Vector bundles over noncommutative nodal curves
We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative curve is a wild problem. Описано векторнi розшарування н...
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Drozd, Y.A. Voloshyn, D.E. 2020-02-08T16:51:26Z 2020-02-08T16:51:26Z 2012 Vector bundles over noncommutative nodal curves / Д.Є. Волошин, Ю.А. Дрозд // Український математичний журнал. — 2012. — Т. 64, № 2. — С. 185-199. — Бібліогр.: 14 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164136 512.723 We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative curve is a wild problem. Описано векторнi розшарування над деяким класом некомутативних кривих, а саме, над нодальними некомутативними кривими струнного та майже струнного типу. Встановлено також, що в iнших випадках класифiкацiя векторних розшарувань над некомутативною кривою є дикою задачею. en Інститут математики НАН України Український математичний журнал Статті Vector bundles over noncommutative nodal curves Векторнi розшарування над некомутативними нодальними кривими Article published earlier |
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Vector bundles over noncommutative nodal curves |
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Vector bundles over noncommutative nodal curves Drozd, Y.A. Voloshyn, D.E. Статті |
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Vector bundles over noncommutative nodal curves |
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Vector bundles over noncommutative nodal curves |
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Vector bundles over noncommutative nodal curves |
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Vector bundles over noncommutative nodal curves |
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vector bundles over noncommutative nodal curves |
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Drozd, Y.A. Voloshyn, D.E. |
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Drozd, Y.A. Voloshyn, D.E. |
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Векторнi розшарування над некомутативними нодальними кривими |
| description |
We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative curve is a wild problem.
Описано векторнi розшарування над деяким класом некомутативних кривих, а саме, над нодальними некомутативними кривими струнного та майже струнного типу. Встановлено також, що в iнших випадках класифiкацiя векторних розшарувань над некомутативною кривою є дикою задачею.
|
| issn |
1027-3190 |
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https://nasplib.isofts.kiev.ua/handle/123456789/164136 |
| citation_txt |
Vector bundles over noncommutative nodal curves / Д.Є. Волошин, Ю.А. Дрозд // Український математичний журнал. — 2012. — Т. 64, № 2. — С. 185-199. — Бібліогр.: 14 назв. — англ. |
| work_keys_str_mv |
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| first_indexed |
2025-11-25T20:31:33Z |
| last_indexed |
2025-11-25T20:31:33Z |
| _version_ |
1850524364551749632 |
| fulltext |
UDC 512.723
Y. A. Drozd, D. E. Voloshyn (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES
ВЕКТОРНI РОЗШАРУВАННЯ НАД НЕКОМУТАТИВНИМИ
НОДАЛЬНИМИ КРИВИМИ
We describe vector bundles over a class of noncommutative curves, namely, over noncommutative nodal curves of string
type and of almost string type. We also prove that, in other cases, the classification of vector bundles over a noncommutative
curve is a wild problem.
Описано векторнi розшарування над деяким класом некомутативних кривих, а саме, над нодальними некомута-
тивними кривими струнного та майже струнного типу. Встановлено також, що в iнших випадках класифiкацiя
векторних розшарувань над некомутативною кривою є дикою задачею.
Introduction. Classification of vector bundles over algebraic curves is a popular topic in mod-
ern mathematical literature. It is due to their importance for many branches of mathematics and
mathematical physics. Vector bundles over the projective line were described by Birkhoff [2] and
Grothendieck [11], vector bundles over elliptic curves were classified by Atiyah [1]. In the paper [9]
Greuel and the first author described vector bundles over a class of singular curves (line configura-
tions of types A and Ã) and showed that in all other cases a complete classification of vector bundles
is a “wild problem” in the sense of representation theory of algebras.
This paper is devoted to analogous questions for noncommutative curves. Perhaps, the first results
in this direction were obtained by Geigle and Lenzing [10] who considered the so called weighted
projective lines. Though the original definition of this paper was in the frames of “usual” (com-
mutative) algebraic geometry, these curves are actually of noncommutative nature. They can be
considered as such noncommutative curves that the underlying algebraic curve is a projective line
and all localizations of the structure sheaf are hereditary. In some sense, it is the simplest example of
noncommutative curves, though their theory is far from being simple.
We consider the “next step,” namely the case when the localizations of the structure sheaf are
nodal in the sense of [5]. In particular, this class contains all line configurations in the sense of [9].
We reduce the description of vector bundles over such curves to the study of a bimodule category
in the sense of [8, 9]. Using this reduction, we describe vector bundles in two cases: string type and
almost string type, see Sections 3 and 4. Note that the string type is an immediate generalization of
line configurations of types A and Ã. The main tool in this description is a special sort of bimodule
problems, namely, the so called bunches of chains. Fortunately, these problems are well elaborated
and a good description of representations is given in [4]. We also show that in all other cases the
classification of vector bundles is a wild problem (Section 5). Thus, in some sense, the question
about the “representation type” of the category of vector bundles over noncommutative curves is
completely solved.
1. Noncommutative curves, vector bundles and categories of triples. We call a noncommu-
tative variety a pair (X,A), where X is an algebraic variety over an algebraically closed field k
(reduced, but maybe reducible) and A is a sheaf of OX -algebras which is coherent as a sheaf of
OX -modules. We often speak about a “noncommutative variety A” not mentioning explicitly the
c© Y. A. DROZD, D. E. VOLOSHYN, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2 185
186 Y. A. DROZD, D. E. VOLOSHYN
underlying variety X. We denote by KX (or K) the sheaf of total rings of fractions of OX (it is
locally constant) and set K(A) = A
⊗
OX
KX . Without loss of generality we may (and usually will)
suppose that A is central, i.e., OX,x = center(Ax) for each x ∈ X. Otherwise we can replace X by
the variety X ′ = spec C, where C = center(A). We define a noncommutative curve as a noncom-
mutative variety (X,A) such that X is a curve (that is all its components are 1-dimensional) and A
is reduced, that is has no nilpotent ideals. A coherent sheaf of A-modules F is said to be a vector
bundle over (X,A) if it is locally projective, i.e. the Ax-module Fx is projective for every x ∈ X.
