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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


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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


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”.



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 = (Cn, dn) that Imdn ⊆ radCn−1

for all n. If C and C0 are two minimal complexes, they are isomorphic in D(A) if and only if they are isomorphic as complexes; moreover, any morphismC →C0 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→P0−→α P →0, where P and P0 are projective modules andα is a monomorphism with Imα⊆radP (maybe P0 = 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


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⊗HA fromD(S) such that the induced morphismιR:R⊗SB →R⊗HA is an isomorphism inD(R).

• A morphism from a triple (A, B, ι) to a triple (A0, B0, ι0) is a pair (Φ, φ), where

– Φ :A →A0 is a morphism fromD(H);

– φ:B →B0 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


B ι

−−−−→ R⊗HA


 y

 y1⊗Φ B0 −−−−→ι0 R⊗HA0 commutes inD(S).

One can define a functor F : D(A) → T(A) setting F(C) = (H⊗A C,S⊗AC, ι), whereι:S⊗AC →R⊗H(H⊗AC)'R⊗ACis 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(C0)is of the formF(γ)for some γ :C →C0;

• 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⊗HA is also a complex with zero differential. We have an exact sequence of complexes

0−→IA −→A −→R⊗HA −→0.

Together with the morphismι:B →R⊗HAit gives rise to a commutative diagram in the category of complexes Com(A-mod)

0 −−−−→ IA −−−−→ A −−−−→ R⊗HA −−−−→ 0





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 ιR is an isomorphism,IA=IC. It implies thatCconsists of projectiveA-modules and H⊗AC 'A, wherefromT 'FC.

Let now (Φ, φ) : FC → FC0. We suppose again that both C and C0 are minimal, while Φ : H⊗AC → H⊗AC0 and φ:S⊗AC → S⊗AC0 are morphisms of complexes. Then the diagram (1.1) is commutative in the category of complexes, so Φ(C)⊆C0 and Φ induces a morphismγ :C → C0. 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) = C0, i.e. Imγ =C0. 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) lengthA(H⊗AU)≤2 for every simple left A-moduleU and lengthA(V ⊗AH)≤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[[x1]]×k[[x2]] 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/(x2, y2) 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


0 t 0 0

, y 7→

0 0 1 0


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









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 (aij) with a12(0) = a13(0) = a23(0) = a32(0) = 0. The arrows of the diagram correspond to the following matrices:

x+ 7→te12, x7→e21, y+ 7→te13, y7→e31,

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 (aij) witha12(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 ofZ22×Mat(2,Z2) consisting of all triples

a, b,

c1 2c2 c3 c4

with a≡c1(mod 2) andb≡c4(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=Z22×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⊗AUi'Vj for somej andVj does not occur as a direct summand inH⊗AUk fork6=i;

2) H⊗AUi 'Vj⊕Vj0 (j6=j0) and neitherVj norVj0 occur inH⊗AUk fork6=i;

3) there are exactly two indicesi6=i0such thatH⊗AUi'H⊗AUi0 'Vj

and Vj does not occur inH⊗AUk fork /∈ {i, i0}.

We denote by Hj the indecomposable projective H-module such that Hj/RHj ' Vj. Since H is a semi-perfect hereditary order, any indecom- posable complex from D(H) is isomorphic either to 0 → Hk −→φ Hj → 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 = lengthHCokerφ. 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 Hk 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 ofHj 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, l0, n), which is an isomorphism on the n-th components, if and only if l≤l0 in Z˜. Otherwise the n-th component of any homomorphism C(j, l, n) → C(j, l0, n) is zero modulo R.

We transfer the ordering from ˜Zto the set Ej,n= n

C(j, l, n)|l∈Z˜ o

, so the latter becomes a chain with respect to this ordering. We also denote by Fj,n the set {(i, j, n)|Vj is a direct summand of H⊗AUi}. It has at most two elements. We always consider Fj,n with trivial ordering. Then a triple (A, B, ι) from the category T(A) is given by homomorphisms φijnjln :di,j,nUi →rj,l,nVj, where (i, j, n)∈Fjn, the leftUi comes fromBnand


the right Vj comes from direct summands rj,l,nC(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 =rk,l,n+1. We present φijnjln by its matrix Mjlnijn. Then Lemma 2.3 implies the following

Proposition 2.4. Two sets of matrices n

Mjlnijn o

and n

Njlnijn 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, simultaneouslyMjlnijn7→MjlnijnS for allj, l such that(ijn)∈Fj,n, where S is an invertible matrix of appropriate size.

2) For any given values of j, l, n, simultaneously Mjlnijn 7→ S0Mjlnijn for all (i, j, n)∈ Fjn and Mk,−l,n−sgni,k,n−sgnll 7→ S0Mk,−l,n−sgni,k,n−sgnll for all (i, k, n− sgnl) ∈ Fk,n−sgnl, where S0 is an invertible matrix of appropriate size and C(j, l, n) ∼ C(k,−l, n−sgnl). If l = ∞, it just means Mj∞nijn 7→SMj∞nijn .

