(a, b) =
a2 + b2 − wab = 1. By Theorem 3.8 item (ii) there exists a non-empty
open subset U ⊂ Hom(Ea,Fb) such that for every α ∈ U the associated
bundle S is simple. From (15) we see that Ext1(S,S) = 0. Finally, from
the considerations above, we have Extp(S,S) = 0 for p > 2. Hence S is
exceptional.
Remark 3.14. Soares also proved in [10, Thm 2.2.7], using a different
method, that a generic Steiner bundle of type (E ,F) given by the short
exact sequence (16) is exceptional if and only if a2 + b2 − wab = 1.
M. Jardim, D. M. Prata 235
4. Syzygy bundles and quivers
In this section we relate a different class of vector bundles, the syzygy
bundles, with representations of quivers. A locally free sheaf G given by
the short exact sequence
0 // G // OPn(−d1)a1 ⊕ · · · ⊕ OPn(−dm)am
α // Oc
Pn
// 0 (18)
is called a syzygy bundle. Here, α = (α1, α2, . . . , αm) is a surjective map
of sheaves on Pn given by
α(f1, f2, . . . , fm) =
m
∑
i=1
αifi
where f1, . . . , fm are homogeneous polynomials of degree d1, . . . , dm in
κ[X0, . . . , Xn] and di are distinct positive integers. Let us assume 0 6
dm < · · · < d1.
Note that for m = 1, the dual bundle G∗ is a cokernel bundle. However,
the same is not true for m > 1, since the bundle F = OPn(d1)a1 ⊕ · · · ⊕
OPn(dm)am is not simple.
To relate syzygy bundles with representations of quivers, we restrict
ourselves, for the sake of simplicity, to the case m = 2. The results for the
general case are the same, but the notation becomes more complicated.
Thus we set m = 2, and consider exact sequences of the form
0 // G // OPn(−d1)a ⊕ OPn(−d2)b
α1,α2 // Oc
Pn
// 0 (19)
with d1 > d2. We denote by Syz(d1, d2) the category of syzygy bundles
given by short exact sequences as in (19) above.
Fix, for i = 1, 2, a basis σi = {f i1, . . . , f
i
wi
} of H0(OPn(di)), where
wi =
(n+di
di
)
. Consider the quiver below, which will be denoted by Aw1,w2 :
•
1 //
...
w1
//
• •...
w2
oo
1oo
(20)
If (a, b, c) is a dimension vector of this quiver, its Tits form is given by
qw1,w2(a, b, c) = a2 + b2 + c2 − w1ab− w2bc. (21)
Let R = ({κa, κb, κc}, {Ai}
w1
1 , {Bj}
w2
1 ) be a representation of Aw1,w2 ,
where each Ai is a c×a matrix, and each Bj is a c× b matrix with entries
236 Vector bundles and representations of quivers
in κ. We define
α1 =
w1
∑
i=1
Ai ⊗ f1
i and α2 =
w2
∑
j=1
Bj ⊗ f2
j ,
so that we have a map
(α1, α2) : OPn(−d1)a ⊕ OPn(−d2)b → Oc
Pn . (22)
Definition 4.1. A representation R of Aw1,w2 is (σ1,σ2)-globally surjec-
tive if the map (α1, α2) is surjective.
Denote by R(Aw1,w2)gs the category of (σ1,σ2)-globally surjective
representations of Aw1,w2 . We will now build a functor Gσ1,σ2 between
the R(Aw1,w2)gs and the category of syzygy bundles Syz(d1, d2).
First, let R = ({κa, κb, κc}, {Ai}
w1
i=1, {Bj}
w2
i=1) be a globally surjective
representation of Aw1,w2 . We define the sheaf
Gσ1,σ2(R) := ker(α1, α2),
where (α1, α2) is the map defined above in (22). Note that, since R is
globally surjective, Gσ1,σ2(R) is a vector bundle, and it is given by the
exact sequence (19).
