PROJECTIVE CONFIGURATIONS

IGOR BURBAN AND YURIY DROZD

Contents

Introduction 1

1. Backstr¨om rings 2

2. Nodal rings 4

3. Examples 8

3.1. Simple node 8

3.2. Dihedral algebra 10

3.3. Gelfand problem 11

4. Projective configurations 14

5. Configurations of type A and ˜A 15

6. Application: Cohen–Macaulay modules over surface singularities 21

References 23

Introduction

This paper is devoted to recent results on explicit calculations in derived categories of modules and coherent sheaves. The idea of this approach is actually not new and was effectively used in several questions of module the- ory (cf. e.g. [10, 12, 13, 7]). Nevertheless it was somewhat unexpected and successful that the same technique could be applied to derived categories, at least in the case of rings and curves with “simple singularities.” We present here two cases: nodal rings and configurations of projective lines of types A and ˜A, when these calculations can be carried out up to a result, which can be presented in more or less distinct form, though it involves rather in- tricate combinatorics of a special sort of matrix problems, namely “bunches of semi-chains” [4] (or, equivalently, “clans” [8]). In Sections 1 and 4 we give a general construction of “categories of triples,” which are a connect- ing link between derived categories and matrix problems, while in Sections 2 and 5 this construction is applied to nodal rings and configurations of types ˜A. Section 3 contains examples of calculations for concrete rings and Section 5 also presents those for nodal cubic. We tried to choose typical ex- amples, which allow to better understand the general procedure of passing from combinatorial data to complexes. Section 6 contains an application to

2000Mathematics Subject Classification. 16E05, 16D90.

It is a survey of a research supported by the CRDF Award UM 2-2094 and by the DFG Schwerpunkt “Globale Methoden in der komplexen Geometrie”.

1

Cohen–Macaulay modules over surface singularities, which was in fact the origin of investigations of vector bundles over projective curves in [13].

More detailed exposition of these results can be found in [5, 6, 14].

1. Backstr¨om rings

We consider a class of rings, which generalizes in a certain way local rings of ordinary multiple points of algebraic curves. Following the terminology used in the representations theory of orders, we call them Backstr¨om rings.

Since in the first three sections we are investigating a local situation, all
rings there are supposed to be semi-perfect [3] and noetherian. We denote
by A-mod the category of finitely generated A-modules and by D(A) the
derived category D^{−}(A-mod) of right bounded complexes overA-mod. As
usually, it can be identified with the homotopy categoryK^{−}(A-pro) of (right
bounded) complexes of (finitely generated) projectiveA-modules. Moreover,
since A is semi-perfect, each complex from K^{−}(A-pro) is homotopic to a
minimal one, i.e. to such a complex C• = (C_{n}, d_{n}) that Imd_{n} ⊆ radCn−1

for all n. If C• and C_{•}^{0} are two minimal complexes, they are isomorphic
in D(A) if and only if they are isomorphic as complexes; moreover, any
morphismC• →C_{•}^{0} inD(A) can be presented by a morphism of complexes,
and f is an isomorphism if and only if the latter one is.

Definition 1.1. A ringAis called aBackstr¨om ring if there is a hereditary ring H ⊇ A (also semi-perfect and noetherian) and a (two-sided) H-ideal I⊂A such that bothR=H/I andS=A/I are semi-simple.

For Backstr¨om rings there is a convenient approach to the study of de-
rived categories. Recall that for a hereditary ring Hevery object C• from
D(H) is isomorphic to the direct sum of its homologies. Especially, any
indecomposable object fromD(H) is isomorphic to a shiftN[n] for someH-
module N, or, the same, to a “short” complex 0→P^{0}−→^{α} P →0, where P
and P^{0} are projective modules andα is a monomorphism with Imα⊆radP
(maybe P^{0} = 0). Thus it is natural to study the category D(A) using this
information about D(H) and the functor T : D(A) → D(H) mapping C•

toH⊗_{A}C•.^{1}

Consider a new category T = T(A) (the category of triples) defined as follows:

• Objects ofT are triples (A•, B•, ι), where – A• ∈D(H);

– B• ∈D(S);

– ιis a morphismB• →R⊗_{H}A• fromD(S) such that the induced
morphismι^{R}:R⊗_{S}B• →R⊗_{H}A• is an isomorphism inD(R).

• A morphism from a triple (A•, B•, ι) to a triple (A^{0}_{•}, B_{•}^{0}, ι^{0}) is a pair
(Φ, φ), where

– Φ :A• →A^{0}_{•} is a morphism fromD(H);

– φ:B• →B_{•}^{0} is a morphism fromD(S);

1Of course, we mean here the left derived functor of⊗, but when we consider complexes of projective modules, it restricts indeed to the usual tensor product.

– the diagram

(1.1)

B• ι

−−−−→ R⊗_{H}A•

φ

y

y^{1⊗Φ}
B_{•}^{0} −−−−→^{ι}^{0} R⊗_{H}A^{0}_{•}
commutes inD(S).

One can define a functor F : D(A) → T(A) setting F(C•) = (H⊗_{A}
C•,S⊗_{A}C•, ι), whereι:S⊗_{A}C• →R⊗_{H}(H⊗_{A}C•)'R⊗_{A}C•is induced
by the embedding S→R. The values ofFon morphisms are defined in an
obvious way.

Theorem 1.2. The functor Fis a full representation equivalence, i.e. it is

• dense, i.e. every object fromT is isomorphic to an object of the form F(C•);

• full, i.e. each morphismF(C•)→F(C_{•}^{0})is of the formF(γ)for some
γ :C• →C_{•}^{0};

• conservative, i.e. F(γ) is an isomorphism if and only if so isγ; As a consequence, F maps non-isomorphic objects to non-isomorphic and indecomposable to indecomposable.

Note that in generalFis notfaithful: it is possible that F(γ) = 0 though γ 6= 0 (cf. Example 3.1.3 below).

Sketch of the proof. Consider any triple T = (A•, B•, ι). We may suppose
that A• is a minimal complex from K^{−}(A-pro), while B• is a complex
with zero differential (sinceS is semi-simple) and the morphismιis a usual
morphism of complexes. Note that R⊗_{H}A• is also a complex with zero
differential. We have an exact sequence of complexes

0−→IA• −→A• −→R⊗_{H}A• −→0.

