Bull. Soc. math. France, 125, 1997, p. 529–545.

LINEARIZATION OF ACTIONS AND THE PICARD GROUP OF THE MODULI OF

SLr/µs-BUNDLES ON A CURVE

PAR Yves LASZLO (*)

ABSTRACT. — In this paper, we consider morphisms of algebraic stacks X→Y which are torsors under a group stack G. We show that line bundles on Y correspond exactly with G-linearized line bundles on X (with a suitable definition of a G- linearization). We use this fact to determine the precise structure of the Picard group of the moduli stack of G-bundles on an algebraic curve when G is any group of type An. RESUM´ E´. — Dans cet article, on consid`ere les morphismes de champs alg´ebriques X→Yqui sont des torseurs sous un champ en groupes G. Nous prouvons que les fibr´es en droites sur Y correspondent exactement aux fibr´es en droites sur X munis d’une G-lin´earisation (avec une d´efinition convenable d’une G-lin´earisation). Nous utilisons ceci pour d´eterminer la structure exacte du groupe de Picard du champ des G-fibr´es sur une courbe alg´ebrique lorsque G est un groupe alg´ebrique (non n´ecessairement simplement connexe) de type An.

1. Introduction Let G be a complex simple group and G →→ G the universal covering. For simplicity, let us consider the moduli stack MG (resp. M)ofdegree G 1 ∈ π1(G)principalG-bundles (resp. G-bundles) over a curve X.In [B-L-S], we have studied the natural morphism

ι :Pic(M ) −→ Pic(M ), G G the group Pic(M ) being infinite cyclic by [L-S]. It is proved in G [B-L-S] that the kernel of ι is naturally identified with the finite group 1 ∨ H´et(X, π1(G) ) reducing the study of Pic(MG) to the computation of the cardinality of Coker(ι). Among other things, it has been possible to

(*) Texte re¸cu le 18 aoˆut 1997, accept´e le 10 septembre 1997. Partially supported by the European HCM Project “ in Europe”. Y. LASZLO,DMI,Ecole´ Normale Sup´erieure, URA 762 du CNRS, 45 rue d’Ulm 75230 Paris CEDEX 05. Email : [email protected].

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 0037–9484/1997/529/$ 5.00 c Soci´et´emath´ematique de France 530 Y. LASZLO perform this computation in the case where G = PSLr but not in the case where G = SLr/µs,wheres | r,althoughwewereabletogivepar- tial results. The reason was that the technical background to study the descent of modules through the morphism p : M →M wasn’t at our G G disposal.

The aim of this paper is to compute card Coker(ι)whenG = SLr/µs. It turns out to be that p is a torsor under some group stack, not far from a Galois ´etale cover in the usual schematic picture. Let f : X→Ybe a torsor under a group G. We know that a line bundle on X descends if and only if it has a G-linearization (easy consequence of descent theory). Now, the descent theory of Grothendieckhas been adapted to the set-up offpqcmorphismsofstacksin[L-M].IfG is now only assumed to be a group stack, there is a notion of G-linearization of line bundles on X (see section 2). One obtains (theorem 4.1) that a line bundle on X descends if and only if it admits a linearization. We then use this technical result to compute card Coker(ι)when G = SLr/µs (theorem 5.7 and its corollary).

I would like to thank L. BREEN for having taught me both the notions of torsor and of linearization of a in the set-up of group-stack actions and for his comments on a preliminary version of this paper.