We denote by VB(X,A) or by VB(A) the category of vector bundles over (X,A).
We call a noncommutative curve (X,A) normal if, for every point x ∈ X, the algebra Ax is
a maximal OX,x-order, that is there is no OX,x-subalgebra Ax ⊂ A′ ⊂ Kx which is also finitely
generated as OX,x-module. Since A is reduced, there is a normal curve X̃ = (X, Ã) such that
A ⊆ Ã ⊂ KX . Moreover, Ax = Ãx for almost all x ∈ X (it follows from [7]). We call (X, Ã) a
normalization of X and denote by sgA the set of all points x ∈ X such that Ax 6= Ãx. Note that such
a normalization is, as a rule, not unique, though sgA does not depend on the choice of normalization.
Let C̃ = center(Ã), X̃ = spec C̃. We can (and will) consider à as a sheaf of central OX̃ -algebras,
hence consider the normalization as the noncommutative curve (X̃, Ã). The natural morphism of
ringed spaces π : (X̃, Ã) → (X,A) is defined. We also denote by s̃gA the set-theoretical preimage
π−1(sgA). If X̃1, X̃2, . . . , X̃s are the irreducible components of X̃, we set Ãi = Ã|X̃i
, so consider
the noncommutative curves (X̃i, Ãi). We also set s̃giA = s̃gA ∩ X̃i. Let Xi = π(X̃i). Certainly,
each Xi is an irreducible component of X, but these components need not be different. We set
Ki(A) = K(A)|X̃i
. It is a constant sheaf of central KXi-gebras. Since k is algebraically closed, the
Brauer group of the field Ki = KXi is trivial [13] (Chapter II, § 3), so Ki(A) ' Mat(ni,Ki) for
some ni. We call a noncommutative curve (X,A) rational if so is the curve X, i.e., all components
of X̃ are isomorphic to the projective line P1.
For calculation of vector bundles over noncommutative curves one can use the “sandwich pro-
cedure,” just as it has been done in [9] in the commutative case. Let π : (X̃, Ã) → (X,A) be a
normalization of a noncommutative curve (X,A). We denote by J the conductor of à in A, that is
the maximal sheaf of Ã-ideals contained in A. We consider the noncommutative varieties (sgA,S)
and (s̃gA, S̃), where S = A/J and S̃ = Ã/J . These varieties are 0-dimensional and usually not
reduced. We denote by π̄ : (s̃gA, S̃) → (sgA,S) the restriction of π onto (s̃gA, S̃) and by ι and
ι̃, respectively, the closed embeddings (sgA,S) → (X,A) and (s̃gA, S̃) → (X̃, Ã). So we have a
commutative diagram of morphisms of noncommutative varieties
(s̃gA, S̃)
ι̃
//
π̄
��
(X̃, Ã)
π
��
(sgA,S)
ι
// (X̃, Ã).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 187
Since (sgA,S) and (s̃gA, S̃) are 0-dimensional, coherent sheaves on them can be identified with
finitely generated modules over the algebras, respectively,
S =
∏
x∈sgA
Ax/Jx and S̃ =
∏
y∈s̃gA
Ãy/Jy.
Following [5, 6], we introduce the category of triples T (A) as follows.
The ōbjects of T (A) are triples (G, P, θ), where
G is a vector bundle over Ã,
P is a vector bundle over S, or, the same, a finitely generated projective S-module,
θ is an isomorphism π̄∗P → ι̃∗G, or, the same, an isomorphism of S̃-modules S̃
⊗
S P →
→
∏
y∈s̃gA
Gy/JyGy.
A morphism (G, P, θ) → (G′, P ′, θ′) is a pair (Φ, φ), where Φ ∈ HomÃ(G,G′) and φ ∈
HomS(P, P ′) such that the induced diagram
π̄∗P
π̄∗φ
//
θ
��
π̄∗P ′
θ′��
ι̃∗G
ι̃∗Φ
// ι̃∗G′
is commutative.
One easily sees that T (A) is indeed a full subcategory of a bimodule category in the sense
of [8], namely, the category defined by the VB(S)-VB(Ã)-bimodule HomS̃(π̄∗P, ι̃∗G). It can also
be considered as the push-out of the categories VB(Ã) and VB(S) over the category VB(S̃) with
respect to the functors ι̃∗ and π̄∗. So it is an analogue of Milnor’s construction of projective modules
from [12] (§ 2).
We define the functor F : VB(A) → T (A), which maps a vector bundle F to the triple
(π∗F , ι∗F , θF ), where θF is the natural isomorphism π̄∗ι∗F → ι̃∗π∗F . The same considerations
as in [6, 9] give the following result.
Theorem 1.1. The functor F induces an equivalence of the categories VB(A)
∼→ T (A). The
inverse functor G : T (A)→ VB(A) maps a triple (G, P, θ) to the preimage in G of the S-submodule
θ(1
⊗
P ) ⊆ ι̃∗G.
2. Nodal curves.
Definition 2.1. (1) An algebra R over a local commutative ring O of Krull dimension 1,
which is finitely generated and torsion free as O-module, is said to be nodal [5, 14] if the following
conditions hold:
(a) EndR(radR) = H is hereditary,
(b) radH = radR (under the natural embedding of R into H),
(c) lengthR(H
⊗
R U) ≤ 2 for every simple R-module U . Note that a nodal algebra never has
nilpotent ideals, since it holds for any hereditary O-algebra.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
188 Y. A. DROZD, D. E. VOLOSHYN
(2) A noncommutative curve (X,A) is said to be nodal if every algebra Ax (x ∈ X) is a nodal
OX,x-algebra. If A = OX , so we deal with a “usual” (commutative) curve, it means that all singular
points of X are nodes (ordinary double points).
