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We strengthen some results in etale (and real etale) motivic stable homotopy theory, by eliminating finiteness hypotheses, additional localizations and/or extending to spectra from HZ-modules.
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In this paper we prove a Thomason-style descent theorem for the $rho$-complete sphere spectrum. In particular, we deduce a very general etale descent result for torsion, $rho$-complete motivic spectra. To this end, we prove a new convergence result for slice spectral sequence in the $rho$-complete motivic category, following Levines work. This generalizes and extends previous etale descent results for motivic cohomology theories which, combined with etale rigidity results, gives a complete, structural description of the etale motivic stable category.
Over any field of characteristic not 2, we establish a 2-term resolution of the $eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $eta$-periodized motivic stable stems and the $eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $eta$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $eta$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.
If $f:S to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal norm functor $f_otimes: mathcal H_*(S) tomathcal H_*(S)$, where $mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite etale, we show that it stabilizes to a functor $f_otimes: mathcal{SH}(S) to mathcal{SH}(S)$, where $mathcal{SH}(S)$ is the $mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_infty$-ring spectrum. The main content of this text is a detailed study of the norm functors and of normed motivic spectra, and the construction of examples. In particular: we investigate the interaction of norms with Grothendiecks Galois theory, with Betti realization, and with Voevodskys slice filtration; we prove that the norm functors categorify Rosts multiplicative transfers on Grothendieck-Witt rings; and we construct normed spectrum structures on the motivic cohomology spectrum $Hmathbb Z$, the homotopy K-theory spectrum $KGL$, and the algebraic cobordism spectrum $MGL$. The normed spectrum structure on $Hmathbb Z$ is a common refinement of Fulton and MacPhersons mutliplicative transfers on Chow groups and of Voevodskys power operations in motivic cohomology.
We establish a kind of degree zero Freudenthal Gm-suspension theorem in motivic homotopy theory. From this we deduce results about the conservativity of the P^1-stabilization functor. In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy invariant sheaf in terms of the Rost--Schmid complex. This establishes the main conjecture of [BY18], which easily implies the aforementioned results.
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