3) For any given values of j, l0 < l, n, simultaneously Mjlnijn 7→ Mjlnijn+ RMjlijn0n for all (i, j, n) ∈ Fj,n, where R is an arbitrary matrix of ap- propriate size. Note that, unlike the preceding transformation, this one does not touch the matrices Mk,−l,n−sgni,k,n−sgnll 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 Ejn,Fjn 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 Vj⊕Vk'H⊗AUi for somei;

(d) (j, n)∼(j, n) if Vj 'H⊗AUi'H⊗AUi0 for somei0 6=i.

2. We define anX-wordas a sequencew=x1r1x2r2x3. . . rm−1xm, where xk∈X, rk∈ { −,∼ }such that

(a) xkrkxk+1 in Xfor 1≤k < m;

(b) rk6=rk+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) x1 6∼y for each y6=x1; (c) xm6∼z for eachz6=xm.

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


4. A wordwis calledsymmetric, ifw=w, wherew =xmrm−1xm−1. . . r1x1 (the inverse word), and quasisymmetric, if there is a shorter word v such thatw=v∼v ∼ · · · ∼v ∼v.

5. We call the end x1 (xm) of a word w special if x1 ∼x1 and r1 = − (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 thatr1 =rm−1 =∼andxm−x1. Such a cycle is called non-periodic if it cannot be presented in the formv−v− · · · −vfor a shorter cyclev. For a cyclewwe setrm =−, 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 = x1r1x2r2. . . (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⊗AA'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 xlj. Consider several examples of strings and bands.

Example 3.1. 1. Let wbe the cycle




−(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

H2 x2 //H2

H1 x


1 //



x42 //H2


x1 //


H2 x


2 //


H1 x


1 //H1






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


A y //A




pp pp pp pp pp pp






pp pp pp pp pp

pp y4 //A




pp pp pp pp pp pp

Here each column presents direct summands of a non-zero compo- nent Cn (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


λx2 0 y3

−−−−−→ A⊕A⊕A

y 0

x2 y4

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 aJm(λ), where Jm(λ) is the Jordanm×m cell with eigenvalue λ. So we obtain the complex


x2Jm(λ) 0 y3E

−−−−−−−−−→ mA⊕mA⊕mA

yE 0

x2E y4E

0 xE

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




∼(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 //






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



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

. . . A−→x2 A−→y A−→x2 A−→y A−→0.

Its homologies are not left bounded, so it does not belong toDb(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 Hj is isomorphic to Hk, 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(j0,−l, n−sgnl), where j0 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:

H1→H1 − by tre11 (colength 2r), H1→H2 − by tre12 (colength 2r−1), H2→H1 − by tre21 (colength 2r+ 1), H2→H2 − by tre22 (colength 2r).

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

tre11 − by (xy)r, tre22 − by (yx)r, tre12 − by (xy)r−1x, tre21 − by (yx)ry.


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)∼



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

mA xyE //mA


(xy)n 2xE

nn n


nn n

(yx)2E //mA

xyxJnn m(λ)

nn n


nn nn

2. Letw be the word



Then the string complexC(w) is A e21 //






A te21 //A

3. The factorA/Ris described by the infinite string complex C(w) . . . e21 //A te12 //A e21 //A.

. . . te12 //A e21 //A


pp pp pp


pp pp

The corresponding wordw is

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


∼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 H1 (the first column) and H2 (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⊗AA1'H1 andH⊗AA2 ' H⊗AA3 '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(j0,−l, n−sgnl) (j0 6=j) if l is odd.

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


Example 3.3. 1. Consider the special word w:



−(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

H2 2 //


H2 4 //H2



2 //H2

H1 1 //H2

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 inEj,n) to the smaller one. When we construct the corresponding complex of A- modules, we replace each H2 by A2 and A3 starting with A2 (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 A2 2 //A2



pp pp p


pp pp p

4 //









pp pp p


pp pp p

A2 2 //

2 **





1 //A3

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


2. Letw be the bispecial word



−(2,1)∼(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 //
















2 //mA3

mA3 4 //mA2



ll ll ll


ll ll ll

M2 //aA2⊕bA3

wherea= [(m+ 1)/2], b= [m/2], soa+b=m. (The change ofδ1, δ2 transpose A2 and A3 at the ends.) All arrows are just αlE, where αl is defined by the colengthl, except of the “end” matrices Mi. To calculate the latter, write αlE for one of them (say, M1) and αlJ 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