Together with the morphismι:B• →R⊗_{H}A•it gives rise to a commutative
diagram in the category of complexes Com^{−}(A-mod)

0 −−−−→ IA• −−−−→ A• −−−−→ R⊗_{H}A• −−−−→ 0

^{α}

x

x

^{ι}

0 −−−−→ IA• −−−−→ C• −−−−→ B• −−−−→ 0,

where C• is the preimage in A• of Imι. The lower row is also an exact
sequence of complexes and α is an embedding. Moreover, since ι^{R} is an
isomorphism,IA•=IC•. It implies thatC•consists of projectiveA-modules
and H⊗_{A}C• 'A•, wherefromT 'FC•.

Let now (Φ, φ) : FC• → FC_{•}^{0}. We suppose again that both C• and C_{•}^{0}
are minimal, while Φ : H⊗_{A}C• → H⊗_{A}C_{•}^{0} and φ:S⊗_{A}C• → S⊗_{A}C_{•}^{0}
are morphisms of complexes. Then the diagram (1.1) is commutative in the
category of complexes, so Φ(C•)⊆C_{•}^{0} and Φ induces a morphismγ :C• →
C_{•}^{0}. It is evident from the construction that F(γ) = (Φ, φ). Moreover, if
(Φ, φ) is an isomorphism, so are Φ andφ(since our complexes are minimal).

Therefore Φ(C•) = C_{•}^{0}, i.e. Imγ =C_{•}^{0}. But kerγ = ker Φ∩C• = 0, thus γ

is an isomorphism too.

2. Nodal rings

We apply these considerations to the class of rings first considered in [10], where the second author has shown that they are unique pure noetherian rings such that the classification of their modules of finite length is tame (all others being wild).

Definition 2.1. A ring A (semi-perfect and noetherian) is called a nodal ring if it is pure noetherian, i.e. has no minimal ideals, and there is a hereditary ring H⊇A, which is semi-perfect and pure notherian such that

1) radA= radH; we denote this common radical by R.

2) length_{A}(H⊗_{A}U)≤2 for every simple left A-moduleU and
length_{A}(V ⊗_{A}H)≤2 for every simple right A-moduleV.

Note that condition 2 must be imposed both on left and on right modules.

It is known that such a hereditary ring His Morita equivalent to a direct product of rings H(D, n), whereD is a discrete valuation ring (maybe non- commutative) and H(D, n) is the subring of Mat(n,D) consisting of all matrices (aij) with non-invertible entriesaij fori < j. Especially,HandA are semi-prime (i.e. without nilpotent ideals)

Example 2.2. 1. The first example of a nodal ring is the completion
of the local ring of a simple node (or a simple double point) of an
algebraic curve over a field k. It is isomorphic to A =k[[x, y]]/(xy)
and can be embedded into H = k[[x_{1}]]×k[[x_{2}]] as the subring of
pairs (f, g) such thatf(0) =g(0): xmaps to (x1,0) and y to (0, x2).

Evidently this embedding satisfies conditions of Definition 2.1.

2. The dihedral algebra A=khhx, yii/(x^{2}, y^{2}) is another example of a
nodal ring. In this case H=H(k[[t]],2) and the embedding A→ H
is given by the rule

x7→

0 t 0 0

, y 7→

0 0 1 0

.

3. The “Gelfand problem” is that of classification of diagrams with re- lations

2

x+

**1

x−

jj

y−

443

y+

tt x+x−=y+y−.

If we consider the case when x+x− is nilpotent (the main part of
the problem), such diagrams are just modules over the ringA, which
is the subring of Mat(3,k[[t]]) consisting of all matrices (a_{ij}) with
a12(0) = a13(0) = a23(0) = a32(0) = 0. The arrows of the diagram
correspond to the following matrices:

x_{+} 7→te_{12}, x−7→e_{21}, y_{+} 7→te_{13}, y−7→e_{31},

where eij are matrix units. It is also a nodal ring with H being the
subring of Mat(3,k[[t]]) consisting of all matrices (a_{ij}) witha_{12}(0) =
a13(0) = 0 (it is Morita equivalent toH(k[[t]],2)).

4. The classification ofquadratic functors, which play an important role
in algebraic topology, reduces to the study of modules over the ring
A, which is the subring ofZ^{2}_{2}×Mat(2,Z2) consisting of all triples

a, b,

c_{1} 2c_{2}
c3 c4

with a≡c_{1}(mod 2) andb≡c_{4}(mod 2),

whereZ2is the ring ofp-adic integers [11]. It is again a nodal ring: one
can take forHthe ring of all triples as above, but without congruence
conditions; thenH=Z^{2}_{2}×H(Z2,2).

Certainly, a nodal ring is always Backstr¨om, so Theorem 1.2 can be ap- plied. Moreover, in nodal case the resulting problem belongs to a well-known type. For the sake of simplicity, we consider now the situation, when Ais a D-algebra finitely generated as D-module, where D is a discrete valuation ring with algebraically closed residue field k. We denote by U1, U2, . . . , Us

indecomposable non-isomorphic projective (left) modules over A and by V1, V2, . . . , Vr those over H. Condition 2 from Definition 2.1 implies that there are three possibilities:

1) H⊗_{A}U_{i}'V_{j} for somej andV_{j} does not occur as a direct summand
inH⊗_{A}Uk fork6=i;

2) H⊗_{A}U_{i} 'V_{j}⊕V_{j}^{0} (j6=j^{0}) and neitherV_{j} norV_{j}^{0} occur inH⊗_{A}U_{k}
fork6=i;

3) there are exactly two indicesi6=i^{0}such thatH⊗_{A}Ui'H⊗_{A}U_{i}^{0} 'Vj

and V_{j} does not occur inH⊗_{A}U_{k} fork /∈ {i, i^{0}}.

