Motivic and Real Etale Stable Homotopy Theory

Let X be a Noetherian scheme of finite dimension and denote by rho the (additive inverse of the) morphism in SH(X) from S to Gm corresponding to the unit -1. Here SH(X) denotes the motivic stable homotopy category. We show that the category obtained by inverting rho in SH(X) is canonically equivalent to the (simplicial) local stable homotopy category of the site X_ret, by which we mean the small real etale site of X, comprised of etale schemes over X with the real etale topology. One immediate application is that SH(RR)[rho^-1] is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the rho-local sphere (over RR). As further applications we improve a result of Ananyevskiy-Levine-Panin, reprove a vanishing result of Roendigs and establish some new rigidity results.