We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.Definition 2.3: The Λ-local stable model structure is the left Bousfield localization of the stable model structure on Spt Σ P 1 MS with respect to the set of naturally induced mapsall integers s, t, and X ∈ Sm S . Denote the corresponding homotopy category by SH Λ . Remark 2.4: A map α : E → F is a weak equivalence in the Λ-local stable model structure if and only if α ∧ 1 Λ : E Λ → F Λ is a stable motivic weak equivalence. When Λ = Q, this defines the rational stable motivic homotopy category. By [20, Theorem 3.3.19(1)] there exists a left Quillen functor from the stable to the Λ-local stable model structure on Spt Σ P 1 MS. We shall refer to its derived functor as the Λ-localization functor. Proof. This follows from Corollaries 2.16, 2.17, Remark 2.18 applied to every connected component of the base scheme, and the computation of s q (KQ) over fields of characteristic unequal to two in [63, Theorem 4.18].Recall that a motivic spectrum E ∈ SH Λ is called slice-wise cellular if s q (E) is contained in the full localizing triangulated subcategory of SH Λ generated by the qth suspension Σ 2q,q MΛ [73, Definition 4.1]. Let D MΛ denote the homotopy category of MΛ − mod. Replacing SH Λ by D MΛ gives an equivalent definition of slice-wise cellular spectra.Corollary 2.16: Every cellular spectrum in SH Λ is slice-wise cellular.
In this article we study locally compact abelian (LCA) groups from the viewpoint of derived categories, using that their category is quasi-abelian in the sense of J.-P. Schneiders. We define a well-behaved derived Hom-complex with values in the derived category of Hausdorff topological abelian groups. Furthermore we introduce a smallness condition for LCA groups and show that such groups have a natural tensor product and internal Hom which both admit derived versions.Comment: 18 pages, AMSLaTe
We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncatedétale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology.As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of Tate sheaves with integral coefficients. These can be modelled as representations of derived fundamental groups.Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle.Further applications include a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.
Coloured operads were introduced in the 1970s for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper, we introduce coloured operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extent the slice filtration preserves highly structured objects such as algebras and modules. We use coloured operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to coloured operads and Bousfield (co)localizations are given in separate appendices. Contents
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