$\mathbb{A}^1$-connected components of classifying spaces and purity for torsors

Abstract

In this paper, we study the Nisnevich sheafification H 1 ét (G) of the presheaf associating to a smooth scheme the set of isomorphism classes of G-torsors, for a reductive group G. We show that if G-torsors on affine lines are extended, then H 1 ét (G) is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for G-torsors. We also identify the sheaf H 1 ét (G) with the sheaf of A 1 -connected components of the classifying space B ét G. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of A 1 -connected components in terms of unramified G-torsors over function fields whenever Nisnevich-local purity holds for G-torsors.

Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups

“…In this section, we briefly review the results of Elmanto, Kulkarni and Wendt [16] that will be used in Section 5. Let Sch k denote the site of schemes of finite type over k with étale topology.…”

“…The following result due to Elmanto, Kulkarni and Wendt [16, Theorem 1.2, Proposition 3.7] will play a key role in the proof of our main theorem. We restate their result here for the convenience of readers, since the result is stated in [16] with additional hypotheses, which are not necessary under the assumption of simple connectedness. Theorem 3.8.…”

“…Proof. The proof of [16,Proposition 3.7] works almost verbatim (with the use of [10] in place of [27]), so we only give a brief sketch.…”

“…By the strong Grothendieck-Serre property over a field (see [17, Corollary 1] and [25, Corollary 1.2]) along with [16,Proposition 3.4], we are reduced to showing that the map…”

“…Acknowledgements. We thank Matthias Wendt for helpful correspondence regarding the results of [16] and Swarnava Mukhopadhyay for pointing us to the literature on principal bundles on curves.…”