D.Avritzer*,H.Lange**andF.A.Ribeiro Torsion-freesheavesonnodalcurvesandtriples

Bull Braz Math Soc, New Series 41(3), 421-447

© 2010, Sociedade Brasileira de Matemática

Torsion-free sheaves on nodal curves and triples

D. Avritzer*, H. Lange** and F.A. Ribeiro

Abstract. Let X be a reduced irreducible curve with at most nodes as singularities with normalizationπ: eX→ X. We study the description of torsion free sheaves on X in terms of vector bundles with an additional structure oneX which was introduced by Seshadri.

Keywords: torsion-free sheaves, moduli vector bundles.

Mathematical subject classification: 14H60, 14F05.

1 Introduction

Let X be an irreducible reduced curve over an algebraically closed field with at most ordinary double points as singularities and let π: eX → X denote its normalization. In [10, Ch. 8] Seshadri described torsion free sheaves on X in terms of vector bundles oneX with some additional structure. Let for simplicity X just have one node x withπ⁻¹(x) = {p1,p2}. Then Seshadri showed that there is a canonical bijection between torsion free sheavesF on X and triples (E, (1₁, 1₂), σ )consisting of a vector bundleEoneX, a pair of vector subspaces (1₁, 1₂)with1_i ⊂ E(pi)and an isomorphismσ : 1₁ → 1₂. To the best of our knowledge, this beautiful and clear result has not been applied very much in the literature (apart from [11]and [1]). The main papers on vector bundles on nodal curves apply different methods (see [2], [3], [4]).

The starting point of this paper was to understand Seshadri’s result. We give a new proof and discuss the functoriality of this construction. We introduce the notion of morphisms of such triples and show that the categories of torsion free sheaves onX and the category of these triples oneX are equivalent. More- over, we define stability of triples overeX using the corresponding notion for

Received 19 August 2009.

*Partially supported by CNPq-Grant 301618/2008-9, DAAD-FAPEMIG.

**Partially supported by DAAD-FAPEMIG cooperation agreement.

torsion free sheaves onX. Using this, we study the relation between the stability of a torsion free sheafF onX and the stability of the vector bundleEoccurring in the corresponding triple.

To be more precise, assume thatx1, . . . ,xn are exactly the nodes of X and π⁻¹(xi)= {pi,qi}fori=1, . . . ,n. For any torsion free sheafFof rankr onX there are unique integersai, 0≤ai ≤rsuch thatFx_i 'O^a_xiⁱ⊕m^r−ax_i ⁱ. Moreover there is a unique subsheafE ⊂F of the formE=π_∗E withEa vector bundle of rankr oneX fitting into an exact sequence

0→E→F → ⊕ⁿ_i=1Wx_i →0

whereWxi is the skyscraper sheaf onX with fibre a vector spaceWxi of dimen- sionai at the node xi. Starting with this exact sequence, the vector bundle E is given by

E =π^∗_E/_T

whereT denotes the torsion subsheaf ofπ^∗(_E). We show then that there is a canonical isomorphism

8= ⊕8_i: Ext¹(⊕ⁿ_i=1Wxi,_E)→^'

→ ⊕' _iⁿ₌₁(Homk(Wx_i,E(pi))⊕Homk(Wx_i,E(qi)))

(Note that this description of Ext¹ is different from the description in [10, Ch. 8, Lemme 15].) Given an exact sequence as above, the vector spaces 1₁ and1₂as well as the isomorphismσ are constructed out of the corresponding homomorphismsWx_i → E(pi)andWx_i → E(qi)(see Remark 2.5).

In Section 3, we introduce morphisms of triples and prove the above men- tioned equivalence of categories (see Theorem 3.2). Since the category of tor- sion free sheaves onXadmits kernels and images, so does the category of triples oneX. We describe them and give a criterion for a cokernel to exist.

In Section 4, we translate the notion of stability of torsion free sheaves to the corresponding triples. This is used to relate the stability properties ofF to the stability of the vector bundle E. We show in particular that if E is stable, so are all triples(E, (1₁, 1₂), σ ) which correspond to a vector bundle on X.

We also give an example of an unstable vector bundle on eX admitting stable triples. Finally in Section 5, we give some applications.

