Motivic and Real Etale Stable Homotopy Theory

TL;DR

This paper establishes an equivalence between the rho-inverted motivic stable homotopy category over a scheme and the local stable homotopy category of its real etale site, with applications to classical stable homotopy and rigidity results.

Contribution

It proves the equivalence of rho-inverted motivic homotopy category with the real etale local stable homotopy category, providing new insights and applications in stable homotopy theory.

Findings

SH(RR)[rho^-1] is equivalent to the classical stable homotopy category

Computed all stable homotopy sheaves of the rho-local sphere over RR

Reproved a vanishing result and established new rigidity results

Abstract

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.