Now let {g1, g2, h} be a morphism between the globally surjective
representations
R = ({κa, κb, κc}, {Ai}
w1
i=1, {Bj}
w2
i=1) and
R′ = ({κa
′
, κb
′
, κc
′
}, {A′
i}
w1
i=1, {B
′
j}
w2
i=1).
The following diagram commutes for i = 1, . . . , w1 and j = 1, . . . , w2.
κa
g1
,,
Ai
��
κa
′
A′
i
κc
h // κc
′
κb
Bi
??
g2
22 κb
′
B′
j
>>
(23)
M. Jardim, D. M. Prata 237
It induces the following diagram:
0 // G
φ
��
i1 // OPn(−d1)a ⊕ OPn(−d2)b
M
��
α1,α2 // Oc
Pn
h⊗1O
Pn
��
// 0
0 // G′ i2 // OPn(−d1)a
′
⊕ OPn(−d2)b
′ α′
1,α
′
2// Oc′
Pn
// 0
(24)
where
M =
(
g1 ⊗ 1OPn (−d1) 0
0 g2 ⊗ 1OPn (−d2)
)
The commutativity of (23) implies the commutativity of the right square
in (24). We then have an induced morphism φ : G = Gσ1,σ2(R) → G′ =
Gσ1,σ2(R′), which we define to be Gσ1,σ2(g1, g2, h).
Lemma 4.2. The functor Gσ1,σ2 is faithful and essentially surjective.
Proof. We prove that Hom(R,R′) → Hom(G(R),G(R′)) is injective. Let
{g1, g2, h} be a morphism between R and R′ such that G({g1, g2, h}) = 0,
that is, φ = 0. Since the diagram (24) commutes if φ = 0 then g1 = g2 =
h = 0, hence G is faithful.
Let G be a syzygy bundle with resolution
0 // G // OPn(−d1)a ⊕ OPn(−d2)b
α1,α2 // Oc
Pn
// 0
Then the maps α1 and α2 are given by
α1 =
w1
∑
i=1
Ai ⊗ f1
i and α2 =
w2
∑
j=1
Bj ⊗ f2
j
with Ai ∈ Hom(ka, kc) and Bj ∈ Hom(kb, kc). Therefore
R = ({ka, kb, kc}, {Ai}
w1
1 , {Bj}
w2
1 )
is a globally surjective representation of (20) such that G(R) = G.
Remark 4.3. Note that Gσ1,σ2 is not full, since not every
M ∈ Hom (OPn(−d1)a ⊕ OPn(−d2)b,OPn(−d1)a
′
⊕ OPn(−d2)b
′
)
is necessarily diagonal. It follows that the categories R(Aw1,w2)gs and
Syz(d1, d2) are not, in general, equivalent.
238 Vector bundles and representations of quivers
This completes the proof of the first part of Theorem 1.2. To establish
its second part, we first need the following two lemmas.
Lemma 4.4. The category of globally surjective representations of
Aw1,w2 is closed under quotients, and hence closed under direct summands.
Proof. Let R = ({κa, κb, κc}, {Ai}
w1
i=1, {Bj}
w2
j=1) be a (σ1,σ2)-globally
surjective representation of Aw1,w2 and R′ = ({κa
′
, κb
′
, κc
′
}, {A′
i}
w1
i=1,
{B′
j}
w2
j=1) be a subrepresentation of R. We want to prove that the quotient
representation
R/R′ = ({κa/κa
′
, κb/κb
′
, κc/κc
′
}, {Ci}
w1
i=1, {Dj}
w2
j=1),
where Ci and Dj are the maps induced by Ai and Bj respectively, is also
globally surjective. We have the diagram
0
��
0
��
0
��
κa
′
l1
��
A′
1 //
...
A′
w1
//
κc
′
l3
��
κb
′
l2
��
B′
w2
oo
...
B′
1oo
κa
p1
��
A1 //
...