Notation: Let X be a curve over k. By anOX-module we always understand a coherentOX-module. Similarly a torsion free OX-module means a coherent torsion freeOX-module. A point of X always means a closed point. Ifx ∈ X

is a point,mx denotes the maximal ideal of the local ringOX,x andm_x denotes the corresponding ideal sheaf inOX. For anyOX-moduleF and anyx ∈ Xwe denote byFx its stalk and byF(x)=Fx/mxFx its fibre at the pointx and we abbreviateOx :=OX,x.

2 Relation to vector bundles on the normalization 2.1 The set up

LetX be an irreducible reduced curve over an algebraically closed field kwith at most ordinary double points as singularities. We assume that X admits ex- actlynordinary double pointsx1, . . . ,xn.

According to [10, Ch. 8, Prop. 2], for any torsion-freeOxi-moduleMof rankr thereisauniquelydeterminednon-negativeintegeraisuchthatM 'O^a_xⁱ_i⊕m^rx^−a_i ⁱ. In particular, for any torsion-free sheafF of rankr and degreedon X and any i = 1, . . . ,n there is an integer ai, uniquely determined with 0 ≤ ai ≤ r such that

Fxi ∼=O^a_xⁱ_i ⊕m^r−a_xi ⁱ. (2.1) This gives surjective homomorphisms

Fxi →Wxi :=k_x^ai_i,

wherekxi 'kdenotes the residue field at the pointxi. If we denote byWxi the skyscraper sheaf concentrated atxi with fibreWxi, we have an exact sequence

0→E →F → ⊕ⁿ_i=1Wxi →0, (2.2) where the kernelEis uniquely determined byF, although the homomorphism F → ⊕ⁿ_i=1Wx_i itself is not. This implies thatF is an extension of⊕ⁿ_i=1Wx_i by a torsion-free sheafEwith

Ex_i ∼=m^rx_i and deg(_E)=deg(_F)− Xn

i=1ai. (2.3) Now consider the normalization map

π :eX→ X

and denote the two points ofπ⁻¹(xi) by pi andqi. According to [10, Ch. 8, Prop. 10], for a torsion free sheaf E of rankr and degree d on X there is a vector bundleEoneX such that

E =π_∗(E) (2.4)

if and only ifExi ∼=m^r_xi for alli. In this caseEis uniquely determined byEand deg(E)=deg(_E)−nr. (2.5) (Note that [10, Ch. 8, Prop. 10] states that deg(E) = deg(_E)+nr. However there is an error in the proof: the degree ofπ_∗OeXis considered to be−1).

We use the following two statements of [11].

Letkxi (respectivelykpi andkqi) be the skyscraper sheaf on X (respectively eX) with fibrek at the pointxi (respectively pi andqi). Then we have

Tor¹O_X kx_i,_OeX

=kp_i ⊕kq_i. (2.6) considered as sheaves oneX. This a consequence of [11, Ch. II, Lemma 2.1].

There exists a locally free sheafGof rankrand degree degF+nr−P_n

i=1ai

onX such that

F ⊂G. (2.7)

and the quotient^G/F is supported at the nodes xi. This is [11, Ch. II, Lem- ma 2.3].

Proposition 2.1. LetF be a torsion free sheaf of rank r on X satisfying(2.1) and letT denote the torsion subsheaf ofπ^∗_F. Then

deg(π^∗_F)=degF +nr− Xn

i=1

ai and degT =2 nr− Xn

i=1

ai

! .

Proof. LetGbe the locally free sheaf of (2.7). Pulling back the exact sequence 0→F →G→G/_F →0 byπ, we get the exact sequence

0→T →π^∗F →π^∗G→π^∗(G/F)→0. withT =Tor¹O_X G/_F,_OeX

. Since

degG/_F =nr− Xn

i=1

ai

we get degπ^∗(_G/_F)=2 nr−P_n

i=1ai

. This implies deg(π^∗_F/_T) = degπ^∗_G−degπ^∗(_G/_F)

= degF+nr− Xn

i=1

ai −2 nr− Xn

i=1

ai

!

= degF− nr− Xn

i=1

ai

!

On the other hand, from (2.6) we deduce

degT =2 degG/_F =2 nr− Xn

i=1

ai

! .

This implies also the first assertion.

For the following corollary see [10, p.175, l.3].

Corollary 2.2. LetF be a torsion free sheaf of rank r satisfying(2.1) and let E = π_∗E its subsheaf as in(2.4). IfT denotes the torsion subsheaf ofπ^∗_E, then we have

π^∗_E/_T 'E.