We denote by Hj the indecomposable projective H-module such that
H_{j}/RH_{j} ' V_{j}. Since H is a semi-perfect hereditary order, any indecom-
posable complex from D(H) is isomorphic either to 0 → H_{k} −→^{φ} H_{j} → 0
or to 0 → Hj → 0 (it follows, for instance, from [9]). Moreover, the for-
mer complex is completely defined by either j or k and the length l =
length_{H}Cokerφ. We shall denote it both byC(j,−l, n) and byC(k, l, n+1),
while the latter complex will be denoted by C(j,∞, n), where n denotes
the place of Hj (so the place of H_{k} is n+ 1). We denote by ˜Z the set
(Z\ {0})∪ { ∞ } and consider the ordering ≤ on ˜Z, which coincides with
the usual ordering separately on positive integers and on negative integers,
but l <∞ <−l for any l∈N. Note that for each j the submodules ofH_{j}
form a chain with respect to inclusion. It immediately implies the following
result.

Lemma 2.3. There is a homomorphismC(j, l, n)→C(j, l^{0}, n), which is an
isomorphism on the n-th components, if and only if l≤l^{0} in Z˜. Otherwise
the n-th component of any homomorphism C(j, l, n) → C(j, l^{0}, n) is zero
modulo R.

We transfer the ordering from ˜Zto the set E_{j,n}=
n

C(j, l, n)|l∈Z˜ o

, so
the latter becomes a chain with respect to this ordering. We also denote
by F_{j,n} the set {(i, j, n)|Vj is a direct summand of H⊗_{A}Ui}. It has at
most two elements. We always consider F_{j,n} with trivial ordering. Then
a triple (A•, B•, ι) from the category T(A) is given by homomorphisms
φ^{ijn}_{jln} :d_{i,j,n}U_{i} →r_{j,l,n}V_{j}, where (i, j, n)∈F_{jn}, the leftU_{i} comes fromB_{n}and

the right V_{j} comes from direct summands r_{j,l,n}C(j, l, n) of A•. Note that
if both C(j,−l, n) and C(k, l, n+ 1) correspond to the same complex (then
we write C(j,−l, n)∼C(k, l, n+ 1)), we haverj,−l,n =r_{k,l,n+1}. We present
φ^{ijn}_{jln} by its matrix M_{jln}^{ijn}. Then Lemma 2.3 implies the following

Proposition 2.4. Two sets of matrices n

M_{jln}^{ijn}
o

and n

N_{jln}^{ijn}
o

describe isomorphic triples if and only if one of them can be transformed to the other by a sequence of the following “elementary transformations”:

1) For any given values ofi, n, simultaneouslyM_{jln}^{ijn}7→M_{jln}^{ijn}S for allj, l
such that(ijn)∈F_{j,n}, where S is an invertible matrix of appropriate
size.

2) For any given values of j, l, n, simultaneously M_{jln}^{ijn} 7→ S^{0}M_{jln}^{ijn} for
all (i, j, n)∈ F_{jn} and M_{k,−l,n−sgn}^{i,k,n−sgn}^{l}_{l} 7→ S^{0}M_{k,−l,n−sgn}^{i,k,n−sgn}^{l}_{l} for all (i, k, n−
sgnl) ∈ F_{k,n−sgn}_{l}, where S^{0} is an invertible matrix of appropriate
size and C(j, l, n) ∼ C(k,−l, n−sgnl). If l = ∞, it just means
M_{j∞n}^{ijn} 7→SM_{j∞n}^{ijn} .

3) For any given values of j, l^{0} < l, n, simultaneously M_{jln}^{ijn} 7→ M_{jln}^{ijn}+
RM_{jl}^{ijn}0n for all (i, j, n) ∈ F_{j,n}, where R is an arbitrary matrix of ap-
propriate size. Note that, unlike the preceding transformation, this
one does not touch the matrices M_{k,−l,n−sgn}^{i,k,n−sgn}^{l}_{l} such that C(j, l, n) ∼
C(k,−l, n−sgnl).

This sequence must contain finitely many transformations for every fixed values of j and n.

Therefore we obtain representations of the bunch of semi-chains E_{jn},F_{jn}
in the sense of [4], so we can deduce from this paper a description of inde-
composables in D(A). We arrange it in terms of strings and bands, often
used in representation theory.

Definition 2.5. 1. We define the alphabet X as the set S

j,n(Ej,n ∪ {(j, n)}). We define symmetric relations ∼ and − on X by the fol- lowing exhaustive rules:

(a) C(j, l, n)−(j, n) for alll∈Z;

(b) C(j,−l, n)∼C(k, l, n+ 1) defined as above;

(c) (j, n)∼(k, n) (k6=j) if V_{j}⊕V_{k}'H⊗_{A}U_{i} for somei;

(d) (j, n)∼(j, n) if Vj 'H⊗_{A}Ui'H⊗_{A}U_{i}^{0} for somei^{0} 6=i.

2. We define anX-wordas a sequencew=x_{1}r_{1}x_{2}r_{2}x_{3}. . . rm−1x_{m}, where
xk∈X, rk∈ { −,∼ }such that

(a) x_{k}r_{k}x_{k+1} in Xfor 1≤k < m;

(b) r_{k}6=r_{k+1} for 1≤k < m−1.

We callx1 and xm the ends of the word w.

3. We call anX-word w full if (a) r1 =rm−1 =−

(b) x_{1} 6∼y for each y6=x_{1};
(c) xm6∼z for eachz6=xm.

Condition (a) reflects the fact thatι^{R}must be an isomorphism, while
conditions (b,c) come from generalities on bunches of semi-chains [4].

4. A wordwis calledsymmetric, ifw=w^{∗}, wherew^{∗} =x_{m}rm−1xm−1. . . r_{1}x_{1}
(the inverse word), and quasisymmetric, if there is a shorter word v
such thatw=v∼v^{∗} ∼ · · · ∼v^{∗} ∼v.

5. We call the end x_{1} (x_{m}) of a word w special if x_{1} ∼x_{1} and r_{1} = −
(respectively, xm∼xm and rm−1=−). We call a word w

(a) usual if it has no special ends;

(b) special if it has exactly one special end;

(c) bispecial if it has two special ends.

Note that a special word is never symmetric, a quasisymmetric word is always bispecial, and a bispecial word is always full.