We recall the construction of nodal algebras over the ring O = k[[t]] from [14]. Up to Morita
equivalence such algebra is given by a tuple N = (s;n1, n2, . . . , ns;∼), where s and n1, n2, . . . , ns
are positive integers, while ∼ is a symmetric relation on the set of pairs I = {(k, i) | 1 ≤ k ≤ s, 1 ≤
≤ i ≤ nk} satisfying the following conditions:
(N1) # { (l, j) ∈ I | (l, j) ∼ (k, i) } ≤ 1 for each pair (k, i) ∈ I.
(N2) If (k, i) ∼ (k, i), then i < nk and (k, i+ 1) 6∼ (l, j) for any (l, j) ∈ I.
Namely, define R(N) as the subring of M(N) =
∏s
k=1
Mat(nk, O) consisting of such collec-
tions of matrices (A1, A2, . . . , As) , where Ak = (akij) ∈ Mat(nk, O), that
akij ≡ 0 (mod t) if i > j or i = j − 1 and (k, i) ∼ (k, i), (2.1)
akii ≡ aljj (mod t) if (k, i) ∼ (l, j). (2.2)
Theorem 2.1 [14]. (1) Every ring R(N) is a nodal O-algebra.
(2) Every nodal O-algebra is Morita equivalent to one of the rings R(N).
(3) radR(N) consists of such collections (A1, A2, . . . , Ak) that the condition (2.1) holds and
also akii ≡ 0 (mod t) for all k, i.
(4) The hereditary algebraH(N) = EndR(N)(radR(N)) consists of such collections (A1, A2, . . .
. . . , Ak) that
akij ≡ 0 (mod t) if i > j, except the case when
i = j − 1 and (k, i) ∼ (k, i).
(5) M(N) is a maximal order containing R(N) such that J(N) = radM(N) is the conductor
of M(N) both in R(N) and in H(N), and J(N) ⊆ radR(N).
(6) R(N)/J(N) is the subring of M(N)/J(N) =
∏s
k=1
Mat(nk,k) consisting of such collec-
tions of matrices (A1, A2, . . . , As) that
akij = 0 if i > j or i = j − 1 and (k, i) ∼ (k, i),
akii = aljj if (k, i) ∼ (l, j).
In particular, R(N) is hereditary if and only if the relation ∼ is empty. (Then we write R =
= R(s;n1, n2, . . . , ns).)
Actually, to define a ring Morita equivalent to R(N), one only has to prescribe positive integers
m(k, i) for each pair (k, i) ∈ I so that m(k, i) = m(l, j) if (k, i) ∼ (l, j), and consider akij in the
definition of R(N) not as elements of k, but as matrices from Mat(m(k, i)×m(k, j),k), preserving
all congruences modulo t. We denote such data by (N,m), where m is the function (k, i) 7→ m(k, i),
and the corresponding algebra by R(N,m). Note that different data N or (N,m) can describe
isomorphic algebras, even if they do not only differ by a permutation of indices (k, i). We extend
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 189
the relation ∼ to an equivalence relation ≈ setting (k, i) ≈ (l, j) if and only if (k, i) = (l, j) or
(k, i) ∼ (l, j).
From the well-known properties of torsion free modules over reduced rings of Krull dimension 1
(see, for instance, [7]) it follows that, given a torsion free coherent sheaf F over a noncommutative
curve (X,A), a finite set of closed points x1, x2, . . . , xm ∈ X and a set of coherent Axi-submodules
Gi ⊂ Fxi
⊗
OX
K, there is a unique coherent sheaf G ⊂ F
⊗
OX
K such that Gxi = Gi and Gy = Fy
if y 6= xi for all i. In particular, since almost all localizations Ax are maximal, one can construct a
normalization à of A locally, choosing arbitrary normalizations Ãx of Ax for x ∈ sgA. Therefore,
given a nodal noncommutative curve (X,A), we can (and will) suppose that the normalizations of its
local components are chosen as in Theorem 2.1. Thus, if x ∈ sgX, y ∈ π−1(x) = { y1, y2, . . . , yr } ,
we identify Ãy with a full matrix ring Mat(ny,OX̃,y) and suppose that the ring Ax is given by some
data (N,m) as above. In what follows, we write (yk, i) instead of (k, i), so the local embeddings
Ax → Ãx =
∏r
k=1
Ãyk for x ∈ sgA are described by the data N(A) consisting of integers ny and
m(y, i) for y ∈ s̃gA, 1 ≤ i ≤ ny, and an equivalence relation ∼ on the set of pairs (y, i) satisfying
the above conditions (N1) and (N2) and such that
(N3) the sum my =
∑ny
i=1
m(y, i) is the same for all points y belonging to the same component
of X̃ .
The last condition just expresses the fact that the sheaf K(Ã) is locally constant. One easily sees
that π(y) = π(y′) if and only if there is at least one relation (y, i) ∼ (y′, j). Moreover, if we suppose
that X is connected and A is central, the set π−1(x) for each x ∈ sgX must be connected as the
graph defined by the symmetric relation y ∼ y′ which means that there is at least one pair i, j such
that (y, i) ∼ (y′, j).
From now on we fix a connected central noncommutative nodal curve (X,A) and its normal-
ization π : (X̃, Ã) → (X,A) chosen as described above. We write O instead of OX and Õ in-
stead of OX̃ . If X̃1, X̃2, . . . , X̃s are the irreducible components of X̃, Xk = π(X̃k), we write
Õk = ÕX̃k
, Ãk = Ã|X̃k
, Ok = OXk
and Ak = A|Xk
. Recall that the sheaves of rings Õk and
Ãk are Morita equivalent. Namely, there is a vector bundle Lk over Ãk such that EndÃk
Lk ' Õk,
EndÕk
Lk ' Ãk, so the functors HomÃk
(Lk, ) and L
⊗
Õk−
establish an equivalence between
Coh(Ãk) and Coh(Õk). We call Lk a basic vector bundle over Ãk. (Note that it is not uniquely
defined.)