Proof. As an immediate consequence of Proposition 2.1 and equation (2.5) we get

deg π^∗_E/_T

=degE. (2.8)

Now consider the canonical mapπ^∗_E →π^∗_E/_T. Adjunction gives a map π_∗E =E →π_∗ π^∗_E/_T

which is of maximal rank outside the nodes xi. Since E is a vector bundle on eX, we have

Hom π_∗E, π_∗(π^∗_E/_T)

= H⁰ π_∗E^∗⊗π_∗(π^∗_E/_T)

⊂ H⁰ π_∗(E^∗⊗π^∗_E/_T) ' H⁰ E^∗⊗π^∗_E/_T

= Hom E, π^∗_E/_T .

Hence we get a homomorphismE →π^∗_E/_T which clearly is of maximal rank.

Hence it is injective and thus an isomorphism by (2.8).

2.2 Extensions

Recall that the extensions 0 → E → G → ⊕ⁿ_i=1Wxi → 0 are classified by the group Ext¹(⊕ⁿ_i=1Wx_i,E).The next proposition gives a characterization of Ext¹(⊕ⁿ_i=1Wxi,_E). Since Ext¹is additive, it suffices to do this for each nodexi

separately. The following proposition appeared in [10, Ch. 8, Lemme 12] in a slightly different form.

Proposition 2.3. LetE = π_∗E with E a vector bundle of rank r oneX. Then there is a canonical isomorphism

8= ⊕8_i: Ext¹ ⊕ⁿ_i=1Wxi,_E '

→

→ ⊕' ⁿ_i=1 Homk(Wxi,E(pi))⊕Homk(Wxi,E(qi))

whereWxi is the skyscraper sheaf concentrated at xi with fibre Wxi := k^a_xiⁱ for integers ai, 0≤ai ≤r.

For the proof we need the following lemma.

Lemma 2.4. There is a canonical isomorphism HomO_X m_xi,_E

'HomO_eX OeX,E(Di) .

where Di := pi +qi and as usual E(Di):= E⊗OeX(Di).

Proof. LetTi denote the torsion subsheaf ofπ^∗(mx_i). Certainly the divisor Di

induces a canonical isomorphism π^∗ mx_i

/_Ti 'OeX(−Di) .

Now the exact sequence 0 →Ti →π^∗(m_xi)→ π^∗(m_xi)/_Ti →0 induces an isomorphism

HomOeX π^∗(m_xi)/Ti,E

→HomOeX π^∗(m_xi),E

since HomOeX(_Ti,E) = 0. Using moreover adjunction, we finally get the fol- lowing canonical isomorphisms

HomOX(m_xi,_E) ∼=HomOeX π^∗(m_xi),E

∼=HomOeX π^∗(m_xi)/_Ti,E

∼=HomOeX OeX(−Di),E

∼=HomOeX OeX,E(Di) .

Proof of Proposition 2.3. We may assumeh¹(_E) = 0, since tensorizingE andWxi by a line bundle changes both sides of the equation only by a canonical isomorphism. Using this, the exact sequence

0→Wxi ⊗k m_xi →Wxi ⊗kOX →Wxi →0

induces an exact sequence

0→HomOX(Wxi ⊗OX,_E)→HomOX(Wxi ⊗m_xi,_E)→

→Ext¹(_Wxi,_E)→0, (2.9) where the last 0 usesh¹(E)=0.

The canonical sequence 0 → E → E(Di) → E(pi)⊕ E(qi) → 0 (here we identify the fibresE(pi)andE(qi)with the corresponding skyscraper sheaf concentrated at pi andqi respectively) induces an exact sequence

0→HomOeX(Wx_i ⊗OeX,E)→HomOeX(Wx_i ⊗OeX,E(Di))→

→HomOeX(Wx_i ⊗OeX,E(pi)⊕E(qi))→0 (2.10) where we used Ext¹(Wxi⊗OeX,E)=Wxi⊗H¹(eX,E)=Wxi⊗H¹(X,_E)=0.

Moreover, by adjunction we have a canonical isomorphism

HomO_X(Wx_i⊗OX,_E)'HomOeX(π^∗(Wx_i⊗OX),E)=HomOeX(Wx_i⊗OeX,E).

