2018/03/20 by Tom Bachmann · 7 citations
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · doi:10.1112/s0010437x17007710
Let S be a Noetherian scheme of finite dimension and denote by \unicode[STIX]x1D70C∈ [\unicode[STIX]x1D7D9,\mathbbGm]SH(S) the (additive inverse of the) morphism corresponding to -1∈ O× (S) . Here SH(S) denotes the motivic stable homotopy category. We show that the category obtained by inverting \unicode[STIX]x1D70C in SH(S) is canonically equivalent to the (simplicial) local stable homotopy category of the site S_r\acuteet , 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 SH(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 ℝ ). As further applications we show that D_\mathbbA1(k,ℤ[1/2])-≃ DMW(k)[1/2] (improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that \underline\unicode[STIX]x1D70Bi(\unicode[STIX]x1D7D9[1/\unicode[STIX]x1D702,1/2])=0 for i=1,2 and establish some new rigidity results.