Notations. Throughout this paper, all the stacks will be implicitely assumed to be algebraic over an algebraically closed field k and the morphisms locally of finite type. We fix once and for all a projective, smooth, connected genus g curve X and a closed point x of X. For simplicity, we assume g>0(see remarks 5.6 and 5.10 for the case of P1). The Picard stackparametrizing families of line bundles of degree 0 on X will be denoted by J (X)andthe of X by JX.IfG is an algebraic group over k,thequotient stackSpec( k)/G (where G acts trivially on Spec(k)) whose category over a k-scheme S is the category of G-torsors (or G-bundles) over S will be denoted by BG.Ifn is an integer and A = J (X),JX or BGm we denote th n by nA the n -power morphism a → a .WedenotebyJn (resp. Jn)the 0-fiber A ×A Spec(k)ofnA when A = J (X)(resp.A = JX). 1. Generalities. — Following [Br], for any diagram

g −−−→ h A−−−→B λ C−−−→D −−−→ f

◦ TOME 125 — 1997 — N 4 LINEARIZATION OF GROUP STACK ACTIONS 531 of 2-categories, we will denote by

 ∗ λ :  ◦ f =⇒  ◦ g (resp. λ ∗ h : f ◦ h =⇒ g ◦ h) the 2-morphism deduced from λ. 1.1. — For the convenience of the reader, let us prove a simple formal lemma which will be usefull in section 4. Let A, B, C be three 2-categories. Let diagram

C δ0 →  − d0 −−−−− A −−−− f→ B −−− −−−  d1 δ1 → C (1.1.1) be a 2-commutative diagram and let µ : δ0 =⇒ δ1 be a 2-morphism.

LEMMA 1.2. — Assume that f is an equivalence. There exists a unique 2-morphism −1 µ ∗ f : d0 =⇒ d1 such that (µ ∗ f −1) ∗ f = µ.

Proof.—Letk,fork =0, 1 be the 2-morphism dk ◦ f =⇒ δk.Letb be ∼ an object of B. Pickan object a of A and an isomorphism α : f(a) −→ b. ∼ Let ϕα : d0(b) −→ d1(b) be the unique isomorphism making the diagram

0(a) d0(α) δ0(a) −−−−→ d0 ◦ f(a) −−−−→ d0(b)     µa   ϕα

1(a) d1(α) δ1(a) −−−−→ d1 ◦ f(a) −−−−→ d1(b)

commutative. We have to show that ϕα does not depend on α but only ∼ on b.Letα : f(a) −→ b be another isomorphism. There exists a unique ∼ isomorphism ι : a −→ a such that α◦f(ι)=α. Then one has the equality −1 ϕα = d1(α) ◦ Φ ◦ d0(α) where       −1 −1 Φ= d1 ◦ f(ι) ◦ 1(a ) ◦ µa ◦ 0(a ) ◦ d0 ◦ f(ι) .

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 532 Y. LASZLO

The functoriality of i and µ ensures that one has the equalities  dk ◦ f(ι) ◦ k(a )=k(a) ◦ δk(ι) and −1 µa = δ1(ι) ◦ µa ◦ δ0(ι) . This shows the equality

−1 Φ=1(a) ◦ µa ◦ 0(a) which proves the equality ϕα = ϕα . We can therefore define µb to be the ∼ isomorphism ϕα for one isomorphism α : f(a) −→ b. We checkthat the construction is functorial in b and the lemma follows.

2. Linearizations of line bundles on stacks. Let us first recall the notion of torsor in the stackcontext according to [Br]. 2.1. — Let f : X→Ybe a faithfuly flat morphism of stacks. Let us assume that an algebraic gr-stack G acts on f (the product of G is denoted by mG and the unit object by 1). Following [Br], this means that there exists a 1-morphism of Y-stacks m : G×X →X and a 2-morphism µ : m ◦ (mG × IdX )=⇒ m ◦ (IdG ×m) such that the obvious associativity condition (see diagram (6.1.3) in [Br]) is satisfied and such that there exists a 2-morphism  : m ◦ (1 × IdX )=⇒ IdX which is compatible to µ in the obvious sense (see (6.1.4) of [Br]).

REMARK 2.2. — To say that m is a morphism of Y-stacks means that the diagram G×X −−−−m→X  Y is 2-commutative. In other words, if we denote for simplicity the image of a pair of objects m(g,x)byg·x. This means that there exists a functorial isomorphism ιg,x : f(g·x) → f(x). 2.3. — Suppose that G acts on such another morphism f  : X  →Y. A morphism p : X  →Xwill be said equivariant if there exists a 2- morphism q : m ◦ (Id ×p)=⇒ p ◦ m which is compatible to µ (as in [Br, (6.1.6)]) and  (which is implicit in [Br]) in the obvious sense.

