A Tannakian approach to patching

Adelic double cosets over semi-global fields

“…This gives rise to a closer analog of the classical adelic double coset space, classifying locally trivial torsors over X rather than over F . Whereas Section 3 relied on results about patching over fields (from [HHK15] and [HHK14]), here we draw on patching results for torsors over schemes that were proven from a Tannakian point of view in [HKL20].…”

“…We next turn to the case of a smooth affine group scheme G over a model X of a semiglobal field, obtaining an associated Mayer-Vietoris sequence in étale cohomology and a description of the associated double coset space as a local-global obstruction, paralleling the above case of a one-dimensional scheme. As in that case, we rely on [HKL20].…”

“…Afterwards, Section 4 establishes an adelic double coset space in terms of local rings, which is thus closer to the situation in the global field case. This is done for reasonable one-dimensional schemes and for models of semi-global fields, using a Tannakian approach to patching that was recently established in [HKL20]. In each of these situations, the double coset space is interpreted as a local-global obstruction, and examples are given that compute the double coset spaces explicitly.…”

“…Proceeding as in the proof of Proposition 4.2, we let P = Spec( P ∈P R P ), let U = Spec( U ∈U R U ), and let B = Spec( ℘∈B R ℘ ). Then patching holds for coherent sheaves on X with respect to P, U , B by [Pri00, Theorem 3.4] (see also [HKL20,Theorem 3.1.4]), as a consequence of [FR70, Proposition 4.2] or [Art70, Theorem 2.6] together with Grothendieck's Existence Theorem ([Gro61, Théorème 5.1.6]). By [HKL20, Corollary 3.0.2], it follows from this patching property that we have the exact sequence asserted in the proposition.…”