Let J be the conductor of à in A. If x ∈ sgA, then Jx =
⊕
π(y)=x rad Ãy, S̃y = Ãy/Jy '
' Mat(my,k) and Ly/JyLy ' myUy, where Uy is the unique simple S̃y-module. For any vector
bundle G over Ãi we define its rank: rkG = r if Gy/JyGy ' rUy for some (then for any) y ∈ X̃i.
Every pair (y, i), where π(y) = x, 1 ≤ i ≤ ny, defines a simple Sx-module Vy,i, where
Sx = Ax/Jx, and Vy,i ' Vy′,j if and only if (y, i) ≈ (y′, j). Moreover, Uy '
⊕ny
i=1 Vy,i as Sy-
module. We denote by Py,i the projective Sy-module such that Py,i/ radPy,i ' Vy,i. In particular,
Py,i ' Py′,j if and only if (y, i) ≈ (y′, j).
To describe the category of triples T (A) it is convenient to introduce new symbols eyij , where
1 ≤ i ≤ j ≤ ny, and the sets Ey,iy′,j consisting of all ezi′j′ such that one of the following conditions
hold:
z = y, (y, j′) ∼ (y′, j) and either i = i′ or (y, i) ∼ (y, i′);
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 2
190 Y. A. DROZD, D. E. VOLOSHYN
z = y′, (y′, i′) ∼ (y, i) and either j = j′ or (y′, j) ∼ (y′, j′).
We also set eyi =
∑
(z,j)≈(y,i)
ezjj and consider the copies Ueyii of the simple modules Uy. Then
S̃
⊗
S
Py,i '
⊕
(z,j)≈(y,i)
U zj e
z
jj ,
EndS Py,i '
keyi if (y, i) 6∼ (y, j) for any j 6= i,
keyi ⊕ ke
y
ij if (y, i) ∼ (y, j) and i < j,
keyi ⊕ ke
y
ji if (y, i) ∼ (y, j) and j < i
and, for (y, i) 6∼ (y′, j),
HomS(Py,i, Py′,j) '
⊕
Ey,i
y′,j
kezi′j′ .
Under such notations the maps S̃
⊗
S Py,i → S̃
⊗
S Py′,j induced by the homomorphisms Py,i →
→ Py′,j as well as the multiplication of homomorphisms are given by the “matrix multiplication” on
the right, i.e., by the rules:
eyii′e
y′
j′j =
{
0 if y 6= y′ or i′ 6= j′,
eyij if y = y′ and i′ = j′.
Let (G, P, θ) be a triple from T (A). Decompose G and P :
G =
⊕
k,l gklGkl, where Gkl are nonisomorphic indecomposable vector bundles over Ãk,
P =
⊕
y,i py,iPy,i.
Set rkl = rkGkl. Then the isomorphism θ : π̄∗P → ι̃∗G is given by a set Θ = {Θy | y ∈ s̃gA} of
invertible block matrices Θy = (Θy,i
kl ), where y ∈ s̃gkA, the block Θy,i
kl has coefficients from k and is
of size rklgkl×py,i. If another triple (G′, P ′, θ′) is given by the matrices Θ′y, a morphism (G, P, θ)→
→ (G′, P ′, θ′) is given by block matrices Φk = (Φkl
kl′) and φy = (φy,iy,j) such that Φk(y)Θy = Θ′yφy
for every y ∈ s̃gkA, where the elements of Φkl
kl′ are from HomÃk
(Gkl,Gkl′), elements of φy,iy,j are
from k, φy,iy,i = φy
′,j
y′,j if (y, i) ∼ (y′, j) and φy,iy,j = 0 if i > j or i = j − 1, (y, i) ∼ (y, i). This
morphism is an isomorphism if and only if all “diagonal” blocks Φkl
kl and φy,iy,i are invertible.
Let N (A) be the ideal in T (A) consisting of all morphisms (Φ, φ) such that all values Φk(y),
where y ∈ s̃gkA, are zero. In the matrix presentation it means that Φkl
kl′(y) = 0 for all possible
triples (k, l, l′) and all y ∈ X̃k. Denote T (A) = T (A)/N (A). These categories have the same
objects and the natural functor T (A) → T (A) is full (not faithful), maps nonisomorphic objects to
nonisomorphic and indecomposable objects to indecomposable. Therefore, to obtain a classification
of vector bundles, we actually have to study the category T (A). Nevertheless, passing from T to
T we can lose some information. It is important, for instance, if we are looking for stable vector
bundles (see, for instance, [3]).
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VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 191
3. String case.
Definition 3.1. A noncommutative nodal curve (X,A) is said to be of string type if it is
rational and every set s̃gkA contains at most 2 points.
If (X,A) is of string type, we identify all components X̃k with P1 and fix an affine part A1 ⊂ X̃k
containing s̃gkA .
In this case the category of triples T (A) can be treated as the category of representations of a
certain bunch of chains B(A) in the sense of [5] (Appendix B) 1. Namely, if Lk is a basic vector
bundle over Ãk, then every indecomposable vector bundle over Ãk is isomorphic to Lk(d) for some
d, which is called the degree of Lk(d) 2. Moreover,
HomÃ(Lk(d),Lk(d′)) '
{
0 if d > d′,
k[t]d′−d if d ≤ d′,
where k[t]m denotes the set of polynomials f(t) such that deg f(t) ≤ m. Therefore, in the decom-
position of a vector bundle Gk over Ãk we can suppose that Gkl = Lk(l). Then the elements of the
matrices Φkl
kl′ can be considered as the polynomials of degree l′− l if l′ ≥ l; they are zero if l′ < l. If
y 6= y′ are two points from s̃gkA and l′ > l, we can always choose a polynomial f(t) ∈ k[t]l′−l such
that f(y) = a, f(y′) = b for any prescribed values a, b ∈ k. It means that the values of the matrices
Φkl
kl′ at the points y and y′ can be prescribed arbitrary. Therefore, the rule Φk(y)Θy = Θ′yφy from the
matrix description of morphisms in T (A) can be rewritten as F (y)Θy = Θ′yφy, where F (y) is an
arbitrary lower block triangular matrix F (y) = (F (y)klkl′) (F (y)klkl′ = 0 if l < l′) over the field k and
the only restrictions for these blocks is that F (y)klkl = F (y′)klkl if y and y′ are in the same component
X̃k.