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DEFINITION 2.4. — With the above notations, we say that f (or X ) is a G-torsor over Y if the morphism pr2 ×m : G×X →X×Y X is an isomorphism (of stacks) and the geometrical fibers of f are not empty.

REMARK 2.5. — In down to earth terms, this means that if

ι : f(x) −→ f(x) is an isomorphism in Y (x, x being objects of X ), there exists an object g ∼ of G and a unique isomorphism (x, g·x) −→ (x, x) which induces ι by way of ιg,x (cf.2.2).

EXAMPLE 2.6. — Let MX (Gm) be the Picard stackof X. Then, the morphism MX (Gm) −→ M X (Gm) of multiplication by n ∈ Z is a torsor under Bµn × Jn(X)(cf. (3.1)). 2.7. — Let L be a line bundle on X . By definition, the datum L is equivalent to the datum of a morphism  : X→BGm (see [L-M, prop. 6.15]). If L, L are two line bundles on X defined by , , we will ∼ view an isomorphism L −→ L  as a 2-morphism  =⇒ . DEFINITION 2.8. — A G-linearization of L is a 2-morphism

λ :  ◦ m =⇒  ◦ pr2 such that the two diagrams of 2-morphisms ∗µ  ◦ m ◦ (mG × IdX ) ======⇒  ◦ m ◦ (IdG ×m) λ∗(mG ×IdX ) λ∗(IdG ×m)

(2.8.1) λ∗pr23  ◦ pr2 ◦(mG × IdX ) ⇐=====  ◦ pr2 ◦(IdG ×m) || ||  ◦ pr2 ◦ pr23  ◦ m ◦ pr23 and ∗  ◦ m ◦ (1 × IdX ) ====⇒  (2.8.2) λ∗(1×IdX )  ======

(strictly) commute.

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 534 Y. LASZLO

REMARK 2.9. — If g1,g2 are objects of G and d is an object of X ,the commutativity of diagram (2.8.1) means that the diagram L −−−−−−→ L (g1·g2)x ∼ g1(g2·x)     L ←−−−−−−−−∼ −L x g2·x is commutative and the commutativity of (2.8.2) that the two isomor- phisms L1·x Lx defined by the linearization λ and  respectively are the same.

3. An example. Let me recall that a closed point x of X has been fixed. Let S be a k-scheme. The S-points of the jacobian variety of X are by definition isomorphism classes of line bundles on XS together whith a trivialization along {x}×S (such a pair will be called a rigidified line bundle). For the convenience of the reader, let me state this well known lemma which can be found in SGA4, exp. XVIII, (1.5.4). LEMMA 3.1. — The Picard stack J (X) is canonically isomorphic (as a k-group stack) to JX × BGm.

Proof.—Letf : J (X) → JX×BGm be the morphim which associates −1 •   to the line bundle L on XS the pair (L L|{x}×S,L|{x}×S)where denotes the external (this pair is thought of as an object of JX × BGm over S); ∼  • to an isomorphism L −→ L on XS its restriction to {x}×S.  Let f : JX × BGm →J(X) be the morphism which associates

• to the pair (L, V )whereL is a rigidified bundle on XS and V a line bundle on S (thought of as an object of JX × BGm over S), the line  bundle L XS V ; ∼   • to an isomorphism (, v):(L, V ) −→ (L ,V ) the tensor pro-  duct  XS v. The morphisms f and f  are (quasi)-inverse to each other and are morphisms of k-stacks.

We will identify from now J (X)andJX×BGm.LetL (resp. P and T ) be the universal bundle on X ×J(X)(resp.onX × JX and BGm)and letΘ=(detRΓP)−1 be the theta line bundle on JX. The isomorphism ∼ L −→ P  T yields an isomorphism ∼ 2 (3.1.1) det RΓLn(m·x) −→ Θ−n  T (m+1−g).

