NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS

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1 NORI FUNDAMENTAL GERBE OF ESSENTIALLY FINITE COVERS AND GALOIS CLOSURE OF TOWERS OF TORSORS MARCO ANTEI, INDRANIL BISWAS, MICHEL EMSALEM, FABIO TONINI, AND LEI ZHANG Abstract. We prove the existence o a Galois closure or towers o torsors under inite group schemes over a proper, geometrically connected and geometrically reduced algebraic stack X over a ield k. This is done by describing the Nori undamental gerbe o an essentially inite cover o X. A similar result is also obtained or the S-undamental gerbe. Introduction Let G and H be inite group schemes over a ield k. A pg, Hq-tower o torsors over a k-algebraic stack X consists in maps Z ÝÑ h Y ÝÑ g X where h and g are an H-torsor and a G-torsor respectively. The initial motivation or this work is the ollowing question: given a tower o torsors as beore does there exist a "Galois closure" under a inite group scheme which would dominate the tower? A Galois closure is a torsor P ÝÑ X under a inite group scheme over k together with a map λ: P ÝÑ Z over X satisying some equivariance conditions (see Deinition 3.8). Previous attempts to give an airmative answer to this question ailed: both [Ga] and (unpublished) [ABE] contain mistakes in the proo o their main theorems. The étale case is not diicult and one can prove that there is even a canonical Galois closure. Theorem I. I X is an algebraic stack over k and G or H is étale then all pg, Hqtowers o torsors Z ÝÑ Y ÝÑ X admit a canonical Galois closure P ÝÑ λ Z by a torsor under the group scheme Q representing the shea o automorphisms o the trivial tower G ˆ H ÝÑ G ÝÑ Spec k. Moreover λ is a torsor under a inite subgroup scheme o Q. Thus the problem o inding a Galois closure is a positive characteristic problem and it is related with the notion o essentially inite covers. A cover : Y ÝÑ X o an algebraic stack (i.e. an aine map such that O Y is a vector bundle) is essentially inite i O Y is essentially inite (in the category o vector bundles o X (see Deinition 1.5)). In order to study Galois closures we introduce the stack BpG, Hq o pg, Hq-tower o torsors, which is locally o inite type over k (see Proposition 3.3). We say that a tower Date: May 29, Mathematics Subject Classiication. 14F35, 14D23. Key words and phrases. Nori undamental gerbe, essentially inite bundle, essentially inite cover, algebraic stack. This work was supported by the European Research Council (ERC) Advanced Grant , the Einstein Foundation and the Labex CEMPI (ANR-11-LABX-01). 1

2 2 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG Z ÝÑ Y ÝÑ X is pointed i Z has a given rational point; it is Nori reduced i the torsors are Nori reduced (see Deinition 1.12). A Galois closure P ÝÑ λ Z is pointed i P comes with a rational point and λ preserves the given rational points. The ollowing is a general result proved here. Theorem II. Let X be a pseudo-proper (see Deinition 1.6) and inlexible (see Deinition 1.1) algebraic stack o inite type over k. I char k ą 0 assume that dim k H 1 px, Eq ă 8 or all vector bundles E on X. I Z ÝÑ h Y ÝÑ g X is a pg, Hq-tower o torsors then Z ÝÑ X is an essentially inite cover. I z P Zpkq then the tower admits a pointed Nori reduced λ Galois closure pp, pq ÝÑ pz, zq such that: (1) λ is aithully lat i and only i the torsors in the tower are Nori reduced, or equivalently Z and Y are inlexible, and in this case it is a Nori reduced torsor under a inite group scheme. (2) P ÝÑ X is a torsor under a inite subgroup o the aine and o inite type k-group scheme Aut BpG,Hq pξq, where ξ P BpG, Hqpkq is the tower iber o the given tower over x ghpzq: Spec k ÝÑ X. Examples 5.3 and 5.4 shows that there are examples o towers without a Galois closure i we remove the pseudo-properness assumption or the condition on the irst cohomology groups. This last condition is needed to ensure that every G a -torsor over a cover o X is induced by a torsor under a inite subgroup o G a. Notice that the cohomological assumption is met or proper algebraic stack o inite type over k ([Fal]). Moreover a geometrically connected and geometrically reduced algebraic stack over k is inlexible (see Remark 1.2). The idea o the proo o Theorem II is simple. Given an essentially inite cover one can deine its Galois closure and, applying this procedure on gh: Z ÝÑ X, one get the desired closure. So one must show that Z ÝÑ X is essentially inite and, since all torsors are essentially inite, the key point is to prove that the pushorward o an essentially inite vector bundle under an essentially inite cover is again essentially inite. This motivates the study o essentially inite covers in general and, because o their strict relation with the notion o essentially inite vector bundles, their Nori undamental groups or gerbes, which are denoted by π N p, q and Π Ń respectively. In particular we use the machinery introduced in [No1] and later generalized in [BV] (see also Deinition 1.1 and Theorem 1.9). Theorem III. Let X be a pseudo-proper and inlexible algebraic stack o inite type over k and let : Y ÝÑ X be an essentially inite cover. I char k ą 0 assume that either is étale or dim k H 1 px, Eq ă 8 or all vector bundles E on X. Then there exists a unique (up to equivalence) inite map Π ÝÑ Π N X {k whose base change along X ÝÑ Π N X {k is Y ÝÑ X and Y is inlexible over k i and only i Π is a gerbe over k.

3 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 3 In this case Y ÝÑ Π is the Nori undamental gerbe o Y, that is the diagram Y Π N Y{k X Π N X {k is 2-Cartesian, and we have EFinpVectpYqq tv P VectpYq V P EFinpVectpX qqu. I is étale then Y is inlexible over k i and only i H 0 py, O Y q k and in this case the ollowing diagrams are 2-Cartesian Y Π N Y{k Π N,ét Y{k X Π N X {k Π N,ét X {k I is a torsor under a inite group scheme G over k the ollowing are equivalent: Y is inlexible over k; H 0 py, O Y q k; is Nori reduced over k. Under these conditions the ollowing diagrams are 2-Cartesian β Y Π N Y{k Spec k α X Π N X {k B G For pointed covers Theorem III also yields the ollowing Galois correspondence. π Corollary I. Let X be a pseudo-proper and inlexible algebraic stack o inite type over k with a rational point x P X pkq. I char k ą 0 assume that dim k H 1 px, Eq ă 8 or all vector bundles E on X. Then there is an equivalence o categories " Pointed essentially inite covers py, yq ÝÑ px, xq with Y inlexible py, yq ÝÑ px, xq * " Subgroups H ă π N px, xq with π N px, xq{h inite π N py, yq where in the right side we consider inclusions as arrows. Moreover an essentially inite cover py, yq ÝÑ px, xq with Y inlexible is a torsor under a inite group G i and only i π N py, yq is a normal subgroup o π N px, xq and in this case there is an exact sequence 1 ÝÑ π N py, yq ÝÑ π N px, xq ÝÑ G ÝÑ 1 an essentially inite cover py, yq ÝÑ px, xq with Y inlexible is étale i and only i the inite scheme π N px, xq{π N py, yq is étale over k. *

