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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.
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 f
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
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 th
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 cert
We define a notion of colimit for diagrams in a motivic category indexed by a presheaf of spaces (e.g. an etale classifying space), and we study basic properties of this construction. As a case study, we construct the motivic analogs of the classical