2019/12/24 by Elmanto, Elden, Shah, Jay
#14F42 #14P10 #55P91 #Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.1912.11557
We give an algebro-geometric interpretation of C2-equivariant stable homotopy theory by means of the b-topology introduced by Claus Scheiderer in his study of 2-torsion phenomena in étale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a ∞-topos \mathscrX with G-action a presentable stable ∞-category SpG(\mathscrX), which recovers the ∞-category SpG of genuine G-spectra when \mathscrX is the terminal G-∞-topos. Given a scheme X with 1/2 ∈ OX, our construction then specializes to produce an ∞-category SpC2b(X) of "b-sheaves with transfers" as b-sheaves of spectra on the small étale site of X equipped with certain transfers along the extension X[i] → X; if X is the spectrum of a real closed field, then SpC2b(X) recovers SpC2. On a large class of schemes, we prove that, after p-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the b-variant SHb(X) of the ∞-category SH(X) of motivic spectra over X (in the sense of Morel-Voevodsky), and produce a natural equivalence of ∞-categories SHb(X)\wedgep ≃ SpC2b(X)\wedgep through amalgamating the étale and real étale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the C2-Tate construction, which may be of independent interest. Finally, as applications, we deduce a "b-rigidity" theorem, use the Segal conjecture to show étale descent of the 2-complete b-motivic sphere spectrum, and construct a parametrized version of the C2-Betti realization functor of Heller-Ormsby.