6. We define acycle as a word wsuch thatr_{1} =rm−1 =∼andx_{m}−x_{1}.
Such a cycle is called non-periodic if it cannot be presented in the
formv−v− · · · −vfor a shorter cyclev. For a cyclewwe setr_{m} =−,
xqm+k =xk and rqm+k=rk for any q, k∈Z.

7. A (k-th) shift of a cycle w, where k is an even integer, is the cycle
w^{[k]} = xk+1rk+1xk+2. . . rk−1xk. A cycle w is called symmetric if
w^{[k]}=w^{∗} for somek.

8. We also consider infinite words of the sorts w = x_{1}r_{1}x_{2}r_{2}. . . (with
one end) andw=. . . x0r0x1r1x2r2. . . (with no ends) with restrictions
(a) every pair (j, n) occurs in this sequence only finitely many times;

(b) there is an n0 such that no pair (j, n) withn < n0 occurs.

We extend to such infinite words all above notions in the obvious manner.

Definition 2.6 (String and band data). 1. String data are defined as follows:

(a) ausual string datum is a full usual non-symmetricX-word w;

(b) a special string datum is a pair (w, δ), where w is a full special word andδ∈ {0,1};

(c) abispecial string datum is a quadruple (w, m, δ1, δ2), where wis a bispecial word that is neither symmetric nor quasisymmetric, m∈Nand δ1, δ2 ∈ {0,1}.

2. A band datum is a triple (w, m, λ), where w is a non-periodic cycle,
m∈Nand λ∈k^{∗}; ifw is symmetric, we also suppose thatλ6= 1.

The results of [4, 8] imply

Theorem 2.7. Every string or band datum d defines an indecomposable object C•(d) fromD(A), so that

1) Every indecomposable object from D(A) is isomorphic to C•(d) for some d.

2) The only isomorphisms between these complexes are the following:

(a) C(w)'C(w^{∗});

(b) C(w, m, δ1, δ2)'C(w^{∗}, m, δ2, δ1);

(c) C(w, m, λ)'C(w^{[k]}, m, λ)'C(w^{∗[k]}, m,1/λ) if k≡0 (mod 4);

(d) C(w^{∗}, m, λ)'C(w^{[k]}, m,1/λ)'C(w^{∗[k]}, m, λ) if k≡2 (mod 4).

3) Every object from D(A) uniquely decomposes into a direct sum of indecomposable objects.

The construction of complexesC•(d) is rather complicated, especially in the case, when there are pairs (j, n) with (j, n) ∼(j, n) (e.g. special ends

are involved). So we only show several examples arising from simple node, dihedral algebra and Gelfand problem.

3. Examples

3.1. Simple node. In this case there is only one indecomposable projective A-module (Aitself) and two indecomposable projective H-modules H1, H2

corresponding to the first and the second direct factors of the ring H. We
have H⊗_{A}A'H'H1⊕H2. So the∼-relation is given by:

1) (1, n)∼(2, n);

2) C(j, l, n)∼C(j,−l, n−sgnl) for any l∈Z\ {0}.

Therefore there are no special ends at all. Moreover, any end of a full string
must be of the formC(j,∞, n). Note that the homomorphism in the complex
corresponding to C(j,−l, n) and C(j, l, n+ 1) (l∈N) is just multiplication
by x^{l}_{j}. Consider several examples of strings and bands.

Example 3.1. 1. Let wbe the cycle

C(2,1,1)∼C(2,−1,0)−(2,0)∼(1,0)−C(1,−2,0)∼C(1,2,1)−

−(1,1)∼(2,1)−C(2,4,1)∼C(2,−4,0)−(2,0)∼(1,0)−

−C(1,−1,0)∼C(1,1,1)−(1,1)∼(2,1)−C(2,−3,1)∼C(2,3,2)−

−(2,2)∼(1,2)−C(1,2,2)∼C(1,−2,1)−(1,1)∼(2,1) Then the band complex C•(w,1, λ) is obtained from the complex of H-modules

H_{2} ^{x}^{2} ^{//}H_{2}

H_{1} ^{x}

2

1 //

H_{1}

H2

x^{4}_{2} //H2

H1

x1 //

H1

H_{2} ^{x}

3

2 //

H_{2}

H_{1} ^{x}

2

1 //H_{1}

λ

,.

*

&(

!#

by gluing along the dashed lines (they present the∼relations (1, n)∼ (2, n)). All glueings are trivial, except the last one marked with ‘λ’;

the latter must be twisted byλ. It gives theA-complex

(3.1)

A ^{y} ^{//}A

A

λx^{2}

88p

pp pp pp pp pp pp

yN^{3}NNNNNNN&&

NN NN

N A

x^{2}

88p

pp pp pp pp pp

pp ^{y}^{4} //A

A

x

88p

pp pp pp pp pp pp

Here each column presents direct summands of a non-zero compo-
nent C_{n} (in our case n = 2,1,0) and the arrows show the non-zero
components of the differential. According to the embeddingA→H,
we have to replace x1 by x and x2 by y. Gathering all data, we can
rewrite this complex as

A

λx^{2}
0
y^{3}

−−−−−→ A⊕A⊕A

y 0

x^{2} y^{4}

0 x

−−−−−−−→ A⊕A,

though the form (3.1) seems more expressive, so we use it further. If
m >1, one only has to replaceAbymA, each elementa∈AbyaE,
where E is the identity matrix, and λa by aJ_{m}(λ), where J_{m}(λ) is
the Jordanm×m cell with eigenvalue λ. So we obtain the complex

mA

x^{2}J_{m}(λ)
0
y^{3}E

−−−−−−−−−→ mA⊕mA⊕mA

yE 0

x^{2}E y^{4}E

0 xE

−−−−−−−−−−→ mA⊕mA. 2. Letw be the word

C(1,∞,1)−(1,1)∼(2,1)−C(2,2,1)∼C(2,−2,0)−(2,0)∼

∼(1,0)−C(1,−3,0)∼C(1,3,1)−(1,1)∼(2,1)−C(2,−1,1)∼

∼C(2,1,2)−(2,2)∼(1,2)−C(1,1,2)∼C(1,−1,1)−(1,1)∼

∼(2,1)−C(2,2,1)∼C(2,−2,0)−(2,0)∼(1,0)−C(1,∞,0) Then the string complexC•(w) is

A ^{y}

2 //A

A ^{y} ^{//}

xNNNNNNN&&

NN NN

NN A

x^{3}

88p

pp pp pp pp pp pp

A ^{y}

2 //A

Note that for string complexes (which are always usual in this case) there are no multiplicities mand all glueings are trivial.