Aw1
//
κc
p3
��
κb
p2
��
Bw2
oo
...
B1oo
κa/κa
′
��
C1 //
...
Cw1
//
κc/κc
′
��
κb/κb
′
��
Dw2
oo
...
D1oo
0 0 0
where li are the inclusions and pi the projections i = 1, 2, 3. Now consider
the commutative diagram
OPn(−d1)a ⊕ OPn(−d2)b
M
��
(α1,α2) // Oc
Pn
p3⊗1O
Pn
��
OPn(−d1)(a−a′) ⊕ OPn(−d2)(b−b′) (γ1,γ2)// O
(c−c′)
Pn
M. Jardim, D. M. Prata 239
where
M =
(
p1 ⊗ 1OPn (−d1) 0
0 p2 ⊗ 1OPn (−d2)
)
and γ1 =
∑w1
i=1Ci ⊗ f1
i , γ2 =
∑w2
j=1Dj ⊗ f2
j .
Since pi is surjective and (α1, α2) is surjective for every P ∈ Pn, we
have that the map (γ1, γ2) is also surjective for every point P ∈ Pn, hence
the quotient representation R/R′ is globally surjective.
Lemma 4.5. Let R be a decomposable globally surjective representation
of Aw1,w2. Then Gσ1,σ2(R) is also decomposable.
Proof. Let R ≃ R1 ⊕R2 be a decomposable globally surjective represen-
tation. From Lemma 4.4 we have that R1 and R2 are globally surjective.
Let Gi = Gσ1,σ2(Ri), i = 1, 2 be given by the short exact sequence
0 // Gi // OPn(−d1)ai ⊕ OPn(−d2)bi
αi
// Oci
Pn
// 0
where αi = (αi1, α
i
2), i = 1, 2. Since
G = Gσ1,σ2(R) = ker(α1 ⊕ α2) ≃ kerα1 ⊕ kerα2 =
= Gσ1,σ2(R1) ⊕ Gσ1,σ2(R2),
it follows that G is decomposable.
We are finally in position to complete the proof of Theorem 1.2.
Indeed, fix bases σj for H0(OPn(dj)), j = 1, 2. For every syzygy bundle G
given by a short exact sequence of the form (19), one can find a (σ1,σ2)-
globally surjective representation R of Aw1,w2 with dimension vector
(a, b, c) with Gσ1,σ2(R) = G. If qw1,w2(a, b, c) > 1, then R is decomposable,
by Lemma 2.2, and it must decompose as a sum of (σ1,σ2)-globally
surjective representations by Lemma 4.4. Therefore Lemma 4.5 implies
that G is also decomposable.
Remark 4.6. All the results can be generalized for syzygy bundles with
m > 2. To build the associated quiver, we add a vertex to the quiver with
wi = dimH0(OPn(di)) arrows from this vertex to the vertex associated
to O⊕c
Pn , for each term OPn(−di)
⊕ai .
240 Vector bundles and representations of quivers
5. Monads and representations of quivers
Recall that a monad M• on a projective variety X is a complex of
locally free sheaves
M• : A⊕a α // B⊕b β // C⊕c (25)
where α is injective and β is surjective. The coherent sheaf
E := kerβ/im α is called the cohomology of M•; note that E is locally
free if and only if the map αP on the fibers is injective for every point
P ∈ X.
Now let m = dim Hom(A,B) and n = dim Hom(B, C). We also assume
that A,B, C are simple vector bundles, and that the cohomology sheaf E
is locally free. We will denote the category of such monads by MA,B,C,
regarding it as a full subcategory of the category of complexes of coherent
sheaves on X.