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4. Descent of G-line bundles. The object of this section is to prove the following statement. G THEOREM 4.1. — Let f : X→Ybe a G-torsor as above. Let Pic (X ) be the group of isomorphism classes of G-linearized line bundles on X .Then, ∼ the pull-back morphism f ∗ :Pic(Y) −→ PicG(X ) is an isomorphism. The descent theory of Grothendieckhas been adapted in the case of algebraic 1-stacks in [L-M], essentially in proposition (6.23).

Let X• →Ybe the (augmented) simplicial complex of stacks coskeleton of f (as defined in [De, (5.1.4)] for instance). By proposition (6.23) of O L L [L-M], one just has to construct a cartesian D• -module • such that 0 is the OX -module L to prove the theorem. The n-th piece Xn is inductively defined by X0 = X , Xn = X×Y Xn−1 for n>0.

Let pn : Xn →X be the projection on the first factor. It is the simplicial morphism associated to the map

∆0 → ∆n, p˜n : 0 −→ 0.

Let Ln be the line bundle defined by the morphism (see (2.7))

pn (4.1.1) n : Xn −−−→X −−−→ BGm.

4.2. — Let δi (resp. sj) be the face (resp. degeneracy) operators (see for instance [De, 5.1.1]). By abuse of notation, we use the same notation for

δj, sj and their image by X ). The category (∆•) is generated by the face and degeneracy operators with the following relations (see for instance the proposition VII.5.2, p. 174 of [McL])

(4.2.1) δi ◦ δj = δj+1 ◦ δi if i ≤ j,

(4.2.2) sj ◦ si = si ◦ sj+1 if i ≤ j,   δi ◦ sj−1 if ij+1.

Therefore, the datum of a cartesian OX• -module L• is equivalent to the data of isomorphisms ∗L −→∼ L αj : δj n n+1,j=0,...,n+1, ∗L −→∼ L βj : sj n+1 n,j=0,...,n,

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 536 Y. LASZLO

(where n is a non negative integer) which are compatible with relations (4.2.1), (4.2.2) and (4.2.32). Let n be a non negative integer. 4.3. — We have first to define for j =0,...,n+ 1 an isomorphism ∗L −→∼ L αj : δj n n+1. ∗L ◦ ◦ X → The line bundle δj n is defined by the morphism  pn δj : n+1 BGm andp ˜n ◦ δj is associated to the map ∆0 −→ ∆n+1,

0 −→ δj(0). ∗ • ◦ L If j = 0, one has thereforep ˜n δj =˜pn+1 and δj n = Ln+1.We define αj by the identity in this case. • Suppose now that j =0.Letπn : Xn →X1 be the projection on the two first factors (associated to the canonical inclusion ∆1 -→ ∆n). The commutativity of the two diagrams δ0 pn+1 X +1 −−−−→X X +1 −−−−−→X n n n     πn+1  pn  and  πn+1  δ1 δ0 X1 −−−−−→X X1 ===== X1 allows to reduce the problem to the construction of an isomorphism ∗ ∼ ∗ δ0 L −→ δ1 L where δi : X1 →X for i =0, 1 are the face morphisms or, what amounts to the same, to the construction of a 2-morphism ν :  ◦ δ0 =⇒  ◦ δ1 (the morphism αj will be αj = ν ∗ πn+1). Now the diagram

BGm m → −  −− −−−−  δ0 pr2 ×m

G×X −−−−−→ X×Y X

− −−−−  − − 

 δ1 pr2 →

BGm (4.3.1) is strictly commutative and pr2 ×m is an equivalence by definition of a torsor. According to lemma 1.2, the 2-morphism λ induces a canonical 2-morphism −1 λ ∗ (pr2 ×m) :  ◦ δ0 =⇒  ◦ δ1 which is the required 2-morphism ν.