3. Seta=x+y. Then the factorA/aA is represented by the complex
A−→^{a} A, which is the band complexC•(w,1,1), where

w=C(1,1,1)∼C(1,−1,0)−(1,0)∼(2,0)−

−C(2,−1,0)∼C(2,1,1)−(2,1)∼(1,1).

Consider the morphism of this complex toA[1] given on the 1-compo-
nent by multiplication A −→^{x} A. It is non-zero in D(A), but the
corresponding morphism of triples is (Φ,0), where Φ arises from the
morphism of the complex H −→^{a} H to H[1] given by multiplication
withx1. But Φ is homotopic to 0: x1 =e1a, where e1 = (1,0)∈H,
thus (Φ,0) = 0 in the category of triples.

4. The string complexC•(l,0), where wis the word

C(1,∞,0)−(1,0)∼(2,0)−C(2,−1,0)∼C(2,1,1)−(2,1)∼

∼(1,1)−C(1,−2,1)∼C(1,1,2)−(1,2)∼(2,2)−C(2,−1,2)∼

∼C(2,1,3)−(2,3)∼(1,3)−C(1,−2,3)∼C(1,2,4)−. . . , is

. . . A−→^{x}^{2} A−→^{y} A−→^{x}^{2} A−→^{y} A−→0.

Its homologies are not left bounded, so it does not belong toD^{b}(A-mod).

3.2. Dihedral algebra. This case is very similar to the preceding one.

Again there is only one indecomposable projective A-module (Aitself) and
two indecomposable projectiveH-modulesH1, H2corresponding to the first
and the second columns of matrices from the ring H, and we have H⊗_{A}
A 'H ' H1⊕H2. The main difference is that now the unique maximal
submodule of H_{j} is isomorphic to H_{k}, where k 6= j. So the ∼-relation is
given by:

1) (1, n)∼(2, n);

2) C(j, l, n) ∼C(j,−l, n−sgnl) ifl∈Z\ {0} is even, and C(j, l, n) ∼
C(j^{0},−l, n−sgnl), where j^{0} 6=j, ifl∈Z\ {0} is odd.

Again there are no special ends. The embeddings Hk → Hj are given by right multiplications with the following elements from H:

H_{1}→H_{1} − by t^{r}e_{11} (colength 2r),
H_{1}→H_{2} − by t^{r}e_{12} (colength 2r−1),
H_{2}→H_{1} − by t^{r}e_{21} (colength 2r+ 1),
H2→H2 − by t^{r}e22 (colength 2r).

When gluing H-complexes into A-complexes we have to replace them re- spectively

t^{r}e_{11} − by (xy)^{r},
t^{r}e_{22} − by (yx)^{r},
t^{r}e12 − by (xy)^{r−1}x,
t^{r}e21 − by (yx)^{r}y.

The glueings are quite analogous to those for simple node, so we only present the results, without further comments.

Example 3.2. 1. Consider the band datum (w,1, λ), where w=C(1,−2,0)∼C(1,2,1)−(1,1)∼(2,1)−C(2,−5,1)∼

∼C(1,5,2)−(1,2)∼(2,2)−C(2,4,2)∼C(2,−4,1)−(2,1)∼

∼(1,1)−C(1,3,1)∼C(2,−3,0)−(2,0)∼(1,0).

The corresponding complex C•(w, m, λ) is

mA ^{xyE} ^{//}mA

mA

(xy)n ^{2}xE

nn n

77n

nn n

(yx)^{2}E //mA

xyxJnn m(λ)

nn n

77n

nn nn

2. Letw be the word

C(2,∞,0)−(2,0)∼(1,0)−C(1,−1,0)∼C(2,1,1)−(2,1)∼(1,1)−C(1,3,1)∼

∼C(2,−3,0)−(2,0)∼(1,0)−C(1,−3,0)∼C(2,3,1)−(2,1)∼(1,1)−C(1,∞,1).

Then the string complexC•(w) is
A ^{e}^{21} ^{//}

t^{2}e12

NN NN N

&&

NN NN N

A

A ^{te}^{21} ^{//}A

3. The factorA/Ris described by the infinite string complex C•(w)
. . . e21 //A ^{te}^{12} ^{//}A ^{e}^{21} ^{//}A.

. . . te12 //A ^{e}^{21} ^{//}A

te12

pp pp pp

77p

pp pp

The corresponding wordw is

· · · −C(2,1,2)∼C(1,−1,1)−(1,1)∼(2,1)−

−C(2,1,1)∼C(1,−1,0)−(1,0)∼(2,0)−C(2,−1,0)∼

∼C(1,1,1)−(1,1)∼(2,1)−C(2,−1,1)∼C(1,1,2)−. . .
3.3. Gelfand problem. In this case there are 2 indecomposable projective
H-modules H_{1} (the first column) and H_{2} (both the second and the third
columns). There are 3 indecomposable A-projectives Ai (i = 1,2,3); Ai

correspond to thei-th column ofA. We haveH⊗_{A}A_{1}'H_{1} andH⊗_{A}A_{2} '
H⊗_{A}A3 'H2. So the relation ∼is given by:

1) (2, n)∼(2, n);

2) C(j, l, n)∼C(j,−l, n−sgnl) if l is even;

3) C(j, l, n)∼C(j^{0},−l, n−sgnl) (j^{0} 6=j) if l is odd.

So a special end is always (2, n).

