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

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 Y→X be a G-torsor and let Z→Y 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

such that the composite Z→X 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/K↦AutXWTW⁢(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

Indeed, it is well known that the functor W/Y↦AutWTW⁢(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,

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

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

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

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 Z→Y comes by pullback from a T-torsor Z′→X. 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 ℰ′⊂Aut¯XT⁢(Z) to be the group sheaf corresponding to the preimage of G via π. We get an exact sequence

Since A is abelian, the extension ℰ′ 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

Then, if one forgets its group structure, ℱ corresponds to an M-torsor over the scheme G. By [15, Exposé 8, Proposition 5.1], we know that H1⁢(G,ℤ)=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 ℱ 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

from where we get a new exact sequence of group sheaves

Once again, since T is an affine group-scheme, [11, Section III.4, Proposition 1.9] tells us that ℰ′ 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

Since E⊂E′ is a subgroup of Aut¯XT⁢(Z), it is immediate then to check that E acts on Z and that Z→X is an E-torsor.

Note finally that the whole construction is clearly canonical. Indeed, the extension ℰ′ is obtained by pullback from the canonical extension of sheaves of automorphism groups. Then we get E′ as theK-group scheme representing ℰ′ 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]. ∎