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

, M26

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 M1 and rows for M2 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






mA2 2 //







nn nn n


nn nn n

4 //












mA3 6 //mA2



nn nn nn


nn nn n

4λ //mA3

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

4. The projective resolution of the simpleA-moduleU1 is A2 1 //A1



pp pp p


pp pp p

1 //A3 p1

pp pp p


pp pp p

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


The projective resolution ofU2 (U3) is A1 →A2 (respectively A1 → A3), 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) ={y1, y2, . . . , ym}, the image of OX,pinQm


i contains Qm

i=1mi, wheremi is the maximal ideal ofOX,y˜


We denote by S = {p1, p2, . . . , ps} the set of singular points of X and by ˜S ={y1, y2, . . . , yr}its preimage in ˜X. We also putO =OX, ˜O =OX˜ and denote by J the conductor of ˜O in O, i.e. the maximal sheaf of πO-˜ ideals contained in O. Set S = O/J and R = πO/J '˜ O/π˜ −1J. 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(pi) and R=Qr


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 withπC.

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 morphismιR:R⊗SB →R⊗O˜A is an isomorphism inD(R).

• A morphism from a triple (A, B, ι) to a triple (A0, B0, ι0) is a pair (Φ, φ), where

– Φ :A → A0 is a morphism from D( ˜X);

– φ:B →B0 is a morphism fromD(S);

– the diagram


B ι

−−−−→ R⊗O˜A


 y

 y1⊗Φ B0 ι


−−−−→ R⊗O˜A0 commutes inD(S).

We define a functor F:D(X) → T(X) setting F(C) = (πC,S⊗OC, ι), where ι:S⊗OC →R⊗O˜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 X1, X2, . . . , Xs and the singular points p1, p2, . . . , ps can be so arranged that pi ∈ Xi ∩Xi+1 for i < s and ps ∈ Xs ∩X1. 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 X1 = ˜X if s = 1. We also denote Xqs+i =Xi. Note that Xi ' P1 for all i. Every singular point pi has two preimages p0i, p00i in ˜X; we suppose that p0i ∈ Xi corresponds to the point ∞ ∈ P1 and p00i ∈Xi+1 corresponds to the point 0∈P1. Recall that any indecomposable vector bundle over P1 is isomorphic to OP1(d) for some d ∈ Z. So every indecomposable complex fromD( ˜X) is isomorphic either to 0→ Oi(d)→0 or to 0 → Oi(−lx)→ Oi →0, where Oi =OXi,d∈ Z,l ∈N and x ∈Xi. 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 → Oi(d) → is denoted by C(p0i, dω, n) and by C(p00i−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(p0i, dω, n)∼C(p00i−1, dω, n).

Let Zω = (Z⊕ {0})∪Zω, where Zω = {dω|d∈Z}. We introduce an ordering on Zω, 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, z0, n) such that its nth component induces a non-zero mapping on Cn(x) if and only if z≤z0 in Zω.

We introduce the ordered setsEx,n ={C(x, z, n)|z∈Zω}with the order- ing inherited from Zω, We also put Fx,n ={(x, n)} and (p0i, n) ∼(p00i−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 {Ex,n,Fx,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 pointsp0s, p00s and the corresponding symbols C(p0s, z, n), C(p00s, z, n),(p0s, n),(p00s, n). Thus in this caseC(p00s−1, dω, n) and C(p01, 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

(p0s,0)∼(p00s,0)−C(p00s, d1ω,0)∼C(p01, d1ω,0)−

−(p01,0)∼(p001,0)−C(p001, d2ω,0)∼C(p02, d2ω,0)−

−(p02,0)∼(p002,0)−C(p002, d3ω,0)∼ · · · ∼C(p0s, drsω,0) (obviously, its length must be a multiple of s, and we can start from any placep0k, p00k). ThenC(w, m, λ) is actually a vector bundle, which can be schematically described as the following gluing of vector bun- dles over ˜X.















Here horizontal lines symbolize line bundles over Xi 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 e1, e2, . . . , em and f1, f2, . . . , fm are bases of the corresponding spaces, one has to identifyf1 withλe1 andfk withλek+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 mPr

i=1di. 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=1di). Thus the Picard group isZs×k.

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

C(p001, d1ω,0)−(p01,0)∼(p001,0)−C(p001, d2ω,0)∼

∼C(p02, d2,0)−(p02,0)∼(p002,0)−C(p002, d3,0)∼

· · · ∼C(p0s−1, ds−1ω,0)−(p0s−1,0)∼(p00s−1,0)−C(p00s−1, dsω,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 Zs. In the picture above one has to set r= 1 and to omit the last gluing (marked with


2. From now ons= 1, so we writep instead ofp1. Let wbe the cycle





Then the band complex C(w, m, λ) can be pictured as follows:


N2 //


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





3 //

p p p pp pp

1 //



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 (p0, n) ∼(p00, 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).



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