4 4 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG The above exact sequence has already been proved in [EHS, Thoerem 2.9] under the assumption that G is étale. The crucial point in the above results is again to show that, i : Y ÝÑ X is an essentially inite cover, then preserves essentially inite vector bundles. This is alse without some properness and cohomological assumptions on X (see Lemmas 2.8, 2.10 and, or a counter-example, Example 5.5). A key tool in the proo is a characterization o essentially inite vector bundles given in [TZ2] which generalizes previous results o [BdS] and [AM]. We also study a similar problem or the S-undamental gerbe (see [BPS], [La1], [La2] and Deinition 4.1). The ollowing analog o Theorem III is proved or S-undamental gerbes. Theorem IV. Let X be a pseudo-proper algebraic stack o inite type over k and with an S-undamental gerbe. Then X is inlexible and the proinite quotient o the S-undamental gerbe o X over k is the Nori undamental gerbe o X over k. Let also : Y ÝÑ X be an essentially inite cover with Y inlexible and, i char k ą 0, assume that dim k H 1 px, Eq ă 8 or all vector bundles E on X. Then NspYq tv P VectpYq V P NspX qu and Y has an S-undamental gerbe itting in the 2-Cartesian diagram Y Π S Y Π N Y X Π S X Π N X In particular i y P Ypkq and x pyq P X pkq then there is a Cartesian diagram o aine group schemes π S py, yq π N py, yq π S px, xq π N px, xq When : Y ÝÑ X is pointed torsor under a inite group scheme G, then the ollowing sequence is exact 1 ÝÑ π S py, yq ÝÑ π S px, xq ÝÑ G ÝÑ 1. The paper is divided as ollows. In the irst section we recall part o the machinery about Nori undamental gerbes and prove some preliminary results. In the second section we study essentially inite covers and their Nori undamental gerbes, proving in particular Theorem III and Corollary I. In the third section we study towers o torsors and their Galois closures, proving Theorems I and II, while in the ourth section we study the S-undamental gerbe and prove Theorem IV. Finally in the last section we collect some counter-examples. Acknowledgement We would like to thank Hélène Esnault and Angelo Vistoli or helpul conversations and suggestions received.

5 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 5 1. Notation and Preliminaries 1.1. Notation. By a ibered category over a scheme S we will always mean a category ibered in groupoids over the category A{S o aine schemes over S. A cover o a ibered category X is a inite, lat and initely presented morphism or, equivalently, an aine map : Y ÝÑ X with the property that O Y is locally ree o inite rank. I X is deined over a ield k a pointed cover over k py, yq ÝÑ px, xq is a cover : Y ÝÑ X with x P X pkq and y P Y x pkq, where Y x denotes the iber o over x, which is a inite k-scheme; equivalently y P Ypkq with a given isomorphism pyq» x. Given a morphism o schemes U ÝÑ V and a unctor F : A{U ÝÑ psetsq, the Weil restriction o F along U ÝÑ V is the unctor W U pf q: A{V ÝÑ psetsq, Z ÞÝÑ HompZ ˆV U, F q. Given any unctor G: A{V ÝÑ psetsq, we set W U pgq W U pg ˆV Uq. Injectivity and surjectivity o morphisms o group schemes always mean the corresponding properties or pqc sheaves. For aine group schemes over a ield an injective morphism is a closed immersion and a surjective morphism is aithully lat ([Wat, Theorem 15.5] or surjectivity) Preliminaries. We will now recall some results used in later sections. Fix a base ield k. For properties o aine gerbes over a ield (oten improperly called just gerbes) and Tannakian categories used here the reader is reerred to [TZ1, Appendix B]. Deinition 1.1 ([BV, Deinition 5.1, Deinition 5.3]). For a ibered category X over k, the Nori undamental gerbe (respectively, Nori étale undamental gerbe) o X {k is a proinite (respectively, proétale) gerbe Π over k together with a map X ÝÑ Π such that or all inite (respectively, inite and étale) stacks Γ over k the pullback unctor Hom k pπ, Γq ÝÑ Hom k px, Γq is an equivalence o categories. Furthermore, i this gerbe exists, it is unique up to a unique isomorphism and in that case it will be denoted by Π N X {k (respectively, ΠN,ét X {kq; sometimes {k will be dropped i it is clear rom the context. We call X inlexible i it is non-empty and all maps rom it to a inite stack over k actors through a inite gerbe over k. Remark 1.2. By [BV, p. 13, Theorem 5.7] X admits a Nori undamental gerbe i and only it is inlexible; in this case, the Nori étale undamental gerbe o X is the maximal proétale quotient o the Nori undamental gerbe o X. I X is reduced, quasi-compact and quasi-separated, then X is inlexible i and only i k is algebraically closed in H 0 po X q [TZ1, Theorem 4.4]. In particular i X is geometrically connected and geometrically reduced, then it is inlexible.

6 6 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG Deinition 1.3. I X is an inlexible ibered category over k with a rational point x P X pkq and Nori gerbe ψ : X ÝÑ Π N X {k, the Nori undamental group scheme πn px {k, xq o px, xq over k is the shea o automorphisms o ψpxq P Π N X {kpkq. Again {k will oten be dropped i it is clear rom the context. Remark 1.4. The Nori undamental group scheme π N px, xq is a proinite group scheme and B π N px, xq» Π N X (the trivial torsor is sent to ψpxq). The universal property o ΠN X translates into the ollowing: or all inite group schemes G over k the map is bijective. Hom k-groups pπ N px, xq, Gq tpointed G-torsors pp, pq ÝÑ px, xqu{» π N px, xq ÝÑ G X ÝÑ B π N px, xq ÝÑ B G Deinition 1.5 ([BV, p. 21, Deinition 7.7]). Let C be an additive and monoidal category. An object E P C is called inite i there exist polynomials g P NrXs and an isomorphism peq» gpeq; the object E is called essentially inite i it is a kernel o a homomorphism o inite objects o C. Let EFinpCq denote the ull subcategory o C consisting o essentially inite objects. Deinition 1.6 ([BV, p. 20, Deinition 7.1]). A category X ibered in groupoid over a ield k is pseudo-proper i it satisies the ollowing two conditions: (1) there exists a quasi-compact scheme U together with a morphism U ÝÑ X which is representable, aithully lat, quasi-compact, and quasi-separated, and (2) or all vector bundles E on X the k-vector space H 0 px, Eq is inite dimensional. Example 1.7 ([BV, p. 20, Example 7.2]). Examples o pseudo-proper iber categories are proper algebraic stacks, inite stacks and aine gerbes. Remark 1.8. Let X be a pseudo-proper algebraic stack o inite type over k. I X is inlexible then H 0 po X q k (see [BV, Lemma 7.4]), while the converse holds i X is reduced (see Remark 1.2). Theorem 1.9 ([BV, p. 22, Theorem 7.9, Corollary 7.10]). Let X be an inlexible pseudoproper ibered category over a ield k. Then the pullback along X ÝÑ Π N X {k induces an equivalence o categories VectpΠ N X {kq ÝÑ EFinpVectpX qq. Let C be a Tannakian category. Then EFinpCq is the Tannakian subcategory o C o objects whose monodromy gerbe is inite. Deinition A cover : Y ÝÑ X is essentially inite i O Y is an essentially inite vector bundle. Deinition Let X be an inlexible and pseudo-proper iber category over a ield k. Given an object V o EFinpVectpX qq, the gerbe corresponding to the ull Tannakian subcategory o EFinpVectpX qq generated by V will be called the monodromy gerbe o V. When : Y ÝÑ X is an essentially inite cover, the monodromy gerbe o the cover is by deinition the monodromy gerbe o O Y.

