Motivic and real étale stable homotopy theory

Abstract

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 thesmallreal 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.