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Register a new userStable motivic invariants are eventually etale local
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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.
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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.
For an infinity of number rings we express stable motivic invariants in terms of topological data determined by the complex numbers, the real numbers, and finite fields. We use this to extend Morels identification of the endomorphism ring of the motivic sphere with the Grothendieck-Witt ring of quadratic forms to deeper base schemes.
Let k be a field and denote by SH(k) the motivic stable homotopy category. Recall its full subcategory HI_0(k) of effective homotopy modules. Write NAlg(HI_0(k)) for the category of normed motivic spectra with underlying spectrum an effective homotopy module. In this article we provide an explicit description of NAlg(HI_0(k)) as the category of sheaves with generalized transfers and etale norms, and explain how this is closely related to the classical notion of Tambara functors.
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.
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