7 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 7 Deinition 1.12 ([BV, Deinition 5.10]). A map X ÝÑ Γ rom a ibered category over k to a inite gerbe over k is called Nori reduced over k i any aithul morphism Γ 1 α ÝÑ Γ that its in a actorization X ÝÑ Γ 1 α ÝÑ Γ, where Γ 1 is a gerbe, is an isomorphism. A torsor P ÝÑ X under a inite group scheme G over k is called Nori reduced over k i the map X ÝÑ B G is Nori reduced over k. Remark I X is inlexible, then any map rom X to a inite gerbe actors uniquely through a Nori reduced map (see [BV, Lemma 5.12]). Moreover Π N X can be seen as the projective limit o the Nori reduced maps X ÝÑ Γ (see [BV, Theorem 5.7] and its proo). I X is an inlexible and pseudo-proper ibered category, and φ: X ÝÑ Γ is a map to a inite gerbe, then φ is Nori reduced i and only i the induced map Π N X ÝÑ Γ is a quotient; in this case VectpΓq ÝÑ EFinpVectpX qq is a sub Tannakian category. This is a direct consequence o Theorem 1.9 and the universal property o Π N X. Moreover φ O X» O Γ (see [BV, Lemma 7.11]). One o the key ingredient in the paper is the ollowing result. Theorem 1.14 ([TZ2, Corollary I]). Let X be a pseudo-proper and inlexible algebraic stack o inite type over a ield k o positive characteristic, and let : Y ÝÑ X be a surjective cover. I V P VectpX q, and V is ree, then V is essentially inite in VectpX q. Remark I X is a pseudo-proper and inlexible algebraic stack o inite type over a ield k (the characteristic is allowed to be 0), : Y ÝÑ X is a surjective étale cover and V P VectpX q is trivialized by, then it ollows that V is essentially inite with étale monodromy gerbe in EFinpVectpX qq. Indeed, by standard theory o étale covers one can assume that is an étale Galois cover. This case is exactly [TZ3, Lemma 1.4]. Lemma Let T 2 and let a ÝÑ T and T 1 b ÝÑ T be two maps o aine group schemes over k, R B T 1 B T 2 be the corresponding 2-Cartesian diagram. Then the ollowing two hold. (1) The unctor Ψ: BpT 2 ˆT T 1 q ÝÑ R mapping a T 2 ˆT T 1 -torsor to the associated T 2 and T 1 torsors is ully aithul and it is an equivalence i and only i the map T 1 ˆ T 2 ÝÑ T, pt 1, t 2 q ÞÑ bpt 1 qapt 2 q is an pqc epimorphism (e.g. i a or b is surjective). In this case a quasi-inverse is obtained mapping an object o R given by torsors P 2, P 1, P under T 2, T 1, T respectively and equivariant maps P 1 ÝÑ P and P 2 ÝÑ P to the iber product P 2 ˆP P 1. (2) I T 2 Spec k, so that B T 2 Spec k, and T 1 ÝÑ T is injective, then R T {T 1, where T {T 1 ÝÑ B T 1 is induced by the T 1 -torsor T ÝÑ T {T 1. In particular, i T 1 is a inite subgroup o T, then B T 1 ÝÑ B T is an aine map. Proo. The unctor Ψ maps the trivial torsor to pt 2, T 1, idq P Rpkq. A direct computation shows that the shea o automorphisms o this object is exactly T 2 ˆT T 1 (via Ψ). This B T

8 8 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG means that Ψ is an equivalence onto the ull-substack R 1 o R o objects locally isomorphic to pt 2, T 1, idq. Thus we have to understand when R 1 R. All objects o R are locally isomorphic to an object o the orm pt 2, T 1, cq P RpUq where U is an aine scheme and c P T puq is thought o as multiplication on the let T ÝÑ T. An isomorphism pt 2, T 1, 1q ÝÑ pt 2, T 1, cq is given by t 2 P T 2 puq and t 1 P T 1 puq such that capt 2 q bpt 1 q. Thus pt 2, T 1, cq is locally isomorphic to pt 2, T 1, 1q i and only i c is in the (pqc) image o T 1 ˆ T 2 ÝÑ T. The last claim o p1q ollows because i P is a T 2 ˆT T 1 -torsor inducing torsors P 2, P 1, P under T 2, T 1, T respectively then the commutative diagram P P 1 P 2 P is automatically Cartesian: locally, ater choosing a section o P, the above diagram is the one yielding T 2 ˆT T 1. Notice that the T 2 ˆT T 1 -space given in the last part o p1q is not a torsor in general because it may not have sections locally. For p2q, R is the shea o T 1 -torsors P together with an equivariant map P ÝÑ T, which is represented by T {T 1. Remark I : Y ÝÑ X is a cover o algebraic stacks then preserves vector bundles. Moreover since is exact we have H i py, Eq H i px, Eq or all i ě 0, E P QCohpYq. In particular, i X is pseudo-proper over k, then Y is pseudo-proper over k. Moreover we will oten use also the ollowing property: i or all vector bundles E on X one has dim k H 1 peq ă 8 then the same holds or vector bundles on Y. Lemma 1.18 ([TZ2, Lemma 2.5]). Let X be an algebraic stack over a ield k, o positive characteristic, such that dim k H 1 px, Eq ă 8 or all vector bundles E on X. Let G 0 ÝÑ G 1 ÝÑ ÝÑ G N 1 ÝÑ G N 0 be a sequence o surjective maps o quasi-coherent sheaves on X such that KerpG l 1 ÝÑ G l q is ree o inite rank or all 1 ď l ď N. Then there exists a surjective cover : X 1 ÝÑ X such that G l is ree o inite rank or all l. Remark 1.19 ([TZ1, Example 1.5, Corollary 1.7]). Let Γ be a inite stack or an aine gerbe over k. For all iber categories Z over k, the pullback o vector bundles establishes an equivalence o categories between Hom k pz, Γq and the groupoid o unctors VectpΓq ÝÑ VectpZq which are k-linear, monoidal and preserves short exact sequences in the category o quasi-coherent sheaves. Remark Let us comment on the relationship between the essentially inite vector bundles on a ibered category X over k and the vector bundles pullback rom a inite stack. When X is inlexible and pseudo-proper, these two notions agree as a consequence o [BV]. One o the key observation in [BV] is that i Γ is a inite stack over k and V P VectpΓq,

