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Local-global principles for homogeneous spaces over some two-dimensional geometric global fields

Diego Izquierdo and Giancarlo Lucchini Arteche ORCID logo

Abstract

In this article, we study the obstructions to the local-global principle for homogeneous spaces with connected or abelian stabilizers over finite extensions of the field ((x,y)) of Laurent series in two variables over the complex numbers and over function fields of curves over ((t)). We give examples that prove that the Brauer–Manin obstruction with respect to the whole Brauer group is not enough to explain the failure of the local-global principle, and we then construct a variant of this obstruction using torsors under quasi-trivial tori which turns out to work. In the end of the article, we compare this new obstruction to the descent obstruction with respect to torsors under tori. For that purpose, we use a result on towers of torsors, that is of independent interest and therefore is proved in a separate appendix.

Funding statement: The second author’s research was partially supported by ANID via FONDECYT Grant 1210010 and PAI Grant 79170034.

A A result on towers of torsors

The goal of this appendix is to prove the following result, which is needed in the proof of Theorem 6.4 above. We are greatly indebted to Mathieu Florence for his help with the proof.

Theorem A.1.

Let K be a field of characteristic 0. Let G be a connected linear K-group and T an algebraic K-torus. Let YX be a G-torsor and let ZY be a T-torsor. Assume that Pic(G¯)=0 (which holds, for instance, if G is a torus) and that X is geometrically integral. Then there exists a canonical extension

1TEG1

such that the composite ZX is an E-torsor. Moreover, if G is a torus, then E is a torus as well.

Remark A.2.

One can find similar results in [1, Appendix A] and [4, Lemma 2.13], but none of these seems to be general enough for our purposes. We hope to generalize this result even further in the future.

Proof.

For a K-scheme W/K, let XW,YW,ZW,TW,GW be the W(-group)-schemes obtained by base change from X,Y,Z,T,G, respectively. Consider the group AutXWTW(ZW) of XW-automorphisms φ of ZW that are compatible with the action of TW in the sense that the following diagram commutes:

where a denotes the morphism defining the action of T on Z and aW the corresponding morphism after base change. The functor W/KAutXWTW(ZW) defines a group presheaf over the big étale site over K. Denote by Aut¯XT(Z) the corresponding sheaf and consider the subsheaf Aut¯YT(Z) defined by taking the subgroup AutYWTW(ZW) of AutXWTW(ZW) for each W/K. We have

AutYWTW(ZW)=T(YW).

Indeed, it is well known that the functor W/YAutWTW(Z×YW) over the big étale site of Y is represented, as a Y-scheme, by TY (cf. for instance [13, Chapter III, Section 1.5]). Moreover, a direct application of Rosenlicht’s Lemma gives us, for geometrically irreducible W,

T(YW)=T(W)×M(W),

where M is a free and finitely generated abelian constant sheaf. We deduce then that the quotient sheaf M associated to the presheaf

WT(YW)/T(W)

is a locally free, locally constant sheaf that is finitely generated and abelian. In other words, we have an exact sequence of abelian sheaves

1TAut¯YT(Z)M1.

In particular, since M is locally constant, it is representable. And since T is affine, we get by [11, Section III.4, Proposition 1.9] that Aut¯YT(Z) is represented by an abelian K-group scheme A. We abusively denote by A the sheaf Aut¯YT(Z) as well.

Since every element in AutXWTW(ZW) induces an XW-automorphism of YW, we have an exact sequence of sheaves

1AAut¯XT(Z)𝜋Aut¯X(Y),

where Aut¯X(Y) denotes the sheaf of X-automorphisms of Y.

We claim that π is surjective. In order to prove this, we can replace K by a finite extension. We may assume then that both T and G are split over K. Since G is split and connected and Pic(G¯)=0, we have that Pic(G)=0 (cf. [20, Lemma 6.9]). By [20, Proposition 6.10], we get then that the map Pic(X)Pic(Y) is surjective. Since T is split, we deduce then that the torsor ZY comes by pullback from a T-torsor ZX. In other words, Z=Y×XZ. The surjectivity is then evident.

Note now that G is clearly a subgroup of Aut¯X(Y). Define then Aut¯XT(Z) to be the group sheaf corresponding to the preimage of G via π. We get an exact sequence

1AG1.

Since A is abelian, the extension induces an action of the sheafG on the sheafA. By Yoneda’s Lemma, this action is actually an action of the K-group G on the K-group A. Note that T corresponds to the neutral connected component of A and thus it is preserved by the G-action since G is connected. In particular, we may quotient by T in order to get an exact sequence

1MG1.

Then, if one forgets its group structure, corresponds to an M-torsor over the scheme G. By [15, Exposé 8, Proposition 5.1], we know that H1(G,)=0 and hence, since M is locally free and finitely generated, H1(G,M)=0 up to taking a finite extension of K. Since representability can be checked over a finite extension (once again by Yoneda’s Lemma), this tells us that is represented by a K-scheme F (which is a K-form of M×G). Thus we have an exact sequence of K-group-schemes

1MFG1

from where we get a new exact sequence of group sheaves

1TF1.

Once again, since T is an affine group-scheme, [11, Section III.4, Proposition 1.9] tells us that is in fact a scheme E and hence a K-group. But now, since M is discrete, it suffices to define E as the identity component of E, which clearly fits into an exact sequence

1TEG1.

Since EE is a subgroup of Aut¯XT(Z), it is immediate then to check that E acts on Z and that ZX is an E-torsor.

Note finally that the whole construction is clearly canonical. Indeed, the extension is obtained by pullback from the canonical extension of sheaves of automorphism groups. Then we get E as theK-group scheme representing and finally we take the identity component, which is a characteristic subgroup. Also, the last assertion when G is a torus follows from [11, Section IV.1, Proposition 4.5]. ∎

Acknowledgements

The authors would like to thank Mathieu Florence for his enormous help with the result in the appendix. They would also like to thank Jean-Louis Colliot-Thélène and an anonymous referee for all their suggestions that allowed them to considerably improve this text. The second author would like to thank Michel Brion for helpful discussions.

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Received: 2021-08-30
Published Online: 2021-10-16
Published in Print: 2021-12-01

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