Next, consider the quiver Km,n given by the graph
•
1 //
...
m
//
• ...
n
//
1 //
•
The category of representations of Km,n is denoted by R(Km,n). Note
that its Tits form is given by
qm,n(a, b, c) = a2 + b2 + c2 −mab− nbc. (26)
5.1. Proof of Theorem 1.3
We begin by describing a functor from MA,B,C to R(Km,n) in a
manner similar to what was done in the previous sections. Choose bases
γ = {γ1, · · · , γm} of Hom(A,B) and σ = {σ1, · · · , σn} of Hom(B, C). We
can write
α =
m
∑
i=1
Ai ⊗ γi and β =
n
∑
j=1
Bj ⊗ σj
where each Ai is a b× a matrix with entries in κ, and each Bj is a c× b
matrix with entries in κ.
Now let
Gγ,σ : MA,B,C → R(Km,n) (27)
M. Jardim, D. M. Prata 241
be the functor that to each monad M• as in (25) with maps α and β,
associates the representation R = ({κa, κb, κc}, {Ai}
m
i=1, {Bj}
m
j=1). Let
ϕ• = (f, g, h) be a morphism between the monads M•
1 and M•
2 below
Aa1
f
��
α1 // Bb1
g
��
β1 // Cc1
h
��
Aa2
α2
// Bb2
β2
// Cc2
Since A, B and C are simple, it follows that
(f, g, h) = (A⊗ 1A, B ⊗ 1B, C ⊗ 1C)
where A,B and C are, respectively, a2 × a1, b2 × b1 and c2 × c1 matrices
with entries in κ. If
Gγ,σ(M•
1 ) = ({κa1 , κb1 , κc1}, {A1
i }
m
i=1, {B
1
j }nj=1)
and Gγ,σ(M•
2 ) = ({κa2 , κb2 , κc2}, {A2
i }
m
i=1, {B
2
j }nj=1),
we then have
κa1
A
��
A1
1 //
...
A1
m
//
κb1
B
��
...
B1
n
//
B1
1 //
κc1
C
��
κa2
A2
1 //
...
A2
m
//
κb2 ...
B2
n
//
B2
1 //
κc2
(28)
BA1
i = A2
iA and CB1
j = B2
jB for i = 1, · · · ,m and j = 1, · · · , n.
Hence the matrices A,B and C define a morphism between the rep-
resentations. From the construction of the functor we see that Gγ,σ :
Hom(M•
1 ,M
•
2 ) → Hom(Gγ,σ(M•
1 ),Gγ,σ(M•
2 )) is an isomorphism, thus
we have the following result, which corresponds to the first part of Theo-
rem 1.3.
Proposition 5.1. The category MA,B,C is equivalent to a full subcategory
of R(Km,n).
242 Vector bundles and representations of quivers
Let us further characterise the subcategory of R(Km,n) obtained in
this way. The monad conditions imply that α(P ) is injective and β(P ) is
surjective for every P ∈ X. Therefore we say that a representation R =
({κa, κb, κc}, {Ai}
m
i=1, {Bj}
n
j=1) is (γ,σ)-globally injective and surjective
if α(P ) =
∑m
i=1Ai ⊗ γi(P ) is injective and β(P ) =
∑n
j=1Bj ⊗ σj(P ) is
surjective, for every P ∈ X. In addition, the matrices Ai and Bj must
satisfy quadratic equations imposed by the condition βα = 0:
∑
16i6j6m
(BiAj +BjAi)(σiγj) = 0;
note that the precise relation depends on the choice of bases γ and σ. We
denote by G
gis
m,n the full subcategory of R(Km,n) consisting of the objects
satisfying the conditions above.
In order to prove the second part of Theorem 1.3, our first goal is to
prove that G
gis
m,n is closed under direct summands.
Lemma 5.2. The category G
gis
m,n is closed under direct summands.
Proof. It is a general fact that if S is a subrepresentation of a quiver
representation R which satisfies the given relations, then S also satisfies
the same relations.
Moreover, every subrepresentation of a γ-globally injective represen-
tation will also be γ-globally injective (cf. Lemma 3.3 above), while any
quotient representation of a σ-globally surjective representation will also
be σ-globally surjective (cf. Lemma 4.4 above).