Thus we define the bunch of chains B(A) as follows. We consider s̃gA as the index set of this
bunch and for every y ∈ s̃gA set
Ey = { (y, i) | 1 ≤ i ≤ ny } \ { (y, i) | (y, i− 1) ∼ (y, i− 1) } ,
Fy = { (d, y) | d ∈ Z } ,
(y, i) < (y, j) if i < j,
(d, y) < (d′, y) if d < d′,
(y, i) ∼ (y′, j) if and only if they are so in the nodal data N(A),
(d, y) ∼ (d′, z) if and only if d = d′, y 6= z but y and z belong
to the same component X̃k.
Recall [4, 5] that a representation M of this bunch of chains is given by a set of block matrices
My = (Myi
dy), where y ∈ s̃gA, 1 ≤ i ≤ ny, M
yi
dy ∈ Mat(mdy × nyi,k) for some integers mdy, nyi
1 Or a bundle of semi-chains in the terms of [4].
2 Note that it is not the degree of Lk(d) as of Õk-sheaf; the latter equals dnk.
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192 Y. A. DROZD, D. E. VOLOSHYN
such that mdy = mdy′ if (d, y) ∼ (d, y′) and nyi = ny′j if (y, i) ∼ (y′, j). Here we identify the
symbols (y, i)′ and (y, i)′′ from [5] (Definition B.1), where (y, i) ∼ (y, i), with the pairs (y, i) and
(y, i + 1). A morphism α : M → M ′ given by a set of block matrices α′y, α
′′
y , where y ∈ s̃gA,
α′y = (αdyd′y), α
′′
y = (αyiyi′), such that
αdyd′y ∈ Mat(md′y ×mdy,k),
αyiyi′ ∈ Mat(myi′ ×myi,k),
αdyd′y = 0 if d > d′,
αyiyi′ = 0 if i > i′ or i′ = i+ 1 and (y, i) ∼ (y, i),
αdydy = αdy
′
dy′ if (d, y) ∼ (d, y′),
αyiyi = αy
′j
y′j if (y, i) ∼ (y′, j),
and
α′yMy = M ′yα
′′
y for all y ∈ s̃gA.
The matrix presentations described above imply the following fact.
Proposition 3.1. Let the noncommutative nodal curve (X,A) is of string type, B = B(A).
Then the category T (A) is equivalent to the full subcategory rep0(B) of the category of represen-
tations of the bunch of chains B consisting of such representations M that all matrices My are
invertible.
In particular, the category T (A) and hence the category VB(A) are tame in the sense that they
have at most 1-parameter families of indecomposable objects. Moreover, from the description of
representations of a bunch of chains given in [4] one can deduce a description of vector bundles
over a noncommutative nodal curve of string type. For the corresponding combinatorics we use the
terminology from [5] adopted to our situation.
Definition 3.2. (1) Let E =
⋃
y Ey, F =
⋃
y Fy, X = E ∪ F. We define the symmetric relation
− on X setting (d, y) − (y, i) for all possible d, i, y. We also write ξ ‖ ξ′ if either both ξ and ξ′
belong to E or both of them belong to F, and ξ ⊥ ξ′ if one of them belongs to E while the other
belongs to F.
(2) We define a word (more precisely, an X-word) as a sequence ξ1r1ξ2r2 . . . ξl−1rl−1ξl such that
(a) ξi ∈ X, ri ∈ {∼,−} ;
(b) ξiriξi+1 for each 1 ≤ i < l accordingly to the definition of the relations ∼ and −;
(c) ri 6= ri+1 for all 1 ≤ i < l− 1. We call l = l(w) the length of the word w and ξ1, ξl the ends
of this word.
(3) We call the word w full if the following conditions hold:
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VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 193
(a) either r1 =∼ or ξ1 6∼ ξ′ for any ξ′ 6= ξ1;
(b) either rl−1 =∼ or ξl 6∼ ξ′ for any ξ′ 6= ξl.
(4) We call the word w terminating if it is full and r1 = rl−1 = −.
(5) The end ξ1 (ξl) is said to be special if r1 = − and ξ1 ∼ ξ1 (respectively, ξl ∼ ξl and
rl−1 = −). Otherwise it is said to be usual.
(6) The terminating word w is said to be
usual if both its ends are usual;
special if one of its ends, but not both, is special;
bispecial if both its ends are special.
(7) The word w∗ = ξlrl−1 . . . ξ2r1ξ1 is called inverse to the word w.
(8) We call w symmetric if w = w∗ and quasisymmetric if it can be presented as v ∼ v∗ ∼ v ∼
. . . ∼ v∗ ∼ v for a shorter word v. Note that a quasisymmetric word is always bispecial.
(9) The word w is said to be cyclic if r1 = rl−1 =∼ and ξl − ξ1 in B. Then we set r0 = − and
ξi+kl = ξi, ri+kl = ri for any k ∈ Z.
(10) A shift of the cyclic word w is the cyclic word
w[k] = ξk+1rk+1ξk+2 . . . r0ξ1r1 . . . ξk,
where k is even. In this case we set ε(w, k) = (−1)k/2.
(11) The cyclic word w is said to be aperiodic if w[k] 6= w for 0 < k < l. It is said to be
cyclic-symmetric if w∗ = w[k] for some k.
Note that the length of a terminating or cyclic word is always divisible by 4.
Definition 3.3. (1) A usual string is a usual nonsymmetric terminating word.
(2) A special string is a pair (w, δ), where w is a special terminating word and δ ∈ { 0, 1 } .
(3) A bispecial string is a quadruple (w,m, δ0, δ1), where w is a bispecial terminating word that
is neither symmetric nor quasisymetric, m ∈ N and δi ∈ { 0, 1 } (i = 0, 1).