9 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 9 then V is essentially inite. More precisely, there is a inite vector bundle E on Γ and an exact sequence in CohpΓq 0 ÝÑ V ÝÑ E a ÝÑ E b ÝÑ E 1 ÝÑ 0 or some a, b P N and E 1 P VectpΓq. In particular, i φ: X ÝÑ Γ is any map rom a ibered category, then φ V is an essentially inite vector bundle on X. The proo o it is the same as that o [BV, Lemma 7.15] together with the ollowing clariication. First we can assume that Γ is connected. Let ρ: T ÝÑ Γ be a surjective cover rom a inite connected k-scheme T. The direct image E ρ O T is inite by [BV, Lemma 7.15]. Since the cokernel o V ÝÑ ρ ρ V» E rk V is a vector bundle one can easily construct the above sequence. Now let X be a ibered category and V P VectpX q. I V is essentially inite, one might argue that there is a homomorphism between two inite vector bundles q : E 1 ÝÑ E 2 whose kernel is V. This is actually misleading. Since the deinition o essentially inite in the category VectpX q is intrinsic to this category, V has to be a kernel o q inside the category VectpX q. This does not imply that V coincides with the kernel K o q in QCohpX q. This equality holds i X is pseudo-proper and inlexible. I X has the resolution property, that is all quasi-coherent sheaves are quotient o sum o vector bundles (e.g. when X is a quasiprojective scheme or a smooth separated scheme), and i the kernel V exists in the category VectpX q, then V K. In order to avoid the above mentioned issue, i X is pseudo-proper but not inlexible it seems to us that the correct essentially inite vector bundles to use are vector bundles coming rom a inite stack or at least that are kernel in QCohpX q o a map o inite vector bundles. Although this is not an intrinsic notion it would be a good working deinition. This should also explain why Lemma 2.10 and Lemma 2.8 should be understood as results assuring that pushorward preserves essentially inite vector bundles. In any case in the present paper we consider essentially inite vector bundles only on pseudo-proper and inlexible ibered categories, but we maintain the notion o essentially inite in Deinition 1.5. There is a partial converse to the act that vector bundles coming rom inite stacks are essentially inite. I X is a ibered category over k with dim k H 0 po X q ă 8 and V P VectpX q is a inite vector bundle, then there exist a map φ: X ÝÑ Φ to a inite stack and W P VectpΦq such that V» φ W. This is essentially proved in [BV, p. 19] (just ater the proo o Lemma 7.11). We recall here the construction or the convenience o the reader. We can assume that V has rank r and take g P Nrxs such that pv q» gpv q. The group GL r acts on the scheme I Isoppk r q, gpk r qq GL N with N prq gprq. The isomorphism pv q» gpv q gives a actorization o the vector bundle V : X ÝÑ B GL r through ri{ GL r s

10 10 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG and we have Cartesian diagrams Ω I X rω{ GL r s ri{ GL r s B GL r Spec H 0 po X q I{GL r Here we are using that I ÝÑ I{ GL r is a geometric quotient and I{ GL r is aine because GL r is geometrically reductive. Thus we must show that Φ rω{ GL r s is a inite stack. As the geometric ibers o I ÝÑ I{ GL r consist (topologically) o one orbit and H 0 po X q is a inite k-algebra one sees that Φpkq has initely many isomorphism classes. The action o GL r on I has inite stabilizers by [BV, Lemma 7.12] and hence it ollows that the diagonal o Φ is quasi-inite. By [BV, Proposition 4.2] it ollows that Φ is a inite stack. Lemma Let X be a quasi-compact and quasi-separated algebraic stack and u: X ÝÑ Γ a map to an aine gerbe. Then u : VectpΓq ÝÑ VectpX q is ully aithul i and only i u # : O Γ ÝÑ u O X is an isomorphism. Proo. For V, V 1 P VectpΓq the composition Hom Γ pv, V 1 q u ÝÑ Hom X pu V, u V 1 q» Hom Γ pv, u u V 1 q» Hom Γ pv, V 1 b u O X q is the map induced by u # : O Γ ÝÑ u O X. In particular, i this map is an isomorphism then u : VectpΓq ÝÑ VectpX q is ully aithul. Conversely, setting V 1 O Γ above, we have that the homomorphism Hom Γ pv, O Γ q ÝÑ Hom Γ pv, u O X q induced by u # is an isomorphism or all vector bundles V over Γ. Since u O X is a quasicoherent shea, it is a quotient o a sum o vector bundles [De, p. 132, Corollary 3.9]. Thereore, above isomorphism implies that u # is surjective. Since u: X ÝÑ Γ is aithully lat we also have that u # injective. Remark I X is a pseudo-proper and inlexible algebraic stack over k and α: X ÝÑ Π N X is the structure map, then using Theorem 1.9 and Lemma 1.21 we conclude that α O X» O Π N X. The same holds or the Nori étale undamental gerbe. 2. Essentially inite covers and their Nori gerbes Let k be a base ield. In this section we study the notion o an essentially inite cover, which generalizes the notion o torsor under a inite group scheme. Moreover we are going to prove Theorem III and Corollary I. Recall that an essentially inite cover : Y ÝÑ X o ibered categories is a cover such that O Y is essentially inite as object o VectpX q (see Deinition 1.10). First observe that a torsor under a inite group scheme is an essentially inite cover. Indeed, let : Y ÝÑ X be a torsor under a inite group scheme G over k corresponding to

11 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 11 u: X ÝÑ B G. Then u pkrgsq» O Y, where krgs is the regular representation. Applying [BV, Lemma 7.15] to Spec k ÝÑ B G we see that krgs is inite in VectpB Gq Rep G and thus O Y is a inite vector bundle. Proposition 2.1. Let X be a pseudo-proper and inlexible ibered category over k. Then there are equivalences o categories " * " * " * Stacks inite Φ Essentially inite Ψ Finite k-schemes with over Π N X covers o X an action o π N px, xq where Φ is the pullback along X ÝÑ Π N x X and Ψ is the pullback along Spec k ÝÑ X with x P X pkq. Furthermore, Ψ extends the correspondence between pointed Nori reduced torsor o X and quotient group schemes o π N px, xq. Proo. The unctor Φ is the equivalence mapping ring objects o VectpΠ N X q to ring objects o EFinpVectpXqq. I x P X pkq, then Π N X B πn px, xq, and the ring objects o VectpΠ N X q Rep π N px, xq are precisely the inite k-algebras with an action o π N px, xq. This easily implies that Ψ Φ is an equivalence. The last claim ollows by construction. Lemma 2.2. Let X be a pseudo-proper and inlexible algebraic stack o inite type over k and : Y ÝÑ X an essentially inite cover; let u: X ÝÑ Γ be the monodromy gerbe o O Y P EFinpVectpX qq. Then there exists a unique extension o and also v O Y» O. I Y is inlexible then is a inite gerbe. Y X v u Proo. The multiplication map o O Y and its unit map lie in VectpΓq Ď VectpX q and thereore determine a cover ÝÑ Γ (1) extending as claimed in the statement. Uniqueness o the extension ollows rom the act that VectpΓq ÝÑ VectpX q is ully aithul. As u is Nori reduced we have that u O X» O Γ. Since ÝÑ Γ is lat we also conclude that v O Y» O. Assume now that Y is inlexible. By deinition, Y ÝÑ actors through a inite gerbe 1, which can be chosen as closed substack 1 Ď. But v O Y» O which implies that 1 as required. Here are some technical lemmas which will be used in proving Theorem III. Lemma 2.3. Consider a 2-Cartesian diagram Y X v u Ψ Φ Γ π