Using this, Lemma 2.4 and the exact sequences (2.9) and (2.10), we obtain the following diagram with exact rows

0→HomO_X(Wx_i ⊗OX,_E)→HomO_X(Wx_i ⊗mx_i,_E)→Ext¹(_Wxi,_E)→0

ko ko

0→HomOeX(Wxi ⊗OeX,E)→HomOeX(Wxi ⊗OeX,E(Di))→^θi (2.11)

θ_i

→HomOeX(Wx_i ⊗OeX,E(pi)⊕E(qi))→0,

Since this diagram is certainly commutative, it induces a canonical isomorphism Ext¹(_Wxi,_E)'HomOeX(Wxi ⊗OeX,E(pi)⊕E(qi)).

Finally, observe that HomOeX(OeX,E(pi)) = Homk(k,E(pi))and similarly for qi. Hence we can identify

HomOeX(Wxi⊗OeX,E(pi)⊕E(qi))=Homk(Wxi,E(pi))⊕Homk(Wxi,E(qi)).

Combining both isomorphisms completes the proof of the proposition.

Remark 2.5. According to the proof and in particular diagram (2.11), the image of an extension(ei): 0→E → F →Wx_i → 0 under the map8_i is given as

follows: choose a preimageψ_ei ∈HomOX(Wxi⊗m_xi,_E)of(ei)∈Ext¹(_Wxi,_E) and consider it as an element in HomOeX(Wxi ⊗OeX,E(Di)). Then

8_i(ei)=θ_i(ψ_ei).

Conversely, given a pair(α_i, β_i)∈ Homk(Wx_i,E(pi))⊕Homk(Wx_i,E(qi)), the corresponding extension8⁻¹_i (α_i, β_i) ∈Ext¹(_Wxi,_E)is constructed as fol- lows: choose a preimage ψ_i of (α_i, β_i) under the map θ_i, considered as an element of HomO_X(Wx_i ⊗mx_i,_E). Then 8⁻¹_i (α_i, β_i) is the push-out of the canonical sequence definingWxi byψ_i:

0 Wxi ⊗m_xi

ψ_i

Wxi ⊗OX Wxi 0

0 ^{E Fi Wx}i 0

It follows from Proposition 2.3 that8_i(ei)and8_i⁻¹(α_i, β_i)do not depend on the choices of the preimages.

The extension8⁻¹((α₁, β₁), . . . , (α_n, β_n)) ∈ Ext¹(⊕_iⁿ₌₁Wxi,_E)is then the sum of the extensions 0→E→Fi→Wx_i→0 in the group Ext¹(⊕ⁿ_i=1Wx_i,_E), where Ext¹(_Wxi,_E)is considered as a subgroup of Ext¹(⊕ⁿ_i=1Wxi,_E).

2.3 Torsion free extensions

Letx denote any of the nodesxiofXand let pandqbe the points ofeX abovex.

Recall that we denote byWxakx-vector space of dimensiona, 1≤a≤rand by Wxthe skyscraper sheaf onXwith fibreWxatx. TheOx-module Ext¹O_x(Wx,_Ex) classifies the extensions 0→Ex →Fx →Wx →0 of modules over the local ringOx. The following lemma shows that every such module is the restriction of a unique exact sequence 0→E →F →Wx →0.

Lemma2.6.ThereisacanonicalisomorphismExt¹(_Wx,_E)'Ext¹_Ox(Wx,_Ex). Proof. The edge homomorphism of the local-global spectral sequence (see [5, (4.2.7)]) is an isomorphism

Ext¹(_Wx,_E)'H⁰(X,Ext¹_OX(_Wx,_E)),

sinceHⁱ(X,Hom_OX(_Wx,_E))=0 fori =1 and 2. This implies the assertion, since the sheaf Ext¹_OX(_Wx,_E)is a skyscraper sheaf with fibre Ext¹_Ox(Wx,_Ex)

concentrated at the pointx.

Combining the isomorphisms of Proposition 2.3 and Lemma 2.6, we get the following description of the extensions ofOx-modules ofWxbyEx:

Let _Oe_x :=Op∩Oq

(intersection in the function field of X) denote the normalization ofOx. It is a semilocal ring with two maximal idealsmpandmqand we have

mx =mp∩mq. Moreover we denote by

Ex := E⊗OeX_Oe_x and E(D)_x := E(D)⊗OeX_Oe_x

the_Oe_x-modules defined byEandE(D), whereDdenotes the divisor p+qon eX. As in the proof of Proposition 2.3, the exact sequences 0→ Wx ⊗kmx → Wx ⊗kOx → Wx →0 and 0→ Ex → E(D)_x → E(p)⊕E(q)→0 induce the following local version of diagram (2.11).