Example 3.3. 1. Consider the special word w:

(2,0)−C(2,−2,0)∼C(2,2,1)−(2,1)∼(2,1)−C(2,−4,1)∼

∼C(2,4,2)−(2,2)∼(2,2)−C(2,2,2)∼C(2,−2,1)−

−(2,1)∼(2,1)−C(2,−1,1)∼C(1,1,2)−(1,2) The complexC•(w,0) is obtained by gluing from the complex ofH- modules

H_{2} 2 //

H_{2}

H2 4 //H2

H2

OO

2 //H2

H_{1} 1 //H_{2}

Here the numbers inside arrows show the colengths of the correspond-
ing images. We mark dashed lines defining glueings with arrows going
from the bigger complex (with respect to the ordering inE_{j,n}) to the
smaller one. When we construct the corresponding complex of A-
modules, we replace each H_{2} by A_{2} and A_{3} starting with A_{2} (since
δ = 0; if δ = 1 we start from A3). Each next choice is arbitrary
with the only requirement that every dashed line must touch both
A2 and A3. (Different choices lead to isomorphic complexes: one can
see it from the pictures below.) All horizontal mappings must be du-
plicated by slanting ones, carried along the dashed arrow from the
starting point or opposite the dashed arrow with the opposite sign
from the ending point (the latter procedure will be marked by ‘−’

near the duplicated arrow). So we get theA-complex
A_{2} 2 //A_{2}

A3

−

p4

pp pp p

88p

pp pp p

4 //

N2

NN NN N

&&

NN NN NN

2

##

A3

p2

pp pp p

88p

pp pp p

A2 2 //

2 **

A2

A_{1}

1 −

44

1 //A_{3}

All mappings are uniquely defined by the colengths in theH-complex, so we just mark them with ‘l.’

2. Letw be the bispecial word

(2,2)−C(2,2,2)∼C(2,−2,1)−(2,1)∼(2,1)−C(2,2,1)∼

∼C(2,−2,0)−(2,0)∼(2,0)−C(2,−4,0)∼C(2,4,1)−

−(2,1)∼(2,1)−C(2,6,1)∼C(2,−6,0)−(2,0) The complex C•(w, m,1,0) is the following one:

aA3⊕bA2 M1 //

−M1 −

RR RR R

((R

RR RR R

mA3

R2

RR RR RR

((R

RR RR RR

− 2

mA2

R2

RR RR RR

−R((

RR RR RR

2 //mA3

mA_{3} 4 //mA_{2}

mA_{2}

l4

ll ll ll

66l

ll ll ll

M2 //aA2⊕bA3

wherea= [(m+ 1)/2], b= [m/2], soa+b=m. (The change ofδ_{1}, δ_{2}
transpose A2 and A3 at the ends.) All arrows are just αlE, where
α_{l} is defined by the colengthl, except of the “end” matrices M_{i}. To
calculate the latter, write α_{l}E for one of them (say, M_{1}) and α_{l}J
for anothher one (say, M2), where J is the Jordan m×m cell with
eigenvalue 1, then put the odd rows or columns into the first part of
Mi and the even ones to its second part. In our example we get

M_{1} =α_{2}

1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0

, M_{2} =α_{6}

1 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 1

.

(We use columns for M_{1} and rows for M_{2} since the left end is the
source and the right end is the sink of the corresponding mapping.)
3. The band complexC•(w,1, λ), where wis the cycle

(2,1)∼(2,1)−C(2,−2,1)∼C(2,2,2)−(2,2)∼(2,2)−

−C(2,4,2)∼C(2,−4,1)−(2,1)∼(2,1)−C(2,6,1)∼

∼C(2,−6,0)−(2,0)∼(2,0)−C(2,−4,0)∼C(2,4,1)

is

mA2 2 //

− 2

mA2

4^{λ}

−

4^{λ}

mA3

n2

nn nn n

77n

nn nn n

4 //

−

P4

PP PP P

''P

PP PP P

− 2

$$

mA2

P6

PP PP P

''P

PP PP P

mA_{3} 6 //mA_{2}

mA_{3}

4^{λ}

nn nn nn

−

77n

nn nn n

4^{λ} //mA_{3}

Superscript ‘λ’ denotes that the corresponding mapping must be twisted by Jm(λ).

4. The projective resolution of the simpleA-moduleU1 is
A_{2} 1 //A_{1}

A_{1}

−

p1

pp pp p

88p

pp pp p

1 //A_{3}
p1

pp pp p

88p

pp pp p

It coincides with the usual string complex C•(w), where wis

(1,0)−C(1,−1,0)∼C(2,1,1)−(2,1)∼(2,1)−C(2,−1,1)∼C(1,1,2)−(1,2).

The projective resolution ofU2 (U3) is A1 →A2 (respectively A1 →
A_{3}), which is the special string complexC•(w,0) (respectivelyC•(w,1)),
where

w= (2,0)−C(2,−1,0)∼C(1,1,1)−(1,1).

Note that gl.dimA= 2.

4. Projective configurations

We can “globalize” the results of the preceding sections. The simplest way is to consider the so calledprojective configurations, which are a sort of global analogues of Backstr¨om rings.

Definition 4.1. Let X be a projective curve over k, which we suppose
reduced, but possibly reducible. We denote by π : ˜X → X its normal-
ization; then ˜X is a disjoint union of smooth curves. We call X a pro-
jective configuration if all components of ˜X are rational curves (i.e. of
genus 0) and all singular points p of X are ordinary. The latter means
that ifπ^{−1}(p) ={y_{1}, y_{2}, . . . , y_{m}}, the image of O_{X,p}inQm

i=1O_{X,y}_{˜}

i contains Qm

i=1m_{i}, wherem_{i} is the maximal ideal ofO_{X,y}_{˜}

i.

We denote by S = {p1, p2, . . . , ps} the set of singular points of X and
by ˜S ={y_{1}, y_{2}, . . . , y_{r}}its preimage in ˜X. We also putO =O_{X}, ˜O =O_{X}_{˜}
and denote by J the conductor of ˜O in O, i.e. the maximal sheaf of π∗O-˜
ideals contained in O. Set S = O/J and R = π∗O/J '˜ O/π˜ ^{−1}J. Both
these sheaves have 0-dimensional support S, so we may (and shall) identify
them with the algebras of their global sections. In the case of projective

configurations both these algebras are semi-simple, namely S =Qs

i=1k(p_{i})
and R=Qr

i=1k(yi).

