Motivic and Real Etale Stable Homotopy Theory

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.

Minor corrections. Accepted for publication in compositio. Comments still welcome!

References in corpus (10)

• Higher Topos Theory
• Triangulated categories of mixed motives
• Étale motives
• On (Enriched) Left Bousfield Localization of Model Categories
• Finite Chow-Witt correspondences
• Cdh descent in equivariant homotopy K-theory
• The Witt group of real algebraic varieties
• On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive
• Witt sheaves and the $\eta$-inverted sphere spectrum
• Coherence for invertible objects and multi-graded homotopy rings

Cited by in corpus (33)

• Triangulated categories of mixed motives
• C_2-equivariant stable homotopy from real motivic stable homotopy
• On the effectivity of spectra representing motivic cohomology theories
• On the Conservativity of the Functor Assigning to a Motivic Spectrum its Motive
• Reconstructing rational stable motivic homotopy theory
• On the rational motivic homotopy category
• On Modules Over Motivic Ring Spectra
• Rigidity in etale motivic stable homotopy theory
• The real cycle class map
• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Real motivic and $C_2$-equivariant Mahowald invariants
• The motivic Adams-Novikov spectral sequence at odd primes over $\mathbb{C}$ and $\mathbb{R}$
• Three real Artin-Tate motives
• The homotopy groups of the η-periodic motivic sphere spectrum
• Motivic Tambara Functors
• The $C_2$-effective spectral sequence for $C_2$-equivariant connective real $K$-theory
• R-motivic stable stems
• Algebraic Cobordism and Étale Cohomology
• Notes on motivic infinite loop space theory
• Remarks on étale motivic stable homotopy theory
• Topological models for stable motivic invariants of regular number rings
• The stable Adams operations on Hermitian K-theory
• Affine quadrics and the Picard group of the motivic category
• Nilpotence in normed MGL-modules
• Algebraic elliptic cohomology and flops II: $SL$-cobordism
• $\mathbb{R}$-motivic $v_1$-periodic homotopy
• Galois reconstruction of Artin-Tate $\mathbb{R}$-motivic spectra
• On some finiteness results in real étale cohomology
• The stable Galois correspondence for real closed fields
• Motivic Pontryagin classes and hyperbolic orientations
• A Gersten complex on real schemes
• Invariants of upper motives
• Odd rank vector bundles in eta-periodic motivic homotopy theory