Linearization of Group Stack Actions and the Picard Group of the Moduli Of

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Linearization of Group Stack Actions and the Picard Group of the Moduli Of 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.
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