◦ TOME 125 — 1997 — N 4 LINEARIZATION OF GROUP STACK ACTIONS 537

4.4. — We then have to define for j =0,...,n an isomorphism ∗L −→∼ L βj : sj n+1 n. ∗L ◦ ◦ The line bundle sj is defined by the morphism  pn+1 sj and pn+1 ◦ sj is associated to the canonical inclusion ∆0 -→ ∆n which means ◦ ∗L L pn+1 sj = pn. Therefore, one has a canonical isomorphism sj n+1 = n and we define βj to be the identity.

4.5. — We have to show that the data L• and αj,βj,forj ≥ 0 define a line bundle on the simplicial stack X• as explained in 4.2. Notice that the fact that the definition of the βj is compatible with relations (4.2.2) is tautological (βj is the identity on the relevant Ln). 4.6. — In terms of , relation (4.2.1) means the following. We have the two strictly 2-commutative diagrams

δi δj X +2 −−−−→ X +1 −−−−−→X n n − n −−−− −  − − −−  − − − pn+1 −  −−− − pn pn+2 → → X −−−−−−−− → BGm

δj+1 δi X +2 −−−−→X+1 −−−−→X n n − n −−−− −  − − −−  − − − pn+1 − −  −−− pn pn+2 → → X −−−−−−−− → BG m and inducing the two 2-morphisms

αj ∗δi αi αi ◦ (αj ∗ δi): ◦ pn ◦ δj ◦ δi =====⇒  ◦ pn+1 ◦ δi ===⇒  ◦ pn+2 and

αi∗δj+1 αj+1 αj+1 ◦ (αi ∗ δj+1): ◦ pn ◦ δi ◦ δj+1 =====⇒  ◦ pn+1 ◦ δj+1 ===⇒  ◦ pn+2. The relation (4.2.1) means exactly the equality  (4.2.1 ) αi ◦ (αj ∗ δi)=αj+1 ◦ (αi ∗ δj+1), for i ≤ j.

 • If j = 0, the relation (4.2.1 ) is just by definition of αj the condition (2.8.1) (see remark2.9).

• If j>0, both isomorphisms αj and αj+1 are the relevant identities and the relation (4.2.1) is tautological.

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 538 Y. LASZLO

4.7. — The only non tautological relation in (4.2.3) corresponds to the equality s0 ◦ δ0 =1in(∆•)whichmeansasbeforethatα0 ∗ δ0 is the identity functor of  ◦ pn =  ◦ pn ◦ δ0 ◦ s0. But, this is exactly the meaning of the relation (2.8.2) (see remark2.9).

5. Application to the Picard groups of some moduli spaces. Let us choose three integers r, s, d such that

r ≥ 2ands | r | ds.

If G is the group SLr/µs we denote as in [B-L-S] by MG(d)the (connected) moduli stackof G-bundles on X of degree exp(2iπd/r) ∈ 2 M H´et(X, µs)=µs and by SLr (d) the moduli stackof rank r vector bundles and determinant O(d·x). If r = s (i.e. G = PSLr), the natural morphism of algebraic stacks

M −→ M π : SLr (d) G(d) is a Jr-torsor (see the corollary of proposition 2 of [Gr] for instance). Let me explain how to deal with the general case.

5.1. — Let E be a rank r vector bundle on XS endowed with an isomorphism ∼ τ : Dr/s −→ det(E) where D is some line bundle. Let me define the SLr/µs-bundle π(E) associated to E (more precisely to the pair (E,τ)).

DEFINITION 5.2.

• An s-trivialization of E on an ´etale neighborhood T → XS is a triple (M,α,σ)where ∼ α : D −→ M s is an isomorphism (M is a line bundle on T ); ⊕r ∼ σ : M −→ ET is an isomorphism; ∼ det(σ) ◦ αr/s : Dr/s −→ det(E)isequaltoτ.    • Two s-trivializations (M,α,σ)and(M ,α,σ )ofE will be said ∼ equivalent if there exists an isomorphism ι : M −→ M  such that ιs ◦ α = α.

