Journal Article

Motivic and real étale stable homotopy theory

Bachmann, Tom

2018

Overview

Let $S$ be a Noetherian scheme of finite dimension and denote by $\\unicode[STIX]{x1D70C}\\in [\\unicode[STIX]{x1D7D9},\\mathbb{G}_{m}]_{\\mathbf{SH}(S)}$ the (additive inverse of the) morphism corresponding to $-1\\in {\\mathcal{O}}^{\\times }(S)$ . Here $\\mathbf{SH}(S)$ denotes the motivic stable homotopy category. We show that the category obtained by inverting $\\unicode[STIX]{x1D70C}$ in $\\mathbf{SH}(S)$ is canonically equivalent to the (simplicial) local stable homotopy category of the site $S_{\\text{r}\\acute{\\text{e}}\\text{t}}$ , by which we mean the small real étale site of $S$ , comprised of étale schemes over $S$ with the real étale topology. One immediate application is that $\\mathbf{SH}(\\mathbb{R})[\\unicode[STIX]{x1D70C}^{-1}]$ is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the $\\unicode[STIX]{x1D70C}$ -local sphere (over $\\mathbb{R}$ ). As further applications we show that $D_{\\mathbb{A}^{1}}(k,\\mathbb{Z}[1/2])^{-}\\simeq \\mathbf{DM}_{W}(k)[1/2]$ (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that $\\text{}\\underline{\\unicode[STIX]{x1D70B}}_{i}(\\unicode[STIX]{x1D7D9}[1/\\unicode[STIX]{x1D702},1/2])=0$ for $i=1,2$ and establish some new rigidity results.

• Publisher: London Mathematical Society,Cambridge University Press
• Subject: Algebra / Applied mathematics / Equivalence / Homotopy theory / Sheaves