Let D(X) = D^{−}(CohX) be the right bounded derived category of co-
herent sheaves over X. As X is a projective variety, it can be identified
with the category of fractions K^{−}(VBX)[Q^{−1}], where K^{−}(VBX) is the
category of right bounded complexes of vector bundles (or, the same, lo-
cally free coherent sheaves) over X modulo homotopy and Q is the set
of quasi-isomorphisms in K^{−}(VBX). So we always present objects from
D(X) and from D( ˜X) as complexes of vector bundles. We denote by
T : D(X) → D( ˜X) the left derived functor Lπ^{∗}. Again if C• is a com-
plex of vector bundles, TC_{•} coincides withπ^{∗}C_{•}.

Just as in Section 1, we define thecategory of triplesT =T(X) as follows:

• Objects ofT are triples (A_{•}, B•, ι), where
– A_{•}∈D( ˜X);

– B• ∈D(S);

– ιis a morphismB• →R⊗_{O}_{˜}A_{•} fromD(S) such that the induced
morphismι^{R}:R⊗_{S}B• →R⊗_{O}_{˜}A_{•} is an isomorphism inD(R).

• A morphism from a triple (A_{•}, B•, ι) to a triple (A^{0}_{•}, B^{0}_{•}, ι^{0}) is a pair
(Φ, φ), where

– Φ :A_{•} → A^{0}_{•} is a morphism from D( ˜X);

– φ:B• →B_{•}^{0} is a morphism fromD(S);

– the diagram

(4.1)

B• ι

−−−−→ R⊗_{O}_{˜}A•

φ

y

y^{1⊗Φ}
B_{•}^{0} ^{ι}

0

−−−−→ R⊗_{O}_{˜}A^{0}_{•}
commutes inD(S).

We define a functor F:D(X) → T(X) setting F(C_{•}) = (π^{∗}C_{•},S⊗_{O}C_{•}, ι),
where ι:S⊗OC• →R⊗_{O}_{˜}(π^{∗}C•) 'R⊗OC• is induced by the embedding
S → R. Just as in Section 1 the following theorem holds (with almost the
same proof, see [6]).

Theorem 4.2. The functor F is a representation equivalence, i.e. it is dense and conservative.

Remark. We do not now whether it isfull, though it seems to be true.

5. Configurations of type A and A˜

As it was shown in [13], even classification of vector bundles is wild for almost all projective curves. Among singular curves the only exceptions are projective configurations of type A and ˜A. These curves only have ordinary double points (so no three components have a common point).

Moreover, in A case irreducible components X1, X2, . . . , Xs and singular
points p1, p2, . . . , ps−1 can be so arranged that pi ∈ Xi ∩Xi+1, while in ˜A
case the components X_{1}, X_{2}, . . . , X_{s} and the singular points p_{1}, p_{2}, . . . , p_{s}
can be so arranged that p_{i} ∈ X_{i} ∩X_{i+1} for i < s and p_{s} ∈ X_{s} ∩X_{1}.
Note that in A case s > 1, while in ˜A case s = 1 is possible: then there

is one component with one ordinary double point (a nodal plane cubic).

These projective configurations are global analogues of nodal rings, and the calculations according Theorem 4.2 are quite similar to those of Section 2. We present here the ˜A case and add remarks explaining which changes should be done for A case.

If s > 1, the normalization of X is just a disjoint union Fs

i=1Xi; for
uniformity, we write X_{1} = ˜X if s = 1. We also denote X_{qs+i} =X_{i}. Note
that X_{i} ' P1 for all i. Every singular point p_{i} has two preimages p^{0}_{i}, p^{00}_{i}
in ˜X; we suppose that p^{0}_{i} ∈ Xi corresponds to the point ∞ ∈ P^{1} and
p^{00}_{i} ∈X_{i+1} corresponds to the point 0∈P^{1}. Recall that any indecomposable
vector bundle over P^{1} is isomorphic to O_{P}1(d) for some d ∈ Z. So every
indecomposable complex fromD( ˜X) is isomorphic either to 0→ O_{i}(d)→0
or to 0 → O_{i}(−lx)→ O_{i} →0, where O_{i} =O_{X}_{i},d∈ Z,l ∈N and x ∈X_{i}.
The latter complex corresponds to the indecomposable sky-scraper sheaf of
length l and support {x}. We denote this complex by C(x,−l, n) and by
C(x, l, n+ 1). The complex 0 → O_{i}(d) → is denoted by C(p^{0}_{i}, dω, n) and
by C(p^{00}_{i−1}, dω, n). As before,n is the unique place, where the complex has
non-zero homologies. We define the symmetric relation ∼for these symbols
setting C(x,−l, n)∼C(x, l, n+ 1) andC(p^{0}_{i}, dω, n)∼C(p^{00}_{i−1}, dω, n).

Let Z^{ω} = (Z⊕ {0})∪Zω, where Zω = {dω|d∈Z}. We introduce an
ordering on Z^{ω}, which is natural onN, on −Nand on Zω, butl < dω <−l
for eachl∈N, d∈Z. Then an analogue of Lemma 2.3 can be easily verified.

Lemma 5.1. There is a morphism of complexes C(x, z, n) → C(x, z^{0}, n)
such that its nth component induces a non-zero mapping on C_{n}(x) if and
only if z≤z^{0} in Z^{ω}.

We introduce the ordered setsE_{x,n} ={C(x, z, n)|z∈Z^{ω}}with the order-
ing inherited from Z^{ω}, We also put F_{x,n} ={(x, n)} and (p^{0}_{i}, n) ∼(p^{00}_{i−1}, n)
for all i, n. Lemma 5.1 shows that the category of triples T(X) can be
again described in terms of the bunch of chains {E_{x,n},F_{x,n}}. Thus we can
describe indecomposable objects in terms of strings and bands just as for
nodal rings. We leave the corresponding definitions to the reader; they are
quite analogous to those from Section 2. If we consider a configuration of
type A, we have to exclude the pointsp^{0}_{s}, p^{00}_{s} and the corresponding symbols
C(p^{0}_{s}, z, n), C(p^{00}_{s}, z, n),(p^{0}_{s}, n),(p^{00}_{s}, n). Thus in this caseC(p^{00}_{s−1}, dω, n) and
C(p^{0}_{1}, dω, n) are not in∼relation with any symbol. It makes possible finite
or one-side infinite full strings, while in ˜Acase only two-side infinite strings
are full. Note that an infinite word must contain a finite set of symbols
(x, n) with any fixed n; moreover there must be n0 such thatn≥n0 for all
entries (x, n) that occur in this word.