(4) A band is a triple (w,m, λ), where w is a cyclic word, m ∈ N, λ ∈ k
× and, if w is
cyclic-symmetric, also λ 6= 1.
(5) The following strings are said to be equivalent:
w and w∗;
(w, δ) and (w∗, δ);
(w,m, δ0, δ1) and (w∗,m, δ1, δ0).
(6) Two bands are said to be equivalent if they can be obtained from one another by a sequence
of the following transformations:
replacing (w,m, λ) by (w[k],m, λε(w,k));
replacing (w,m, λ) by (w∗,m, λ−1).
Note that if w∗ = w[k], then k ≡ 2 (mod 4), so ε(w, k) = −1.
Now the results of [4] imply the following theorem.
Theorem 3.1. The isomorphism classes of indecomposable vector bundles over a noncommu-
tative nodal curve curve of string type (X,A) are in one-to-one correspondence with the equivalence
classes of strings and bands for the bunch of chains B(A). The rank of the vector bundle corre-
sponding to a string or a band equals l/4, where l is the length of the word w entering into this
string or band.
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194 Y. A. DROZD, D. E. VOLOSHYN
We refer to [4] for an explicit construction of representations corresponding to strings and bands,
hence of vector bundles over noncommutative nodal curves of string type.
Note that it can so happen that there are no strings or no bands. For instance, if all localizations
Ax are hereditary, there are no bands as well as no special and bispecial strings. Then there are only
finitely many isomorphism classes of indecomposable vector bundles up to twist, i.e., up to change
of degrees d in the pairs (d, y) occurring in a word. On the other hand, if each s̃gkA consists of 2
points and for every pair (y, i) there is another pair (z, j) 6= (y, i) such that (z, j) ∼ (y, i), then there
are no terminating strings.
Actually, one can easily deduce the following criterion of finiteness.
Corollary 3.1. The following conditions for a noncommutative nodal curve of string type (X,A)
are equivalent:
1. There are only finitely many isomorphism classes of indecomposable vector bundles over A up
to twist.
2. There are no cycles for the bunch of chains B(A).
3. There are no sequences of points y1, y2, . . . , yn, yn+1 = y1 from s̃gA such that, for 1 ≤ k ≤ n,
if k is odd, then the points yk and yk+1 are different and belong to the same component of X̃;
if k is even, there are indices i, j such that (yk, i) ∼ (yk+1, j) (possibly yk = yk+1).
4. Almost string case. We consider one more case when there is a good description of vector
bundles.
Definition 4.1. A noncommutative nodal curve (X,A) is said to be of almost string type if
every set s̃gkA contains at most 3 point, and if it contains three points then for 2 of them the algebra
Aπ(y) is hereditary and Morita equivalent to the algebra R(1; 2) from Theorem 2.1 (with the empty
relation ∼).
Note that if Aπ(y) is hereditary, y is the unique point of s̃gA with the image π(y). Hence, if X
is connected, either X̃ consists of a unique component or there must be another point z on the same
component of X̃ such that Aπ(z) is not hereditary.
Let s̃gkA = { y0, y1, y2 } so that Aπ(y1) and Aπ(y2) are Morita equivalent to R(1; 2). In this
case we call y1, y2 extra points and y0 a marked point. Then the horizontal stripes of the matrices
Θy1 ,Θy2 corresponding to the vector bundle Lk(d) can be reduced to the form
Θy1,1
kd =
0 0
I 0
0 0
0 I
, Θy1,2
kd =
I 0
0 0
0 I
0 0
, Θy2,1
kd =
0 0
0 0
I 0
0 I
, Θy2,2
kd =
I 0
0 I
0 0
0 0
, (4.1)
where I denote identity matrices of some sizes (equal if they are in the same row). From now on
we only consider the objects from T (A) such that these matrices have the form (4.1), calling them
precanonical. If (Φ, φ) is a morphism between precanonical objects, then the matrix Φkd
kd must be of
the 4× 4 block form
Φkd
kd =
∗ 0 0 0
∗ ∗ 0 0
∗ 0 ∗ 0
∗ ∗ ∗ ∗
,
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VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 195
where stars denote arbitrary matrices of appropriate sizes. Moreover, if we consider Φk,d−1
kd also as
a 4 × 4 block matrix (fab) (a, b ∈ { 1, 2, 3, 4 }), where the blocks fab consist of linear polynomials,
then f14(y1) = f14(y2) = 0, so f14 = 0. Note that the values fab(y0) can be chosen arbitrary for
(ab) 6= (14), as well as the values of Φkd′
kd for d′ < d − 1. Therefore, the full subcategory of T (A)
consisting of precanonical objects can again be treated as the category of representations of a bunch
of chains B′ = B′(A). Namely, let exA be the set of all extra points. The index set for the bunch
B′ is s̃gA \ exA. If a point y is not marked, the sets Ey and Fy are defined just as in Section 3
(p. 191). If y is marked, the set Fy is also defined as in Section 3, but the set Ey consists of the triples
(d, y, α), where α ∈ { 0, 1 } , such that
(d′, y, α′) < (d, y, α) if and only if either d′ < d or d′ = d and α′ < α;
(d, y, α) ∼ (d, y, α) for all d, α.
Actually the element (d, y, 0) represents in this bunch the first horizontal row of the stripe (d, y)
and the fourth horizontal row of the stripe (d−1, y) in the precanonical form (4.1), while the element
(d, y, 1) represents the second and the third horizontal rows of the stripe (d, y).
The preceding observations imply the following theorem.
Theorem 4.1. Let (X,A) be a noncommutative nodal curve of almost string type. The category
T (A) is equivalent to the full subcategory of the category of representations of the bunch of chains
B′(A) consisting of such representations M that all matrices My are invertible.
Just as in Section 3, these representations (hence, vector bundles over A) correspond to termi-
nating strings and bands. In particular, the category of vector bundles over a noncommutative nodal
curve of almost string type is also tame.