The principal homogeneous space T −→ equivalence classes of s-trivializations of ET

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† defines the SLr/µs-bundle π(E) . Now, the construction is obviously M →M functorial and therefore defines the morphism π : SLr (d) G(d) M O · r/s (observe that an object E of SLr (d) has determinant (ds/r x) ). Let L be a line bundle and (M,α,τ)ans-trivialization of ET . Then, (M ⊗ L, α ⊗ IdLs ,σ ⊗ IdL)isans-trivialization of E ⊗ L (which has determinant (D ⊗ Ls)r/s). This shows that there exists a canonical functorial isomorphism

∼ (5.2.1) π(E) −→ π(E ⊗ L)

In particular, π is Js-equivariant. LEMMA 5.3. — The natural morphism of algebraic stacks

M −→ M π : SLr (d) G(d) is a Js-torsor.

 Proof.—LetE,E be two rank r vector bundles on XS (with deter- ∼ minant equal to O(d·x)) and let ι : π(E) −→ π(E) be an isomorphism. As in the proof of the lemma 13.4 of [B-L-S], we have the exact sequence of sets   πE,E 1 1 → µs −→ Isom(E,E ) −→ Isom π(E),π(E) −−−−→ H´et(XS,µs).

Let L be a µs-torsor such that πE,E (ι)=[L]. Then, π(E ⊗ L)isequal ∼  to π(E), πE⊗L,E =0andι is induced by an isomorphism E ⊗ L −→ E well defined up to multiplication by µs. The lemma follows.

5.4. — Let U be the universal bundle on X ×MSLr (d). We would like to know which power of the determinant bundle D =(detRΓU)−1 on M M SLr (d) descends to G(d). As in I.3 of [B-L-S], the rank r bundle

F = L⊕(r−1) ⊕L1−r(d·x) on X ×J(X) has determinant O(d·x) and therefore defines a morphism

J × −→ M f : (X)=JX BGm SLr (d) which is Js-equivariant.

† WeseehereaG-bundle as a formal homogeneous space under G.

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 540 Y. LASZLO

The vector bundle

F  = O⊕(r−1) ⊕L−r/s(d·x) on X ×J(X) has determinant [L−1(ds/r · x)]r/s.TheG-bundle π(F )on  X ×J(X) defines a morphism f : J→MG(d). The relation

∗  L⊗(IdX ×sJ ) ∗ (F )=F and (5.2.1) gives an isomorphism

∗  π(F)=(IdX ×sJ ) π(F ) which means that the diagram

f J (X) −−−−−→MSL (d)  r   (5.4.1) sJ   π f  J (X) −−−−−→MG(d) is 2-commutative. Exactly as in I.3 of [B-L-S], let me prove the ∗ k LEMMA 5.5. — Thelinebundlef D on J (X) descends through sJ if and only if k multiples of s/(s, r/s).

Proof.—Letχ = r(g −1)−d be the opposite of the Euler characteristic k M of ( -)points of SLr (d). By (3.1.1), one has an isomorphism

∼ f ∗Dk −→ Θkr(r−1)  T kχ.

kr(r−1) The theory of Mumford groups says that Θ descends through sJ kχ if and only if k is a multiple of s/(s, r/s). The line bundle T on BGm descends through sBGm if and only if kχ is a multiple of s. The lemma follows from the above isomorphism and from the observation that the condition s | r | ds forces sχ to be a multiple of s.

REMARK 5.6. — If g = 0, the jacobian J is a point and the condition on Θ is empty. The only condition in this case is that kχ is a multiple of s. D M Let me recall that is the determinant bundle on SLr (d)and G = SLr/µs. THEOREM 5.7.—Assume that the characteristic of k is 0. The integers k k such that D descends to MG(d) are the multiple of s/(s, r/s). By the proposition 1.5 of [BLS], one gets the

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COROLLARY 5.8. — The natural morphism M −→ M Z·D Pic G(d) Pic SLr (d) =

s/(s,r/s) makes the Picard group of MG(d) an extension of Z = Z·D by 1 ∼ 2g H´et(X, Z/dZ) −→ (Z/dZ) . Proofofthetheorem. — By lemma 5.5 and diagram (5.4.1), we just have to prove that Dk effectively descends when k = s/(s, r/s). By theorem 4.1 k and lemma 5.3, this means exactly that D has a Js-linearization. We know by lemma 5.5 that the pull-back f ∗Dk has such a linearization.