If x /∈ S and z /∈Zω, the complex C(x, z, n) vanishes after tensoring by R, so gives no essential input into the category of triples. It gives rise to the n-th shift of a sky-scraper sheaf with support at the regular point x.

Therefore in the following examples we only consider complexes C(x, z, n) with x∈S. Moreover, we confine most examples to the cases= 1 (soX is a nodal cubic). If s >1, one must distribute vector bundles in the pictures below among the components of ˜X.

Example 5.2. 1. First of all, even a classification of vector bundles is non-trivial in ˜A case. They correspond to bands concentrated at 0 place, i.e. such that the underlying cycle wis of the form

(p^{0}_{s},0)∼(p^{00}_{s},0)−C(p^{00}_{s}, d1ω,0)∼C(p^{0}_{1}, d1ω,0)−

−(p^{0}_{1},0)∼(p^{00}_{1},0)−C(p^{00}_{1}, d_{2}ω,0)∼C(p^{0}_{2}, d_{2}ω,0)−

−(p^{0}_{2},0)∼(p^{00}_{2},0)−C(p^{00}_{2}, d_{3}ω,0)∼ · · · ∼C(p^{0}_{s}, d_{rs}ω,0)
(obviously, its length must be a multiple of s, and we can start from
any placep^{0}_{k}, p^{00}_{k}). ThenC_{•}(w, m, λ) is actually a vector bundle, which
can be schematically described as the following gluing of vector bun-
dles over ˜X.

• d1

O O O O O O

O ^{•}

λ

• d2

O O O O O O

O ^{•}

• d3

;;

;; ^{•}

...

;;

;;

• drs

•

Here horizontal lines symbolize line bundles over X_{i} of the super-
scripted degrees, their left (right) ends are basic elements of these
bundles at the point ∞ (respectively 0), and the dashed lines show
which of them must be glued. One must take m copies of each vec-
tor bundle from this picture and make all glueings trivial, except
one going from the uppermost right point to the lowermost left one
(marked by ‘λ’), where the gluing must be performed using the Jor-
dan m×m cell with eigenvalue λ. In other words, if e_{1}, e_{2}, . . . , e_{m}
and f1, f2, . . . , fm are bases of the corresponding spaces, one has to
identifyf_{1} withλe_{1} andf_{k} withλe_{k}+ek−1 ifk >1. We denote this
vector bundle over X by V(d, m, λ), where d = (d1, d2, . . . , drs); it
is of rank mr and of degree mP_{r}

i=1d_{i}. If r = s = 1, this picture
becomes

• d λ

q k _ S M

•

Ifr =m= 1, we obtain all line bundles: they areV((d1, d2, . . . , ds),1, λ) (of degree Ps

i=1d_{i}). Thus the Picard group isZ^{s}×k^{∗}.

InAcase there are no bands concentrated at 0 place, but there are finite strings of this sort:

C(p^{00}_{1}, d1ω,0)−(p^{0}_{1},0)∼(p^{00}_{1},0)−C(p^{00}_{1}, d2ω,0)∼

∼C(p^{0}_{2}, d2,0)−(p^{0}_{2},0)∼(p^{00}_{2},0)−C(p^{00}_{2}, d3,0)∼

· · · ∼C(p^{0}_{s−1}, ds−1ω,0)−(p^{0}_{s−1},0)∼(p^{00}_{s−1},0)−C(p^{00}_{s−1}, d_{s}ω,0)

So vector bundles over such configurations are in one-to-one corre-
spondence with integral vectors (d1, d2, . . . , ds); in particular, all of
them are line bundles and the Picard group is Z^{s}. In the picture
above one has to set r= 1 and to omit the last gluing (marked with

‘λ’).

2. From now ons= 1, so we writep instead ofp_{1}. Let wbe the cycle

(p^{00},1)∼(p^{0},1)−C(p^{0},−2,1)∼C(p^{0},2,2)−(p^{0},2)∼(p^{00},2)−

−C(p^{00},3ω,2)∼C(p^{0},3ω,2)−(p^{0},2)∼(p^{00},2)−C(p^{00},3,2)∼

∼C(p^{00},−3,1)−(p^{00},1)∼(p^{0},1)−C(p^{0},1,1)∼C(p^{0},−1,0)−

−(p^{0},0)∼(p^{00},0)−C(p^{00},−2,0)∼C(p^{00},2,1).

Then the band complex C_{•}(w, m, λ) can be pictured as follows:

•

N N N N NN

N ◦ 2 //•

λ

.. .. .. .. .. .. ..

. ◦

• 3

N N N N NN

N ^{•}

◦ ^{•} 3 //◦ ^{•}

p p p pp pp

• ◦ 1 //•

NN N N N N

N ◦

◦ • 2 //◦ •

Again horizontal lines describe vector bundles over ˜X. Bullets and
circles correspond to the points ∞ and 0; circles show those points,
where the corresponding complex gives no input intoR⊗_{O}_{˜}A_{•}. Hor-
izontal arrows show morphisms inA_{•}; the numbers l inside give the
lengths of factors. Dashed and dotted lines describe glueings. Dashed
lines (between bullets) correspond to mandatory glueings arising from
relations (p^{0}, n) ∼(p^{00}, n) in the word w, while dotted lines (between
circles) can be drawn arbitrarily; the only conditions are that each
circle must be an end of a dotted line and the dotted lines between
circles sitting at the same level must be parallel (in our picture they
are between the 1st and 3rd levels and between the 4th and 5th levels).

The degrees of line bundles in complexesC(x, z, n) withz∈N∪(−N) (they are described by the levels containing 2 lines) can be chosen as d−l and dwith arbitraryd(we setd= 0), otherwise (in the second row) they are superscripted over the line. Thus the resulting complex is

V((−2,3,−3), m,1)−→ V((0,0,−1,−2), m, λ)−→ V((0,0), m,1) (we do not precise mappings, but they can be easily restored).