0→HomO_x(Wx⊗Ox,_Ex)→HomO_x(Wx ⊗mx,_Ex)→Ext¹O_x(Wx,_Ex)→0

ko ko

0→Hom_Oe_x(Wx⊗_Oe_x,Ex)→Hom_Oe_x(Wx⊗_Oe_x,E(D)_x)→^θ (2.12)

→θ Hom_Oe_x(Wx ⊗_Oe_x,E(p)⊕E(q))→0 ko

Homk(Wx,E(p))⊕Homk(Wx,E(q)).

Now consider(α, β) ∈ Homk(Wx,E(p))⊕Homk(Wx,E(q)). Let (eα,β)e de- note the corresponding element in Hom_Oe_x(Wx ⊗_Oe_x,E(p)⊕E(q)). Choose a preimage ψ of (eα,β)e under the map θ and consider it as an element of HomOx(Wx ⊗mx,_Ex). Then the extension in Ext¹_Ox(_Wx,_Ex) corresponding to(α, β)is the pushout of the canonical sequence definingWx byψ:

0 Wx⊗mx ψ

Wx ⊗Ox Wx 0

0 ^{Ex Fx} Wx 0.

(2.13)

Recall thatEx 'm^r_x. Hence, fixing isomorphismsEx 'm^r_xandWx⊗mx 'm^a_x, any homomorphismψ: Wx⊗mx →Exis given by a matrix

A=(α_{i j})∈ M(r×a,_Oe_x).

We say thatψhas birank(b1,b2), if rkA mod mp =b1and rkA mod mq = b2. This definition does not depend on the choice of the isomorphisms. Certainly ψhas birank(b1,b2)if and only if rkα =b1and rkβ =b2.

Proposition 2.7. Let0→Ex →Fx →Wx →0be the extension correspond- ing to the pair(α, β) ∈ Homk(Wx,E(p))⊕Homk(Wx,E(q)). The following statements are equivalent:

(1) Fx is torsion-free, (2) Fx 'O^a_x ⊕m^r_x^−a, (3) αandβ are injective.

Proof. The equivalence of (1) and (2) is clear, we have already used it. We have to show that (2) is equivalent to (3). Nowαandβare injective if and only if they are both of ranka. As we saw just before the proposition, this is the case if an only ifψis of birank(a,a). SinceWxis of dimensiona, this is the case if and only ifψis injective. From diagram (2.13) we deduce that this is the case if and only if the push-outWx⊗Ox →Fxis injective. But this is injective if and

only ifFxis torsion-free.

As an immediate consequence we get,

Corollary 2.8. Let(e): 0 → E → F → ⊕ⁿ_i=1Wxi → 0be an extension as in subsection2.2and let8((e))=((α₁, β₁), . . . , (α_n, β_n))∈ ⊕ⁿ_i=1(Homk(Wx_i, E(pi))⊕Homk(Wxi,E(qi)))(see Proposition2.3). Then the following condi- tions are equivalent:

(1) F is torsion free;

(2) α_i andβ_i are injective for i =1, . . . ,n.

2.4 Triples oneX

In this subsection we outline the relation between torsion free sheaves onX and vector bundles oneX with an additional structure.

Given a torsion free sheafF on X, letE =π_∗(E)be its subsheaf such that 0→E →F → ⊕ⁿ_i=1Wx_i →0 is exact, where⊕ⁿ_i=1Wx_i is the torsion sheaf as above. Let

((α₁, β₁), . . . , (α_n, β_n))∈ ⊕ⁿ_i=1(Homk(Wx_i,E(pi))⊕Homk(Wx_i,E(qi)))

denote the corresponding element according to Proposition 2.3. Then by Corol- lary 2.8 the homomorphismsα_i andβ_i are all injective and we can consider the following vector spaces:

1ⁱ₁:=Imα_i ⊂E(pi), 1ⁱ₂:=Imβ_i ⊂E(qi), 1₁:= ⊕ⁿ_i=11ⁱ₁, 1₂:= ⊕_iⁿ₌₁1ⁱ₂.

Combining the inverse of the isomorphismα_i onto its image withβ_i we obtain an isomorphism

σ_i =β_i ◦α_i⁻¹: 1ⁱ₁→1ⁱ₂. Finally we denote the direct sum by

σ := ⊕ⁿ_i=1σ_i: 1₁→1₂.