LEMMA.—The pull-back morphism J ×M −→ J ×J Pic s SLr (d) Pic s (X) is injective.

Proof. — By lemma 3.1, one is reduced to prove that the natural morphism ×M −→ ×J Pic Bµs SLr (d) Pic Bµs (X) is injective. Let X be any stack. The canonical morphism X→X×Bµs is a µs-torsor (with the trivial action of µs on X ). By theorem 4.1, one has the equality µs Pic(X×Bµs)=Pic (X ). Assume further that H0(X , O)=k. The later group is then canonically isomorphic to

Pic(X ) × Hom(µs,Gm)=Pic(X ) × Pic(Bµs).

Eventually, we get a functorial isomorphism

∼ (5.9.1) ιX :Pic(X×Bµs) −→ Pic(X ) × Pic(Bµs).

M Z·D By [L-S], the Picard group of SLr (d) is the free abelian group and the formula (3.1.1) proves that ∗ f :Pic MSLr (d) −→ Pic J (X)

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 542 Y. LASZLO is an injection. The diagram Pic MSL (d) × Pic(Bµ ) -−→ Pic J (X) × Pic(Bµ ) r  s  s     ιM  ιJ  M × −−−−−−→ J × Pic( SLr (d) Bµs) Pic( (X) Bµs) is commutative and the lemma follows from this commutative diagram. H H J ×M J ×J Let (resp. J ) be the line bundle on s SLr (d)(resp. s (X))

∗ k ∗ k H = Hom(mMD , pr2 D ) ∗ ∗ k ∗ ∗ k resp. HJ = Hom(mMf D , pr2 f D ) .

∗ k Let us choose a Js-linearization λJ of f D . It defines a trivialization of the line bundle HJ . The equivariance of f implies (cf.2.3)thatthere exists a (compatible) 2-morphism

q : mM ◦ (Id ×f)==⇒ f ◦ mJ making the diagram

mJ Js ×J(X) −−−−−−→J(X)     Id×f   f J ×M −−−−mM→M s SLr (d) SLr (d)

2-commutative. The 2-morphism q defines an isomorphism from the pull- ∗ k ∗ ∗ k ∗ k back mMD on Js ×J(X)tomJ (f D ). The pull-backof pr 2 D on ∗ ∗ k Js ×J(X) is tautologically isomorphic to pr2(f D ). The preceding isomorphisms induce an isomorphism

∗ ∼ (Id ×f) H −→ H J . The later line bundle being trivial, so is (Id ×f)∗H. The lemma above proves therefore that H itself is trivial.Each(k-)point j of Js defines a morphism M →J ×M J →J ×J SLr (d) s SLr (d) resp. (X) s (X) ;

∗ ∗ letmedenotebyHj (resp. f Hj) the pull-backof H (resp. (Id ×f) H) by this morphism. The pull-backmorphism 0 J ×M H −→ 0 J ×J × ∗H H s SLr (d), H s (X), (Id f)

◦ TOME 125 — 1997 — N 4 LINEARIZATION OF GROUP STACK ACTIONS 543 can be identified to the direct sum 0 M H −→ 0 J ∗H H SLr (d), j H (X),f j .

j∈Js(k) As 0 M O 0 J O k (5.9.2) H SLr (d), = H (X), = , this morphism is a direct sum of non-zero homomorphisms of vector spaces of dimension 1 and therefore an isomorphism. In particular, a linearization ∗ k λJ of f D defines canonicaly an isomophism ∗ k ∼ ∗ k λM : mMD −→ pr2 D ∗ such that (Id ×f) λM = λJ . Explicitely, λM is characterized as follows. Let x be an object of M J J k SLr (d) over a connected scheme S and g an object of s(S)= s( ). The preceding dicussion means that the functorial isomorphisms Dk −→∼ D k λM(g,x): g·x x are determined when x lies in the essential image of f.Inthiscase, ∼ let us choose an isomorphism f(x) −→ x (inducing an isomorphism ∼ g·f(x) −→ g·x). Then, the diagram of isomorphisms of line bundles on S



