Localization and nilpotent spaces in A^1-homotopy theory

Abstract

For a subring R of the rational numbers, we study R-localization functors in the local homotopy theory of simplicial presheaves on a small site and then in A 1 -homotopy theory. To this end, we introduce and analyze two notions of nilpotence for spaces in A 1 -homotopy theory paying attention to future applications for vector bundles. We show that R-localization behaves in a controlled fashion for the nilpotent spaces we consider. We show that the classifying space BGL n is A 1 -nilpotent when n is odd, and analyze the (more complicated) situation where n is even as well. We establish analogs of various classical results about rationalization in the context of A 1 -homotopy theory: if −1 is a sum of squares in the base field, A n \ 0 is rationally equivalent to a suitable motivic Eilenberg-Mac Lane space, and the special linear group decomposes as a product of motivic spheres.

Detecting motivic equivalences with motivic homology

“…and m = 2 otherwise. Localization away from a prime p is well-behaved on A 1 -1-connected spaces; see [1]. Perfect fields of finite 2-étale cohomological dimension include algebraically closed fields, finite fields, and totally imaginary number fields.…”

Suslin's cancellation conjecture in the smooth case

“…Using the (iso)morphism of cycle modulesK M d (−)/p → H d (−, µ ⊗d p ), it follows that H d−1 Zar (X , K M d /p) = H d−2 Zar (X , K M d /p) = 0.Remark 5.0.4. Using the localization of the motivic homotopy theory developed in[AFH20], and the subsequent decomposition of the spheres using Suslin matrices, it is possible to prove thatH d Nis (X , π A 1 d (A d 0)) ⊗ Z[ notsufficient to prove Theorem 3.0.3, as we ignore a priori if the group H d Nis (X , π A 1 d (A d 0)) is finitely generated. proof of Theorem 3.0.3.…”