12 12 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG where X is a quasi-compact and quasi-separated algebraic stack, π is aine and aithully lat, Φ is a stack such that u O X» O Φ and u is aithully lat. Then v O Y» O Ψ, and the two unctors u : VectpΦq ÝÑ VectpX q and v : VectpΨq ÝÑ VectpYq are ully aithul. A vector bundle V P VectpX q lies in the essential image o u : VectpΦq ÝÑ VectpX q i and only i V comes rom a vector bundle on Ψ. I π is a surjective cover, a vector bundle V P VectpYq lies in the essential image o v : VectpΨq ÝÑ VectpYq i and only i V comes rom a vector bundle on Φ. Proo. By [BV, Lemma 7.17] and lat base change it ollows that v O Y» O Ψ, u : VectpΦq ÝÑ VectpX q and v : VectpΨq ÝÑ VectpYq are ully aithul. Denote by D and C the essential images o these u and v respectively. Let V P VectpX q. We must show that V P D i and only i V P C. The only i part is clear. Conversely, suppose that V v W with W P VectpΨq, and consider the canonical homomorphism u u V ÝÑ V ; pulling back by one gets v v p V q v v pv W q ÝÑ v W V. This homomorphism is an isomorphism because v O Y» O Ψ. As is aithully lat, one concludes that V» u u V, and as u is aithully lat, it ollows that u V is a vector bundle. Thus we have V P D. Assume now that π is a surjective cover; consequently is also a surjective cover. In particular, π and send vector bundles to vector bundles. Given V P VectpYq we must show that V P C i and only i V P D. The only i part is easy: i W P VectpΨq then pv W q» u π W because π is aine. For the converse, assume that V P D, meaning V comes rom a vector bundle on Φ. Since u O X» O Φ it ollows that u p V q is a vector bundle and the canonical homomorphism u u p V q ÝÑ p V q is an isomorphism. This homomorphism can also be obtained by applying to the canonical homomorphism v v V ÝÑ V. Since is aine this means that the previous homomorphism is an isomorphism. To conclude that V P C it suices to show that v V is a vector bundle. But v is aithully lat and v pv V q is a vector bundle. Now by descent it ollows that v V is also a vector bundle. Remark 2.4. Consider a G-torsor : Y Ñ X or an aine group scheme G, where X is a quasi-compact and quasi-separated algebraic stack, and the corresponding 2-Cartesian diagram Y X v u Spec k We see that H 0 po Y q k i and only i u O X» O B G. In this case, applying Lemma 2.3, we conclude that u : VectpB Gq ÝÑ VectpX q is ully aithul with essential image the category o vector bundles V such that V is trivial. B G

13 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 13 Lemma 2.5 ([No2, p. 264, Lemma 1]). Let X be a quasi-compact and quasi-separated algebraic algebraic stack and Z h X a 2-commutative diagram, where and g are torsors or aine group schemes G and H respectively. Suppose that H 0 po Z q k. Then there exists a homomorphism ϕ: G ÝÑ H inducing h. Moreover, h is aithully lat i and only i ϕ: G ÝÑ H is aithully lat, in which case h: Z ÝÑ Y is a torsor or the kernel o ϕ. I H{ Impϕq is aine (e.g. i H or G are inite) then this is also equivalent to the statement that H 0 po Y q k. Proo. Consider the morphisms u: X ÝÑ B G and v : X ÝÑ B H corresponding to the torsors and g respectively. Since H 0 po Z q k we have that u O X» O B G, and hence the pullback unctor u : VectpB Gq ÝÑ VectpX q is ully aithul. The objects o the essential image o v : VectpB Hq ÝÑ VectpX q are trivialized by g and thus by. From Remark 2.4 we obtain a actorization v : VectpB Hq ÝÑ VectpB Gq Ď VectpX q γ ÝÑ B H. Con- which, by Tannakian duality, is induced by a actorization v : X sider the 2-Cartesian diagrams Z h a Spec k w g Y Y U Spec k g g 1 u X B G B H γ u ÝÑ B G We claim that there exists a dashed arrow w as above making the upper diagram 2- Cartesian. This would imply that the unctor γ : B G ÝÑ B H is induced by a group homomorphism ϕ: G ÝÑ H. Set 1 : Spec k ÝÑ B G. Consider the map o O X -algebras λ: g O Y ÝÑ O Z. Applying u to λ and using a O Y» O U, H 0 po Z q k, we get that g 1 O U g 1 a O Y u g O Y ÝÑ u O Z 1 O Spec pkq. Applying Spec B G p q on both sides we get the arrow w. To prove that h is the pullback o w we just have to show that the adjunction maps u u g O Y ÝÑ g O Y and u u O Z ÝÑ O Z are isomorphisms. But the adjunction map or g O Y coincides with the ollowing composition: u u g O Y u g 1 a O Y u g 1 O U g O Y which is an isomorphism, and the same is true or O Z.

14 14 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG The map h is aithully lat i and only i w is aithully lat. By Lemma 1.16 this is the case i and only i ϕ: G ÝÑ H is surjective, so that U BpKerpϕqq. The condition that H 0 po Y q k is equivalent to the condition that v O X» O B H and, by Lemma 1.21, to the ully aithulness o the unctor γ : VectpB Hq ÝÑ VectpB Gq. This last condition is equivalent to the surjectivity o ϕ: G ÝÑ H when H{ Impϕq is aine (see [TZ1, Remark B.7]). Pointed essentially inite covers admit a natural Galois closure by a torsor, as explained in the ollowing proposition. Proposition 2.6. Let X be a pseudo-proper and inlexible ibered category over k and : Y ÝÑ X an essentially inite cover with a rational point y P Ypkq. Denote by Γ the monodromy gerbe o O Y in EFinpVectpX qq and by ÝÑ Γ the cover in (1) that extends. Then there are Cartesian diagrams p Spec k P Spec k x y Y X The map π : P ÝÑ X is a pointed Nori reduced torsor or the inite group scheme G Aut Γ pupxqq, and Γ is the monodromy gerbe o π O P. The map λ: P ÝÑ Y is aithully lat i and only i is a gerbe in which case it is a Nori reduced torsor or a subgroup scheme o G. In the general case λ: P ÝÑ Y actors as η λ 1, where λ 1 : P ÝÑ Y 1 is aithully lat and a Nori reduced torsor or a inite subgroup scheme o G, and η : Y 1 ÝÑ Y is a closed immersion. Finally the torsor π : P ÝÑ X has the ollowing universal property: or any pointed Nori-reduced torsor g : pt, tq ÝÑ px, xq or a inite group scheme G, and any pointed aithully lat X -morphism h : pt, tq Ñ py, yq, there is a unique actorization h λ j, where j : pt, tq ÝÑ pp, pq is equivariant with respect to a surjective homomorphism G ÝÑ G. Proo. The existence o the diagram is clear. Since u: X ÝÑ Γ B G is Nori reduced so is the G -torsor P ÝÑ X. The claim about the monodromy gerbe o π O P ollows rom the ollowing act: i G is a inite group scheme over k, the regular representation krgs generates Rep G because every inite G-representation is a sub object o some krgs n. The morphism λ is aithully lat i and only i Spec k ÝÑ is aithully lat. This is the case i and only i is a gerbe. In such a situation we have B H, where H is a λ subgroup o G and P ÝÑ Y is an H-torsor. The map v : Y ÝÑ B H is Nori reduced because v O Y» O. The actorization λ η λ 1 arises rom the actorization Spec k ÝÑ 1 ÝÑ, where 1 is a subgerbe o the inite stack. λ u Γ