Next, the previous lemma allows us to relate the decomposability of the
monad with the decomposability of the associated quiver representation.
Proposition 5.3. A monad M• is decomposable if and only if the asso-
ciated quiver representation Gγ,σ(M•) is decomposable. In addition, if
Gγ,σ(M•) is decomposable, then the cohomology of M• is a decomposable
vector bundle.
Proof. We begin by showing that the functor Gγ,σ : MA,B,C → G
gis
m,n
preserves direct sums, that is, Gγ,σ(M•
1 ⊕M•
2 ) ≃ Gγ,σ(M•
1 ) ⊕ Gγ,σ(M•
2 ).
In particular, if M• is decomposable then R = Gγ,σ(M•) is decomposable.
Indeed, consider a monad M• = M•
1 ⊕M•
2 given by
Aa1+a2 α // Bb1+b2
β // Cc1+c2
M. Jardim, D. M. Prata 243
where α = α1 ⊕ α2 and β = β1 ⊕ β2 with αi ∈ Hom(Aai ,Bbi) and
βi ∈ Hom(Bbi , Cci), i = 1, 2. We write αi, βi as
αi =
m
∑
l=1
Ail ⊗ γl and βi =
n
∑
j=1
Bi
j ⊗ σj , i = 1, 2.
Then Gγ,σ(M•
1 ⊕M•
2 ) is the representation
κa1⊕a2
A1
1⊕A2
1//
...
A1
m⊕A2
m
//
κb1⊕b2 ...
B1
n⊕B2
n
//
B1
1⊕B2
n//
κc1⊕c2
and it is clear that
Gγ,σ(M•
1 ⊕M•
2 ) = ({κa1+a2 , κb1+b2 , κc1+c2}, {A1
i ⊕A2
i }
m
i=1, {B
1
j ⊕B2
j }nj=1)
≃ ({κa1 , κb1 , κc1}, {A1
i }
m
i=1, {B
1
j }nj=1)⊕({κa2 , κb2 , κc2}, {A2
i }
m
i=1, {B
2
j }nj=1)
= Gγ,σ((M•
1 ) ⊕ Gγ,σ((M•
2 ).
For the converse, suppose R = Gγ,σ(M•) ≃ R1 ⊕R2. By Lemma 5.2
we know that there are monads M•
i , for i = 1, 2, such that Ri = Gγ,σ(M•
i ).
It follows that
Gγ,σ(M•) ≃ Gγ,σ(M•
1 ) ⊕ Gγ,σ(M•
2 ) ≃ Gγ,σ(M•
1 ⊕M•
2 )
hence M• is decomposable.
The second claim follows easily from the observation that if a monad
is decomposable, then so is its cohomology sheaf.
The completion of the proof of Theorem 1.3 is at hand: if M• is a
monad of the form (25) with (a, b, c) satisfying qm,n(a, b, c) = a2 +b2 +c2 −
mab−nbc > 1, then the associated quiver representation is decomposable,
by Proposition 2.2. This means that M• itself, and hence its cohomology
sheaf, must also be decomposable, as desired.
5.2. Decomposability of bundles vs. decomposability of repre-
sentations
The last goal of this paper will be to examine under which assumption
one does have the converse of the second part of Proposition 5.3, that is, if
the cohomology of a monad is decomposable as a vector bundle, then the
quiver representation associated to the monad is also decomposable. The
244 Vector bundles and representations of quivers
difficulty here, of course, is to argue that if the cohomology of a monad
of the form (25) decomposes, then its summands are also cohomologies of
monads of the same form. Such statement can be proved under the follow-
ing additional assumptions, and using the cohomological characterisation
of monads provided by Theorem 2.11 above.