Summing up, we associated to the sheafF on X the object(E, (1₁, 1₂), σ )in a canonical way. We call these objectstriples oneX in the sequel.

Conversely, given a triple(E, (1₁, 1₂), σ )oneX, we denote fori =1,∙ ∙ ∙,n Wxi :=1ⁱ₁

and ifα_i: Wx_i ,→ E(pi)is the natural inclusion, we defineβ_i: 1ⁱ₂→ E(qi)to be the composition

Wxi =1ⁱ₁→^σi 1ⁱ₂,→ E(qi).

According to Proposition 2.3 there is a unique extension 0 → π_∗E → F →

⊕ⁿ_i=1Wxi → 0 associated to then pairs ((α₁, β₁), . . . , (α_n, β_n)), where F is torsion free of rankr according to Corollary 2.8.

Summing up, we associated to the triple(E, (1₁, 1₂), σ )the torsion free sheaf F in a unique way. The following theorem is due to Seshadri (see [10, p. 178, Theorem 17]).

Theorem 2.9.Given intergers a1, . . . ,anand r such that0≤ai ≤r, there is a canonical bijection between the sets:

(1) of isomorphism classes of torsion free sheavesF of rank r and degree d on X such that for i =1,∙ ∙ ∙,n,

Fx_i ∼= ⊕^a_i=1ⁱ Ox_i ⊕ ⊕⁽_i=1^r−ai)mx_i (2.14) and

(2) of isomorphism classes of triples(E, (1₁, 1₂), σ ), where

• E is a vector bundle of degree d−nr −P_n

i=1ai and rank r oneX,

• for j =1and 2, 1_j = ⊕ⁿ_i=11ⁱ_j with1ⁱ₁and1ⁱ₂vector subspaces of dimension ai of E(pi)and E(qi)respectively,

• σ = ⊕ⁿ_i=1σ_i with isomorphismsσ_i: 1ⁱ₁→1ⁱ₂.

Proof. It is easy to see that the maps of one set to the other set given above are inverse to each other. The statement relating the degrees of E andF is a

consequence of equations (2.3) and (2.5).

As a direct consequence we get the following corollary.

Corollary 2.10. LetF correspond to the triple (E, (1₁, 1₂), σ ) as in Theo- rem2.9. ThenF is a vector bundle if and only if a1= ∙ ∙ ∙ =an=r.

3 The functor9

Let(E, (1₁, 1₂), σ )and(E⁰, (1⁰₁, 1⁰₂), σ⁰)be triples oneX. Ahomomorphism g˜: (E, (1₁, 1₂), σ )→(E⁰, (1⁰₁, 1⁰₂), σ⁰)

of tripleson eX is by definition a homomorphism g: E → E⁰ of the under- lying vector bundles satisfying g(pi)(1ⁱ₁) ⊂ 1⁰₁ⁱ and g(qi)(1ⁱ₂) ⊂ 1⁰₂ⁱ for i=1, . . . ,nsuch that the following diagram commutes

1₁= ⊕i1ⁱ₁

⊕ig(p_i)

σ 1₂= ⊕i1ⁱ₂

⊕ig(q_i)

1⁰₁= ⊕i1⁰₁^{i σ0} 1⁰₂= ⊕i1⁰₂ⁱ.

(3.1)

With this notion of morphisms the set of triples oneX forms a category which we denote byT R.

Let f: F → F⁰ denote a homomorphism of torsion free sheaves on X and denote by

9(_F):=(E, (1₁, 1₂), σ ) and 9(_F⁰):=(E⁰, (1⁰₁, 1⁰₂), σ⁰) the corresponding triples according to Theorem 2.9. LetE ⊂ F andE⁰ ⊂ F⁰ be the subsheaves defined in (2.2). It is easy to see that f mapsEintoE⁰. We then see from Corollary 2.2 that

π^∗(f|E)/torsion: E =π^∗E/T →π^∗E⁰/T⁰ =E⁰

is a homomorphism of vector bundles. Moreover we have,

Lemma 3.1. The homomorphism f induces a homomorphism of triples 9(f): (E, (1₁, 1₂), σ )→(E⁰, (1⁰₁, 1⁰₂), σ⁰)

oneX.