 −−−− −−−−−→

Lx = Lf(x ) Lx  → 

 λM(g,x)

λJ (g,x ) 

−  −

−  q ,x − −−−−−  −−−−−→  −−−−−→ Lg·x = Lf(g·x ) Lg·f(x ) Lg·x is commutative (where L = Dk and L = f ∗Dk).

Now, the pull-backof λM on Js ×J(X) satisfies conditions (2.8.1) and (2.8.2). Using (5.9.2) and the equivariance of f as above, this shows that λM is a linearization. For instance, keeping the notation above, let us verify condition (2.8.2). We have to checkthat the isomorphism ι of L induced by  is the identity. As above, it is enough to checkthat on J (X). With a slight abuse of notations, the two diagrams

 ι  −−−− −−−−→ −−−−→

Lx = Lf(x ) Lx Lx Lx



→  

 

  ( ) and 

λJ (1,x ) λM(1,x)  x  −

−

−   M(1 ) − −−−− λ ,x 

  q1,x   −−−−→  −→−−− L1·x = Lf(1·x ) L1·f(x ) L1·x L1·x

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE 544 Y. LASZLO

are commutative (the commutativity of the first diagram follows from the equivariance of f and the commutativity of the second diagram follows exactly from the definition of ι). Because λJ is a linearization, condition (2.8.2) shows that the diagram

 

Lx ===== Lx



  

 ( )

  x

 

(1 )  λJ ,x     L1·x

is commutative. It follows that the equality ι = Id will follow from the commutativity of the diagram  L (1· ) −−−−→ L ( ) f  x f x    (5.9.3) q1,x  L1·f(x) −−−−→ Lf(x). But q is compatible with  as required in 2.3. The diagram  f(1.x) −−→ f(x)   q1,x  1·f(x) −−→ f(x)

is therefore commutative from which the commutativity of (5.9.3) follows. One would checkcondition (2.8.1) in an analogous way.

REMARK 5.10. — In the case g = 0, the condition is an in remark5.6.

REMARK 5.11. — This linearization can be certainly also deduced from a careful analysis of the first section of [Fa], but the method above seems simpler.

BIBLIOGRAPHIE

[B-L-S] BEAUVILLE (A.), LASZLO (Y.), SORGER (C.). — The Picard group of the moduli of G-bundles on a curve, preprint alg-geom/9608002, to appear in Compositio Math.

◦ TOME 125 — 1997 — N 4 LINEARIZATION OF GROUP STACK ACTIONS 545

[Br] BREEN (L.). — Bitorseurs et cohomologie non ab´elienne, in ‘The Grothendieck Festschrift I’, Progr. Math., Birkh¨auser, t. 86, , p. 401–476. [De] DELIGNE (P.). — Th´eorie de Hodge III, Publ. Math. IHES, t. 44, , p. 5–78. [Fa] FALTINGS (G.). — Stable G-bundles and Projective Connections, JAG, t. 2, , p. 507–568. [Gr] GROTHENDIECK (A.). — G´eom´etrie formelle et g´eom´etrie alg´ebrique, FGA, S´em. Bourbaki, t. 182, /, p. 1–25. [L-M] LAUMON (G.), MORET-BAILLY (L.). — Champs alg´ebriques,preprint Universit´e Paris-Sud, . [L-S] LASZLO (Y.), SORGER (C.). — The line bundles on the moduli of parabolic G-bundles over curves and their sections, Ann. Ecole´ Normale Sup´erieure, 4e s´erie, t. 30, , p. 499–525. [McL] MAC LANE (S.). — Categories for the working mathematician,GTM, Springer-Verlag, t. 5, .

BULLETINDELASOCIET´ EMATH´ EMATIQUE´ DE FRANCE