Let B = (F0, · · · ,Fn) be an n-block collection generating the
bounded derived category Db(X) of coherent sheaves on X, and let ∨B
its left dual n-block collection, as in the statement of Theorem 2.11. Let
E be a vector bundle on X given by the cohomology of type (6), and
assume that E is decomposable: E ≃ E1 ⊕ E2. From Theorem 2.11, since
E has natural cohomology with respect to ∨B we have
dim Extn−i−1(Hn−i
i0
, E) = a,
dim Extn−j(Hn−j
j0
, E) = b,
dim Extn−k+1(Hn−k
k0
, E) = c,
and extp(Hm
q , E) = 0 otherwise. Hence for l = 1, 2,
dim Extn−i−1(Hn−i
i0
, El) = al,
dim Extn−j(Hn−j
j0
, El) = bl,
dim Extn−k+1(Hn−k
k0
, El) = cl, l = 1, 2,
where a = a1 + a2, b = b1 + b2, and c = c1 + c2, with al, bl, cl > 0 and
Extq(Hm
p , El) = 0 otherwise.
Let us prove that M
F i
i0
,F
j
j0
,Fk
k0
is closed under direct summands. From
Lemma 2.10 and Theorem 2.11, El is isomorphic to a Beilinson monad
G•
l , l = 1, 2, where each Gul is given by
Gul =
⊕
p,q
Extn−q+u(Hn−q
p , El) ⊗ Fq
p , l = 1, 2.
Then we have
Gul = 0, l = 1, 2; u < −1, u > 1,
and
G−1
l =
⊕
p,q
Extn−q−1(Hn−q
p , El)⊗Fq
p = Extn−i−1(Hn−i
i0
, El)⊗F i
i0
≃ (F i
i0
)al
G0
l =
⊕
p,q
Extn−q(Hn−q
p , El) ⊗ Fq
p = Extn−j(Hn−j
j0
, El) ⊗ F j
j0
≃ (F j
j0
)bl
M. Jardim, D. M. Prata 245
G1
l =
⊕
p,q
Extn−q+1(Hn−q
p , El)⊗Fq
p = Extn−k+1(Hn−k
k0
, El)⊗Fk
k0
≃ (Fk
k0
)cl
for l = 1, 2. From the definition of Beilinson monad, Definition 2.9, El is
isomorphic to the monad
(F i
i0
)al // (F j
j0
)bl // (Fk
k0
)cl (29)
with l = 1, 2 and al, bl, cl > 0. We have the following cases:
1) If al, bl, cl 6= 0 for l = 1, 2, E1 and E2 are cohomology of a monad of
type (29)
E1 = H0(G•
1), E2 = H0(G•
2).
2) If a1, b1, c1, b2, c2 6= 0 and a2 = 0, then E1 = H0(G•
1) and E2 is given
by the short exact sequence
0 // E2
// (F j
j0
)b2 // (Fk
k0
)c2 // 0.
3) If a1, b1, c1, a2, b2 6= 0 and c2 = 0 then E1 = H0(G•
1) and E2 is given
by the short exact sequence
0 // (F i
i0
)a2 // (F j
j0
)b2 // E2
// 0 .
4) If a1, b1, c1, b2 6= 0 and a2 = c2 = 0, then E1 = H0(G•
1) and E2 ≃
(F j
j0
)b2 .
5) If b1, c1, a2, b2 6= 0 and a1 = c2 = 0 then
0 // E1
// (F j
j0
)b1 // (Fk
k0
)c1 // 0
and
0 // (F i
i0
)a2 // (F j
j0
)b2 // E2
// 0 .
And the symmetric cases to cases 2, 3, 4 and 5.
We have just proved that:
Lemma 5.4. The category M
F i
i0
,F
j
j0
,Fk
k0
is closed under direct
summands.
246 Vector bundles and representations of quivers
Suppose m = dim Hom(F i
i0
,F j
j0
), and n = dim Hom(F j
j0
,Fk
k0
) and
choose γ and σ bases of Hom(F i
i0
,F j
j0
) and Hom(F j
j0
,Fk
k0
), respectively.
