Bull. Soc. math. France, 125, 1997, p. 529–545. LINEARIZATION OF GROUP STACK 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 “Algebraic Geometry 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 scheme 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 vector bundle 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 jacobian variety 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. ◦ TOME 125 — 1997 — N 4 LINEARIZATION OF GROUP STACK ACTIONS 533 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.