15 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 15 For the last statement, h induces an inclusion O Y Ă h O T, and as is aine, an inclusion O Y Ă g O T. I we denote by xg O T y (respectively, xπ O P y) the ull Tannakian subcategory o EFinpVectpX qq generated by the object g O T (respectively, π O P ), one gets the ollowing 2-Cartesian diagram: VectpB G q VectpB Gq u xπ O P y v xg O T y x VectpSpec kq where v : X ÝÑ B G corresponds to the torsor g : T ÝÑ X, the horizontal arrows are inclusions and the vertical arrows are equivalences. As the torsors are pointed above x, the unctors x u and x v are equivalent to the orgetul unctors. Thereore, the commutativity o this last diagram proves the existence o a surjective morphism ϕ: G ÝÑ G such that u» v ϕ : VectpB G q ÝÑ VectpX q. Thus implies that u ϕ 1 v, where ϕ 1 : B G ÝÑ B G the morphism o gerbes induced by ϕ. Lemma 2.7. Let X be an algebraic stack over a ield k, A a inite and local k-algebra with residue ield k and F P VectpX ˆ Spec Aq. Let X ÝÑ i X ˆ Spec A ÝÑ X be the maps corresponding to k ÝÑ A ÝÑ k. Then there is a sequence o surjective maps o vector bundles F G N ÝÑ G N 1 ÝÑ ÝÑ G 1 ÝÑ G 0 0 such that KerpG l ÝÑ G l 1 q» i F or all 1 ď l ď N. Proo. Consider a decomposition series o A-modules A A N ÝÑ A N 1 ÝÑ ÝÑ A 1 ÝÑ A 0 0 ; the above maps are surjective with KerpA l ÝÑ A l 1 q» k as A-modules or all 1 ď l ď N. Let p: X ˆ Spec A ÝÑ Spec A be the projection; consider the unctor Ψ pf b p p qq: ModA ÝÑ QCohpX q. Since p is lat, F is a vector bundle, and as is aine the unctor Ψ is exact. Moreover ΨpAq F and Ψpkq pf b p kq» pf b i O X q» i i F» i F. Applying Ψ to the above sequence o A-modules we ind the desired sequence. Lemma 2.8. Let X be a pseudo-proper and inlexible algebraic stack o inite type over a ield k o positive characteristic, such that dim k H 1 px, Eq ă 8 or all vector bundles E on X. Let : Y ÝÑ X be an essentially inite cover. Then or all maps φ: Y ÝÑ Φ to a inite stack over k, and or all W P VectpΦq, the vector bundle φ W is essentially inite in VectpX q. x

16 16 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG Proo. To avoid problems with dierent ranks, we irst observe that the rank o W can be assumed to be constant, or instance, by considering the connected components o Φ. By Lemma 2.2 we have a Cartesian diagram Y X v u Γ where Γ is a inite gerbe. We will prove that there exists a surjective cover s: X 1 ÝÑ X such that s φ W is ree on the connected components o X 1 ; this will be done by only assuming there is a Cartesian diagram as above with Γ a inite gerbe and inite, but without requiring that X is inlexible, as this will be more useul. In the case when X is inlexible the above claim ollows rom Theorem Having weakened the hypothesis, i s 1 : X 1 ÝÑ X is a surjective cover we can always replace X by X 1. Let L{k be a inite ield extension with a map Spec L ÝÑ Y. The base change o Spec L ÝÑ Y ÝÑ ÝÑ Γ along X ÝÑ Γ is a surjective cover Q ÝÑ X. Replacing X by Q we can assume that X ÝÑ Γ actors as X ÝÑ Spec L ÝÑ Γ. This means that Y ÝÑ X is the projection X ˆL A ÝÑ X, where A{L is a inite L-algebra. Since φ actors as Y ÝÑ ΦˆkL ÝÑ Φ we can assume that L k. Extending again k we can urther assume that A is a product o local k-algebras with residue ield k. Splitting Y according to the decomposition o A we can assume that A is also local. Let i: X ÝÑ Y X ˆ A be the inclusion corresponding to A ÝÑ k. Finally consider a surjective cover T ÝÑ Φ rom a inite scheme and the 2-Cartesian diagram Z T i φ X Y Φ Replacing X by Z we can assume that φ i: X ÝÑ Φ actors through a inite scheme, which in particular implies that i φ W is ree. Applying Lemma 2.7 to F φ W we obtain a sequence o surjective homomorphisms o quasi-coherent sheaves on X pφ W q G N ÝÑ G N 1 ÝÑ ÝÑ G 1 ÝÑ G 0 0 with ree kernels. The inal cover trivializing pφ W q exists due to Lemma Remark 2.9. Assume char k 0. It is unclear whether Lemma 2.8 continues to hold and under which hypothesis on X it may hold. Clearly the initeness o H 1 does not help since Lemma 1.18 only holds in positive characteristic. Let Y ÝÑ X be an essentially inite cover over a pseudo-proper and inlexible algebraic stack o inite type over k. Following the proo o Lemma 2.8, taking into account Remark 1.15 we have the possibility o extending the base ield [BV, Proposition 6.1]. Consequently, considering L k and noticing that the cover constructed Q ÝÑ X is a Nori reduced étale torsor (so that Q is inlexible by Theorem III we are going to prove) it ollows that we can

17 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 17 reduce the problem to the case where is the projection Y X ˆ A ÝÑ X and A is a inite and local k-algebra with residue ield k. I i: X ÝÑ Y corresponds to A ÝÑ k, and F φ W, then i F is essentially inite in X. Thus replacing again X by the total space o an étale Nori reduced torsor we can urther assume that i F ree. By Lemma 2.7 we obtain a sequence o surjective homomorphisms F G N ÝÑ G N 1 ÝÑ ÝÑ G 1 ÝÑ G 0 0 with KerpG l ÝÑ G l 1 q ree or all 1 ď l ď N. Missing Lemma 1.18 we can t go urther. Moreover there are sequences like this with G N not essentially inite, or instance when X B G a with V being the 2-dimensional representation given by G a U 2. The representation V is a nontrivial extension o k by itsel and is not essentially inite because the proinite quotient o G a and thereore the Nori gerbe o B G a are trivial. Set ξ : V ÝÑ k ÝÑ V. Since ξ 2 0, ξ determines a shea F on B G aˆpkrxs{px 2 qq whose pushorward to B G a is V. It is also possible to show that F is an invertible shea. On the other hand it seems unlikely that F is essentially inite. At least with some urther computation one can exclude the possibility o being inite. Lemma Let X be a pseudo-proper and inlexible algebraic stack o inite type over k, and let : Y ÝÑ X be an étale surjective cover. Then or all maps φ: Y ÝÑ Φ to a inite (respectively, inite and étale) stack over k and or all W P VectpΦq, the vector bundle pφ W q is essentially inite (respectively, essentially inite with étale monodromy gerbe) in VectpX q. In particular, is essentially inite. Proo. There exists a Cartesian diagram \ i Z Y Z b r where r : Z ÝÑ X is an étale surjective cover. Since a φ W also comes rom the inite stack Φ, taking a inite atlas o Φ we can ind a surjective cover λ: U Ñ Ů i Z which trivializes it. Denote by U i the inverse image o the i-th piece o Ů i Z under λ. Then r φ W b a φ W is trivialized by the surjective cover U 1 ˆZ ˆ ˆZ U n ÝÑ Z. Thus, by Remark 1.15, φ W is essentially inite and it has an étale monodromy gerbe i Φ is étale (so that λ can also be chosen to be étale). Proo o Theorem III. By Lemma 2.2 there are 2-Cartesian diagrams a X Y Π β α X Π N X Γ where Γ is the monodromy gerbe o O Y P EFinpVectpX qq. Notice that, since Π N X ÝÑ Γ is a quotient, the stack is a gerbe i and only i Π is a gerbe. It ollows rom Lemma 2.2 that Π is a gerbe i Y is inlexible. π