Let Gγ,σ be the functor between M
F i
i0
,F
j
j0
,Fk
k0
and S
gis
m,n described after
equation (27). Note that given a monad of type (29) with al = 0 the
associated representation is
0
0 //
...
0
//
κbl ...
Bl
n
//
Bl
1 //
κcl
which is (γ,σ)-globally injective and surjective. If cl = 0, the associated
representation is
κal
Al
1 //
...
Al
m
//
κbl ...
0
//
0 //
0
that is (γ,σ)-globally injective and surjective. If al = cl = 0, the associated
representation is
0
0 //
...
0
//
κbl ...
0
//
0 //
0
which is also (γ,σ)-globally injective and surjective. Hence we can prove
the following.
Theorem 5.5. Let E be a vector bundle on X given by the cohomology
of a monad in M
F i
i0
,F
j
j0
,Fk
k0
and R the associated (γ,σ)-globally injective
and surjective representation in S
gis
m,n. Then E is decomposable if and only
if R is decomposable.
Proof. We only need to prove the sufficient condition. If E ≃ E1 ⊕ E2 then
from Lemma 5.4, Ei, i = 1, 2, are cohomologies of monads in M
F i
i0
,F
j
j0
,Fk
k0
,
therefore R = Gγ,σ(E) ≃ Gγ,σ(E1⊕E2) ≃ Gγ,σ(E1)⊕Gγ,σ(E2) = R1⊕R2.
5.3. An example: generalized Horrocks–Mumford monads
As an application of Theorem 5.5, let X = P2p with p > 2, κ = C,
and consider the 2p-block collection
B = (Ω2p
P2p(2p),Ω2p−1
P2p (2p− 1), · · · ,Ω1
P2p(1),OP2p)
M. Jardim, D. M. Prata 247
generating the bounded derived category Db(P2p). The complex
OP2p(−1)2p+1 α // Ωp
P2p(p)2 β // O2p+1
P2p (30)
is a monad, for α = (αij) ∈ ∧pC2p+1 ⊗ Mat2×2p+1(C) and β = (βij) ∈
∧pC2p+1 ⊗ Mat2p+1×2(C) given by
βi1 = x1+i ∧ x2+i ∧ · · · ∧ xp+i;
βi2 = xi ∧ xp+1+i ∧ xp+2+i ∧ · · · ∧ x2p−1+i
where i ≡ k(mod 2p + 1) and the matrix α is given by
α = (βQ)t
with
Q =
(
0 1
(−1)p−1 0
)
.
Note that when p = 2, the monad (30) is precisely the one that yields,
as its cohomology, the Horrocks–Mumford rank 2 bundle on P4. For this
reason, monads of the form (30) are called generalized Horrocks–Mumford
monads. The goal of this section is to prove, as an application of Theorem
5.5, that the cohomology of a monad of type (30) is an indecomposable
vector bundle of rank 2
(
(2p
p
)
− 2p− 1
)
on P2p.
To this end, note that one can fix a basis of the vector space ∧pC2p+1
so that the quiver representation associated to the morphism
β ∈ Hom(Ωp
P2p(p)2,O2p+1
P2p )
is a representation of the Kronecker quiver K(2p+1
p ) of the form
R = ({C2,C2p+1}, {φl}
(2p+1
p )
l=1 ) where 4p + 2 elements φl are elementary
matrices of size (2p+ 1) × 2 for some l and null matrices otherwise. The
crucial step is the following result.
Lemma 5.6. The representation R is simple.
In particular, it follows from Theorem 3.5 that the kernel bundle kerβ,
whose dual is a Steiner bundle, is also simple.