Proof. Consider the extension (2.2) defined byF and let 0 →E⁰ → F⁰ →

⊕iW_x⁰_i →0 be the corresponding extension forF⁰. Since f mapsEintoE⁰we get the following commutative diagram

0 ^E

f|E

F

f

⊕iWx_i

⊕if_xi

0

0 E⁰ F⁰ ⊕iW_x⁰_i 0.

(3.2)

Let((α₁, β₁), . . . , (α_n, β_n))∈ ⊕i(Hom(Wxi,E(pi))⊕Hom(Wxi,E(qi)))and similarly

((α₁⁰, β₁⁰), . . . , (α_n⁰, β_n⁰))∈ ⊕i(Hom(W_x⁰_i,E⁰(pi))⊕Hom(W_x⁰_i,E⁰(qi))) denote then-tuples of pairs of homomorphisms associated to the extensions of diagram (3.2) according to Proposition 2.3. By definition the following diagrams commute fori=1, . . . ,n:

Wx_i f_xi

α_i

E(pi)

9(f)(p_i)

Wx_i f_xi

β_i

E(qi)

9(f)(q_i)

W_x⁰_i ^α0i E⁰(pi) W_x⁰_i ^βi0 E⁰(qi).

which implies9(f)(pi)(1ⁱ₁)⊂1⁰₁ⁱ and9(f)(qi)(1ⁱ₂)⊂1⁰₂ⁱ fori =1, . . . ,n.

It remains to show that the diagram (3.1) commutes, but this is straightfor- ward using Proposition 2.7 and σ = ⊕iσ_i = ⊕i β_i ◦ α_i⁻¹

and similarly

forσ⁰.

IfT Sdenotes the category of torsion free sheaves on X, then clearly9 is a functor from the categoryT Sto the categoryT Rof triples oneX.

Theorem 3.2.The functor9: T S→T Ris an equivalence of categories.

Proof. According to Theorem 2.9 the functor 9 is a bijection on the ob- jects. We have to show that it is a bijection on the sets of morphisms. So let g˜: (E, (1₁, 1₂), σ ) → (E⁰, (1⁰₁, 1⁰₂), σ⁰) be a homomorphism of triples on eX. Let F and F⁰ denote the torsion free sheaves on X with 9(_F) = (E, (1₁, 1₂), σ )and9(_F⁰)=(E⁰, (1⁰₁, 1⁰₂), σ⁰).

DefineWxi :=1ⁱ₁, W_x⁰_i :=1⁰₁ⁱ. Soα_i andα⁰_i are its canonical inclusions into E(pi)andE⁰(pi)respectively. Similarlyβ_i =ι_i ◦σ_i andβ_i⁰=ι⁰_i ◦σ_i⁰, whereι_i andι⁰_iare the canonical inclusions1ⁱ₂,→ E(qi)and1⁰₂ⁱ ,→ E⁰(qi)respectively.

Then diagram (3.1) implies that the following diagrams are commutative Wxi

g(pi) α_i

E(pi)

g(pi)

Wxi

g(pi) β_i

E(qi)

g(qi)

W_x⁰_i ^αi0 E⁰(pi) W_x⁰_i ^β0i E⁰(qi)

(3.3)

fori=1, . . . ,n.

Now choose fori = 1, . . . ,n preimagesψ_i and ψ_i⁰ of(α_i, β_i)and (α_i⁰, β_i⁰) under the mapθ_i of diagram (2.11) respectively and consider them as homo- morphismsψ_i : Wxi ⊗m_xi → E = π_∗E andψ_i⁰: W_x⁰_i ⊗m_xi → E⁰ = π_∗E⁰. According to remark 2.5, ^F is given by the sum in Ext¹(⊕ⁿ_i=1Wx_i,E) of the push-outs 0 →E → Fi → Wxi →0 byψ_i of the canonical exact sequences 0→ Wxi ⊗m_xi → Wxi ⊗OX →Wxi → 0 defining the sheafWxi. Similarly F⁰ is defined. For every i = 1, . . . ,n we obtain the following commutative diagram:

Wxi⊗mxi Wxi⊗OX Wxi

Wxi0 ⊗mxi Wxi0 ⊗OX Wxi0

E Fi Wxi

E0 Fi0 W0

xi

where the left hand vertical maps areψ_i andψ_i⁰ and the commutativity of the left hand vertical square is a consequence of the commutativity of diagram (3.3).

The universal property of the push-out gives us a homomorphism fi: Fi →F_i⁰