18 18 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG For the converse assume that Π is a gerbe. By Remark 1.22 and Lemma 2.3, the pullback unctor β : VectpΠq ÝÑ VectpYq is ully aithul with essential image the ull subcategory C o VectpYq o vector bundles V such that V P EFinpVectpX qq. Since Π ÝÑ Π N X is aithul the gerbe Π is proinite, so we have C Ď EFinpVectpYqq. We will show that this is an equality and that Y is inlexible. This would immediately imply that Y ÝÑ Π is the Nori undamental gerbe. Notice that i char k 0 and Π is a gerbe then is a gerbe and thereore ÝÑ Γ and : Y ÝÑ X are étale covers. Let φ: Y ÝÑ Φ be a map to a inite stack. Using Lemma 2.10 and Lemma 2.8 it ollows that φ W P C or all W P VectpΦq. Thus the pullback by φ: Y ÝÑ Φ has a actorization VectpΦq ÝÑ VectpΠq» C Ď VectpYq. By Remark 1.19 one gets a actorization Y ÝÑ Π ÝÑ Φ as required. This shows that Y ÝÑ Π is a Nori undamental gerbe and, in particular, Y is inlexible. Since Y is pseudoproper, all essentially inite vector bundles on Y are pullbacks rom some inite gerbe. Thus the above actorization also implies the equality C EFinpVectpYqq. Notice that i Y is inlexible then one always has H 0 po Y q k (see Remark 1.8). The étale case. Assume that is étale and that H 0 po Y q k. By Lemma 2.3 we have that H 0 po q k. In particular, is geometrically connected. Since ÝÑ Γ is étale, it ollows that is also geometrically reduced. Using [BV, Proposition 4.3] we conclude that and thereore Π are gerbes. Thus Y is inlexible. By Lemma 2.10 we see that Γ is étale. In particular there are 2-Cartesian diagrams Y Π N Y Π 1 X Π N X Π N,ét X We have to show that Π 1 Π N,ét Y. By Lemma 2.3 the pullback VectpΠ1 q ÝÑ VectpYq is ully aithul and its essential image C consists o vector bundles V P VectpYq such that V comes rom Π N,ét X. One must show that the essential image D o the ully aithul map VectpΠ N,ét Y q ÝÑ VectpYq coincides with C. The vector bundles in D are the essentially inite vector bundles with étale monodromy gerbes. Since the map Π 1 ÝÑ Π N,ét X is aithul, it ollows that Π 1 is proétale, so that C Ď D. The opposite inclusion instead ollows rom Lemma The torsor case. Assume that is a torsor or a inite group scheme G. Since the regular representation krgs generates Rep pgq as k-tannakian category, there are 2-Cartesian diagrams Γ Y β Π Spec pkq X α π Π N X Γ B G

19 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 19 and the map Γ ÝÑ B G is aithul and, by [TZ1, Remark B.7], aine. In particular is a inite scheme, and by Lemma 2.3 we can conclude that Spec ph 0 po Y qq. Now the theorem ollows because Y is inlexible i and only i is a gerbe. Proo o Corollary I. The Nori gerbe o X is Π N X B πn px, xq, and X ÝÑ B π N px, xq maps x to the trivial torsor. I py, yq ÝÑ px, xq is an essentially inite cover, then the extension Π ÝÑ Π N X deined in Theorem III is described by the 2-Cartesian diagrams 1 Spec k π N px, xq r X Spec k y Y x Y ry x {π N px, xqs x Spec k X B π N px, xq that is Π ry x {π N px, xqs. Again by Theorem III we have that Y is inlexible i and only i Π is a gerbe. On the other hand, the ollowing three conditions are equivalent: Π is a gerbe, Spec k ÝÑ Π is aithully lat, and the orbit map π N px, xq ÝÑ Y x o y is aithully lat. When these equivalent conditions hold, by Theorem III we have that Π B π N py, yq and Y x» π N px, xq{π N py, yq. The equivalence o categories in the statement ollows easily rom Proposition 2.1. Let : py, yq ÝÑ px, xq be an essentially inite cover with Y inlexible. I is a torsor or a group G, using Lemma 1.16 the last diagram in Theorem III tells that π N py, yq is normal in π N px, xq with quotient G. Conversely, i π N py, yq is normal in π N px, xq with quotient G, then using again Lemma 1.16, B π N py, yq ÝÑ B π N px, xq and its base change : Y ÝÑ X is torsor or G. For the last claim, considering the above Cartesian diagrams we see that the ollowing three are equivalent: is étale, Π B π N py, yq ÝÑ π N px, xq is étale, Y x» π N px, xq{π N py, yq ÝÑ Spec k is étale. This completes the proo. Theorem Let X be a pseudo-proper and inlexible ibered category over k and : Y ÝÑ X be a cover. The ollowing are equivalent: (1) Y is inlexible, is essentially inite and O Y has étale monodromy gerbe in EFinpVectpX qq. (2) is étale and H 0 po Y q k. Proo. The implication p2q ùñ p1q ollows rom Lemma 2.10 and Theorem III. For the converse, let ÝÑ Γ be as in Lemma 2.2. Since Y is inlexible it ollows that is a gerbe.

20 20 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG Moreover by hypothesis Γ is étale. Thus is étale too because ÝÑ Γ is aithul. By base change it ollows that is étale. 3. Tower o torsors and their Galois closure Let k be a base ield and G and H be inite group schemes over k. In this section we introduce the notion o tower o torsors and Galois closure o a tower o torsors. At the end o the section Theorems I and II will be proved. Deinition 3.1. A pg, Hq-tower o torsors over a ibered category X over k is a sequence o map o ibered categories Z ÝÑ h Y ÝÑ g X where g is a G-torsor and h is an H-torsor. When G, H are clear rom the context we will just talk about a tower o torsors. The pg, Hq-tower is called pointed over k i Z ÝÑ X (and thereore Y ÝÑ X ) is a pointed cover over k. We deine the stack BpG, Hq as the stack over A{k whose section over an aine scheme U is the groupoid o pg, Hq-tower o torsors over U. The isomorphisms o towers are deined in the obvious way. Let us start with a preliminary remark: Remark 3.2. A tower o torsors over a ibered category X is the same as a map X ÝÑ BpG, Hq. Moreover BpG, Hq has a universal tower given by W u Spec k Hom k pg, B Hq B H Spec k BpG, Hq B G where u is the restriction along 1 : Spec k ÝÑ G. The pullback o the above tower along a map X ÝÑ BpG, Hq yields exactly the tower encoded in the map X ÝÑ BpG, Hq. Proposition 3.3. The stack BpG, Hq is algebraic, locally o inite type over k and has aine diagonal. Proo. The stack BpG, Hq is algebraic and locally o inite type over the ield k because Hom k pg, B Hq is so by [HR, Theorem 3] and the act that there is a inite and lat map Hom k pg, B Hq ÝÑ BpG, Hq by Remark 3.2. Now let P i ÝÑ Pi 1 ÝÑ T, i 1, 2, be two towers ξ i P BpG, HqpT q or some aine scheme T, and set I Iso BpG,Hq pξ 1, ξ 2 q. We need to show that I is aine. We denote by p q T the base change to T. Base changing along P1 1 ˆT P2 1 ÝÑ T allows us to assume that Pi 1 G T. In particular, there is a map a: I ÝÑ G T Aut G T pg T q and we want to show that it is aine. I W is the iber o a along a map T ÝÑ g G T, it is enough to show that W is aine. Let P 2 ÝÑ G T be the H-torsor base change o P 2 ÝÑ G T along the multiplication