Proof. Suppose without loss of generality that the ordered basis is the
following
{x01···p−1, x12···p, x23···p+1, · · · , x2p0···p−2, · · · }
248 Vector bundles and representations of quivers
where xi1i2···ip = xi1 ∧ xi2 ∧ · · · ∧ xip . Then β can be written as
β = x01···p−1 · E2p,1 + x12···p · E2p+1,1 + x23···p+1 · E1,1 + · · ·
where Ei,j ∈ Mat(2p+1)×2(C) is an elementary matrix. The associated
quiver representation is of the form R = ({C2,C2p+1}, {φl}
(2p+2
2 )
l=1 ) where
φ1 = E2p,1, φ2 = E2p+1,1, ϕ3 = E1,1 and so on.
LetR1 = ({V1, V2}, {ψ}
(2p+1
p )
l=1 ) be a subrepresentation ofR and without
loss of generality suppose V1 6= 0. Then there is v = (a, b) ∈ V1 ⊂ C2,
with a 6= 0 . The following diagram commutes
C2
φ1 //
...
φ(2p+1
p )
//
C2p+1
V1
i1
OO
ψ1 //
...
ψ(2p+1
p )
//
V2
i2
OO (31)
and note that the vectors {φj(v)}2p+1
j=1 are linearly independent, hence
V2 ≃ C2p+1. If R2 = ({W1,W2}, {γl}
(2p+1
p )
l=1 ) is a subrepresentation of R
such that R ≃ R1 ⊕R2, then W2 ≡ 0 and if W1 6= 0 there is k such that
φk 6= 0 and φk |W1 6= 0. Therefore R is simple, hence indecomposable.
Since α = (βQ)t, the representation of the Kronecker quiver K(2p+1
p )
associated to α is of the form R
′
= ({C2p+1,C2}, {φ′
l}
(2p+1
p )
l=1 ), where φ′
l are
elementary matrices or null matrices (the transpose of φl up to sign). Hence
R′ is also simple. By Theorem 5.5, the cohomology of the monad (30) is
an indecomposable vector bundle on P2p.
Acknowledgements
We thank Helena Soares for her help with the results in Section 5,
and Rosa Maria Miró-Roig for describing to us the monads considered
in Section 5.3. We also thank the referee for his help in improving the
presentation of the paper.
M. Jardim, D. M. Prata 249
References
[1] M. C. Brambilla, Cokernel bundles and Fibonacci bundles. Math. Nach. 281 (2008),
499–516.
[2] H. Brenner, Looking out for stable syzygy bundles. Adv. Math. 219 (2008), 401–427.
[3] I. Dolgachev, M. Kapranov, Arrangements of hyperplanes and vector bundles on
Pn. Duke Math. J. 71 (1993), 633–664.
[4] M. Jardim, V. M. F. Silva. Decomposability criterion for linear sheaves. Cent. Eur.
J. Math. 10 (2012), 1292–1299.
[5] V. G. Kac. Infinite root systems, representations of graphs and invariant theory.
Invent. Math. 56 (1980), 57–92.
[6] P. M. Marques, H. Soares. Cohomological caracterisation of monads. Math. Nachr.
287 (2014), 2057–2070.
[7] R. M. Miró-Roig, H. Soares. Cohomological characterisation of Steiner bundles.
Forum Math. 21 (2009), 871–891.
[8] A. Schofield. General representations of quivers. Proc. London Math. Soc. 65
(1992), 46–64.
[9] A. Schofield. Birational classification of moduli spaces of representations of quivers.
Indag. Mathem. 12 (2001), 407–432.
[10] H. Soares. Steiner vector bundles on algebraic varieties. Ph.D. Thesis, University
of Barcelona, 2008.
Contact information
M. Jardim,
D. M. Prata
IMECC–UNICAMP, Departamento de Matemática,
Rua Sérgio Buarque de Holanda, 651, Campinas,
SP, Brazil, CEP 13083-859
E-Mail(s): jardim@ime.unicamp.br
Web-page(s): www.ime.unicamp.br/j̃ardim/
Received by the editors: 01.03.2015
and in final form 13.09.2015.
|