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We advance the understanding of K-theory of quadratic forms by computing the slices of the motivic spectra representing hermitian K-groups and Witt-groups. By an explicit computation of the slice spectral sequence for higher Witt-theory, we prove Milnor's conjecture relating Galois cohomology to quadratic forms via the filtration of the Witt ring by its fundamental ideal. In a related computation we express hermitian K-groups in terms of motivic cohomology.

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.

Let S be an essentially smooth scheme over a field and ℓ = char S a prime number. We show that the algebra of bistable operations in the mod ℓ motivic cohomology of smooth S-schemes is generated by the motivic Steenrod operations. This was previously proved by Voevodsky for S a field of characteristic zero. We follow Voevodsky's proof but remove its dependence on characteristic zero by usingétale cohomology instead of topological realization and by replacing resolution of singularities with a theorem of Gabber on alterations.

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