21 ESSENTIALLY FINITE COVERS AND TOWERS OF TORSORS 21 G T ÝÑ G T by g and set J Iso H G T pp 1, P 2 q ÝÑ G T. It is easy to see that W W GT pjq, where W GT pjq is the Weil restriction o J along G T ÝÑ T. The scheme J is aine and initely presented over G T. Since G T ÝÑ T is a cover using a presentation o J it is easy to write W as a closed subscheme o an aine space over T. Lemma 3.4. Let X be a pseudo-proper and inlexible algebraic stack o inite type over k, and let Z ÝÑ h Y ÝÑ g X be a pg, Hq-tower o torsors. I char k ą 0 assume that either g is étale or dim k H 1 px, Eq ă 8 or every vector bundle E on X. Then g h is an essentially inite cover. Proo. Consider the morphism v : Y ÝÑ B H corresponding to the H-torsor h: Z ÝÑ Y. One knows that h O Z v pkrhsq, where krhs denotes the regular representation. Thus we have pg hq O Z g v pkrhsq. When char k ą 0, one concludes the proo by Lemma 2.8 and when char k 0 by Lemma Lemma 3.5. Let P ÝÑ G ÝÑ Spec k be a rational point ξ o BpG, Hq with a point p P P pkq mapping to 1 P Gpkq, and let Q ξ Aut BpG,Hq pξq. Then Q ξ is an aine group scheme o inite type over k, and there are exact sequences 0 ÝÑ W G paut H GpP qq ÝÑ Q ξ α ÝÑ G, 0 ÝÑ W 1 ÝÑ W G paut H GpP qq β ÝÑ H where α orgets the automorphism o P, W G denotes the Weil restriction and β is the evaluation at 1: Spec k ÝÑ G. There is a ully aithul map B Q ξ ÝÑ BpG, Hq sending the trivial torsor to ξ and whose image consist o the towers locally isomorphic to ξ. Proo. The scheme Q ξ is aine o inite type by Proposition 3.3. The irst sequence is clear. The map β is well deined because the point p P P pkq gives an H-equivariant isomorphism between H and the base change o P ÝÑ G along the identity: the base change o Aut H GpP q ÝÑ G along the identity i Aut H k phq H. The last claim is standard. More can be said in the case o the trivial tower. Lemma 3.6. Let Q be the shea o automorphisms o the trivial tower G ˆ H ÝÑ G ÝÑ Spec k in BpG, Hq. Then Q is an aine group scheme o inite type, Q W G phq G where W G phq is the Weil restriction o the group scheme H over G along G ÝÑ Spec k with G acting on W G phq via automorphisms o the base. Evaluation at 1 P G yields a map W G phq ÝÑ H and, i W 1 is the kernel, then W G phq W 1 H. The ully aithul map B Q ÝÑ BpG, Hq o Lemma 3.5 corresponds to the pointed tower o Nori reduced torsors B W 1 ÝÑ B W G phq ÝÑ B Q.

22 22 M. ANTEI, I. BISWAS, M. EMSALEM, F. TONINI, AND L. ZHANG I G or H is étale over k then Q is a inite group scheme. Proo. Consider the maps α and β deined in Lemma 3.5. We have P G ˆ H, so that Aut H GpP q H ˆ G ÝÑ G. Moreover it is easy to see that both α and β are surjective. The map G ÝÑ Q, g ÞÝÑ pt g ˆ id H, t g q, where t g is the multiplication by g, produces the irst decomposition. The map HpT q ÝÑ HpG ˆ T q W G phqpt q produces the second decomposition. The claim about the tower o B Q is an easy consequence o Lemma Now assume that G or H are étale. We can also assume that k is algebraically closed, so that G or H are constant. I G is constant then W G phq H #G is a inite scheme and thereore Q is inite. Thus assume H constant. The map G red ÝÑ G, where p q red denotes the reduction, is a nilpotent closed immersion and, since H is étale, it ollows that the map W G phq ÝÑ W Gred phq is an isomorphism. But G red G ét and again we conclude that W Gred phq is inite. Proposition 3.7. I G or H is étale then the map B Q ÝÑ BpH, Gq is an equivalence, where Q is the shea o automorphisms o the trivial tower G ˆ H ÝÑ G ÝÑ Spec k in BpG, Hq. In particular, BpG, Hq is a inite neutral gerbe over k. Proo. In view o Lemma 3.6 it suices to show that any tower is pqc locally trivial. Let P ÝÑ P 1 ÝÑ U be a tower over an aine scheme. Using base changing along P ÝÑ U we may assume this map has a section, so that, in particular, P 1 U ˆ G. We can also assume that k is algebraically closed, so that G or H is constant. I G is constant, P ÝÑ U ˆ G is given by #G many H-torsors over U and, trivializing those torsors, one gets a trivialization o P ÝÑ U ˆ G. Now consider H to be étale. Since P ÝÑ U ˆ G is étale and U ˆG red ÝÑ U ˆG is a nilpotent closed immersion we conclude that P ÝÑ U ˆG has a section i and only i its restriction to U ˆ G red is trivial. Thus we reduce to the known case where G G red G ét is étale. We now move to the problem o inding a Galois closure or a given tower o torsors. Deinition 3.8. Let X be a ibered category, and let Z ÝÑ Y ÝÑ X be a pg, Hq-tower o torsors. A Galois closure or the pg, Hq-tower consists o the ollowing data: a inite group scheme G with homomorphisms o group schemes α: G ÝÑ G and Kerpαq ÝÑ H, a G-torsor P ÝÑ X together with a actorization P ÝÑ Z such that P ÝÑ Y is G-equivariant and P ÝÑ Z is Kerpαq-equivariant. We say that the Galois closure is Nori reduced i the G-torsor P ÝÑ Z is Nori reduced. Theorem I is now easy to deduce. Proo o Theorem I. By Lemma 3.6 and Proposition 3.7 we have BpG, Hq B Q, with universal tower B W 1 ÝÑ B W G phq ÝÑ B Q. The Q-torsor Spec k ÝÑ B Q with splitting

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