Corollary 4.1. The following conditions for a noncommutative nodal curve of almost string type
(X,A) are equivalent:
(1) There are only finitely many isomorphism classes of indecomposable vector bundles over A
up to twist.
(2) There are no cycles for the bunch of chains B′(A).
(3) There are no sequences of points y1, y2, . . . , yn, yn+1 = y1 from s̃gA \ exA such that, for
1 ≤ k ≤ n,
if k is odd, then either the points yk and yk+1 are different and belong to the same component
of X̃ or yk = yk+1 is a marked point;
if k is even, there are indices i, j such that (yk, i) ∼ (yk+1, j) (possibly yk = yk+1).
5. Wild cases. If a noncommutative curve (X,A) is rational and connected and all localizations
Ax are hereditary, then X ' P1 and the category Coh(A) is equivalent to the category of coherent
sheaves over a weighted projective line C(p,λ) in the sense of [10]. Here λ = {λ1, λ2, . . . , λs } =
= sgA and p = (p1, p2, . . . , ps) are the integers such that Aλk is Morita equivalent to the hereditary
algebra R(1; pk). Then it is known that VB(A) is of finite type if and only if
∑s
k=1
1/pk > 1
and is tame if
∑s
k=1
1/pk = 1. If
∑s
k=1
1/pk < 1, it is wild. It means that the classification of
vector bundles over such noncommutative curve contains the classification of representations of every
finitely generated k-algebra (see [9] for formal definitions). Note also that if (X,A) is normal, then,
just as X itself, it is of finite type if X ' P1, tame if X is an elliptic curve and wild otherwise [9].
So the next theorem completes the answer to the question about the representation type of VB(A).
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196 Y. A. DROZD, D. E. VOLOSHYN
Theorem 5.1. In the following cases the category VB(A) is wild:
(1) (X,A) is neither rational nor normal.
(2) At least one of the localizations Ax is not nodal.
(3) (X,A) is nodal, at least one of the localizations Ax is not hereditary and (X,A) is neither
of string nor of almost string type.
Proof. The cases (1) and (2) are considered quite analogously to the commutative case [9]
(Proposition 2.5), so we omit their proofs. The proof of (3) we shall give in two cases:
(3a) X = P1, sgA = {x, x2, x3 } , Axk is Morita equivalent to R(1; k) for k = 2, 3, while, Ax is
Morita equivalent to R(1; 2;∼), where either (1, 1) ∼ (1, 2) or (1, 1) ∼ (1, 1).
(3b) X = X1 ∪ X2 so that X1 ' X2 ' P1, X1 ∩ X2 = {x} and this intersection is transversal
(i.e. Ox is nodal), there are two more singular points x2, x3 ∈ X1 and Axk is Morita equivalent to
R(1; k) for k = 2, 3, while Ax is Morita equivalent to R(2; 1, 1;∼), where (1, 1) ∼ (2, 1).
All other cases easily reduce to these ones.
In both cases π−1(xk) = {yk} for k = 2, 3 and the d-th horizontal stripe of the matrices Θyk can
be reduced to the form:
Θ2d =
0 0 0 0 I 0
0 0 I 0 0 0
I 0 0 0 0 0
0 0 0 0 0 I
0 0 0 I 0 0
0 I 0 0 0 0
, Θ3d =
0 0 0 I 0 0
0 0 0 0 I 0
0 0 0 0 0 I
I 0 0 0 0 0
0 I 0 0 0 0
0 0 I 0 0 0
,
where the vertical lines divide these matrices into the stripes corresponding to the projective modules
Pki. In the case (3a) we only consider such triples that the 1st, 5th and 6th horizontal rows of these
matrices are empty. Then the matrix Θy, where y ∈ s̃g1A and π(y) = x, is divided into 3 horizontal
stripes and if (Φ, φ) is a morphism of such representations, then
Φ1d
1d =
∗ 0 0
∗ ∗ 0
0 0 ∗
.
The classification of such triples can be considered as a bimodule problem (see [8, 9] for definitions
and details) so that the corresponding Tits form is either
Q1 = 2t21 + z2
1 + z2
2 + z1z2 + z2
3 − 2t1(z1 + z2 + z3)
or
Q2 = t21 + t22 + z2
1 + z2
2 + z1z2 + z2
3 − (t1 + t2)(z1 + z2 + z3),
where ti are the sizes of vertical stripes and zi are the sizes of horizontal stripes (if (1, 1) ∼ (1, 2),
then t1 = t2). Since Q1(2, 1, 1, 1) = Q2(2, 2, 1, 1, 1) = −1, this bimodule is wild, hence so is the
category VB(A). Note that we need to check that t1 + t2 = z1 + z2 + z3, since the matrix Θy must
be invertible.
In the case (3b) we only omit the 1st and the 6th row of the matrices Θyk . Then the matrix Φ1d
1d
will be of the form
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VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 197
Φ1d
1d =
∗ 0 0 0
∗ ∗ 0 0
0 0 ∗ 0
∗ 0 ∗ ∗
.
We have one more matrix Θz, where z ∈ s̃g2A and π(z) = x. We consider the triples such that
G|Y2 =
⊕8
d=1 rdG2d. The matrix Θz reduces to the form
Θz =
I 0 0 0 0 0 0 0
0 I 0 0 0 0 0 0
0 0 I 0 0 0 0 0
0 0 0 I 0 0 0 0
0 0 0 0 I 0 0 0
0 0 0 0 0 I 0 0
0 0 0 0 0 0 I 0
0 0 0 0 0 0 0 I
.
Then the matrix φy,1y,1 = φz,1z,1 from a morphism (Φ, φ) of such triples must be triangular and we obtain
a matrix problem with the Tits form
Q = t21 + t22 + t23 + t24 + t1t2 + t1t4 + t3t4 +
∑
i≤j
rirj −
∑
i,j
tirj .
Now Q(1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1) = −1, so we again obtain a wild problem.
Theorem 5.1 is proved.
6. Example. We consider a simple but typical example. Let (X,A) be defined as follows.
X = X1 ∪X2 , where X1 ' X2 ' P1, X1 ∩X2 = {x} and the intersection is transversal;
sgA = {x, x1, x2 } , where x1 ∈ X1, x2 ∈ X2;
K(A) = Mat(2,K1)×Mat(2,K2);
The singular localizations are:
Ax = R(2; 2, 2;∼), where (1, 1) ∼ (2, 1),
Ax1 = R(1; 2;∼), where (1, 1) ∼ (1, 1),
Ax2 = R(1; 2;∼), where (1, 1) ∼ (1, 2).
Then
X̃ = X̃1 ∪ X̃2, where X̃1 ' X̃2 ' P1, X̃1 ∩ X̃2 = ∅,
s̃gA = { y1, y2, y3, y4 } , where y1, y3 ∈ X̃1, y2, y4 ∈ X̃2,
π(y3) = π(y4) = x, π(y1) = x1, π(y2) = x2.
Therefore the corresponding bunch of chains is
E1 = { (d1) | d ∈ Z } , F1 = { (1, 1) } ,
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198 Y. A. DROZD, D. E. VOLOSHYN
E2 = { (d2) | d ∈ Z } , F2 = { (2, 1) < (2, 2) } ,
E3 = { (d3) | d ∈ Z } , F3 = { (3, 1) < (3, 2) } ,
E4 = { (d4) | d ∈ Z } , F4 = { (4, 1) < (4, 2) } ,
(1, 1) ∼ (1, 1), (2, 2) ∼ (2, 1), (3, 1) ∼ (4, 1), (d1) ∼ (d3), (d2) ∼ (d4).
(We write (dk) and (k, i) instead of (d, yk) and (yk, i).) We fix a basic vector bundle Lk over Ãk,
k = 1, 2. Then L1(d)/JL1(d) has a k-basis e1
i (d), e3
j (d), 1 ≤ i, j ≤ 2, and L2(d)/JL2(d) has a
k-basis e2
i (d), e4
j (d), 1 ≤ i, j ≤ 2, the upper index showing the point yk where the corresponding
element is supported.
An example of a usual string is given by the word
(4, 2)− (d14) ∼ (d12)− (2, 2) ∼ (2, 1)− (d22) ∼ (d24)− (4, 2)
with d1 6= d2 in order that the word be not symmetric. The corresponding vector bundle F is the
A-submodule in G = L2(d1) ⊕ L2(d2) such that Fx = Gx for x /∈ sgA, Fx2 is generated by the
preimages of e2
2(d1) and e2
1(d2), and Fx is generated by the preimages of e4
2(d1) and e4
2(d2). Since
suppG = X2, Fx1 = 0.
An example of a special string is (w, 1), where
w = (1, 1)− (d1) ∼ (d3)− (3, 2).
Here G = L1(d), Fx1 is generated by the preimage of e1
2 and Fx is generated by the preimage of e3
2.
An example of a bispecial string is (w,m, 1, 0), where
w = (1, 1)− (d11) ∼ (d13)− (3, 1) ∼ (4, 1)− (d24) ∼ (d22)− (2, 1) ∼
∼ (2, 2)− (d32) ∼ (d34)− (4, 1) ∼ (3, 1)− (d43) ∼ (d41)− (1, 1).
The degrees di can be arbitrary with the only restriction that d2 6= d3 or d1 6= d4.
G = m(L1(d1)⊕ L2(d2)⊕ L2(d3)⊕ L1(d4));
Fx is generated by the preimages of the columns of the matrices Ime3
1(d1), Ime
3
1(d4), Ime
4
1(d2)
and Ime4
1(d3), where Im denotes the identity m×m matrix;
Fx2 is generated by the preimages of the columns of the matrices Ime2
1(d2) and Ime2
2(d3);
Fx1 is generated by the preimages of the columns of the matrices(
Iq
0
)
e1
2(d1),
(
0
Im−q
)
e1
1(d1),
(
Iq
Aq
)
e1
1(d4) and
(
Bq
Im−q
)
e1
2(d4),
where q = [(m+ 1)/2] and
if m = 2q, then Aq = Iq, Bq = Jq(0), the Jordan q × q matrix with eigenvalue 0;
if m = 2q − 1, then Aq is of size (q − 1)× q and Bq is of size q × (q − 1), namely,
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VECTOR BUNDLES OVER NONCOMMUTATIVE NODAL CURVES 199
Aq =
1 0 0 . . . 0 0
0 1 0 . . . 0 0
. . . . . . . . . . . . . . . . . . . . . . . . .
0 0 0 . . . 1 0
, Bq =
0 0 . . . 0
1 0 . . . 0
0 1 . . . 0
. . . . . . . . . . . . . . . .
0 0 . . . 1
.
Finally, an example of a band is (w,m, λ), where
w = (2, 2) ∼ (2, 1)− (d12) ∼ (d14)− (4, 1) ∼ (3, 1)− (d23) ∼ (d21)−
− (1, 1) ∼ (1, 1)− (d31) ∼ (d33)− (3, 1) ∼ (4, 1)− (d44) ∼ (d42).
We suppose that d3 < d2 or d3 = d2, d4 ≤ d1. Then
G = m(L1(d1)⊕ L2(d2)⊕ L2(d3)⊕ L1(d4));
Fx1 is generated by the preimages of the columns of the matrices(
Ime
1
1(d2)
Ime
1
1(d3)
)
and
(
0
Ime
1
2(d3)
)
;
Fx is generated by the preimages of the columns of the matrices Ime4
1(d1), Ime
3
1(d2), Ime
3
1(d3)
and Ime4
1(d4);
Fx2 is generated by the preimages of the columns of the matrices Ime2
1(d1) and Jm(λ)e2
2(d4)
(the Jordan m ×m matrix with eigenvalue λ). If d2 < d3 or d2 = d3, d1 < d4, one has to permute
d2 and d3 in the generators of Fx1 , also permuting the rows.
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Received 28.11.11
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