We examine the slice spectral sequence for the cohomology of singular schemes with respect to various motivic T -spectra, especially the motivic cobordism spectrum. When the base field k admits resolution of singularities and X is a scheme of finite type over k, we show that Voevodsky's slice filtration leads to a spectral sequence for MGLX whose terms are the motivic cohomology groups of X defined using the cdh-hypercohomology. As a consequence, we establish an isomorphism between certain geometric parts of the motivic cobordism and motivic cohomology of X.A similar spectral sequence for the connective K-theory leads to a cycle class map from the motivic cohomology to the homotopy invariant K-theory of X. We show that this cycle class map is injective for a large class of projective schemes. We also deduce applications to the torsion in the motivic cohomology of singular schemes.is an isomorphism.We apply our descent result to obtain a similar spectral sequence for the connective KHtheory, KGL 0 (see §5). We use this spectral sequence and the canonical map CKH(−) → KH(−) from the connective KH-theory to obtain the following cycle class map from the motivic cohomology of a singular scheme to its homotopy invariant K-theory.Theorem 1.4. Let k be a field of exponential characteristic p and let X be a separated scheme of dimension d which is of finite type over k. Then the map KGL 0 X → s 0 KGL X ∼ = HZ induces for every integer i ≥ 0, an isomorphismIn particular, there is a natural cycle class mapWe use this cycle class map and the Chern class maps from the homotopy invariant Ktheory to the Deligne cohomology of schemes over C to construct intermediate Jacobians and Abel-Jacobi maps for the motivic cohomology of singular schemes over C. More precisely, we prove the following. This generalizes intermediate Jacobians and Abel-Jacobi maps of Griffiths and the torsion theorem of Roitman for smooth schemes.
Let k be a field with resolution of singularities, and X a separated k-scheme of finite type with structure map g. We show that the slice filtration in the motivic stable homotopy category commutes with pullback along g. Restricting the field further to the case of characteristic zero, we are able to compute the slices of Weibel's homotopy invariant K-theory [24] extending the result of Levine [10], and also the zero slice of the sphere spectrum extending the result of Levine [10] and Voevodsky [23]. We also show that the zero slice of the sphere spectrum is a strict cofibrant ring spectrum HZ sf X which is stable under pullback and that all the slices have a canonical structure of strict modules over HZ sf X . If we consider rational coefficients and assume that X is geometrically unibranch then relying on the work of Cisinski and Déglise [4], we deduce that the zero slice of the sphere spectrum is given by Voevodsky's rational motivic cohomology spectrum HZ X˝Q and that the slices have transfers. This proves several conjectures of Voevodsky [22, conjectures 1, 7, 10, 11] in characteristic zero.
The main goal of this paper is to construct an analogue of Voevodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the existence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy categories constructed in [19]. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much better in the unstable setting. 2010 Mathematics Subject Classification. Primary 14F42. Now we describe the open immersions which will be used to construct the left Bousfield localizations of M.Definition 2.3 (see [20, section 7.5]). Let Y ∈ Sch X , and Z a closed subscheme of Y . The codimension of Z in Y , codim Y Z is the infimum (over the generic points z i of Z) of the dimensions of the local rings O Y,zi .Since X is Noetherian of finite Krull dimension and Y is of finite type over X, codim Y Z is always finite.Definition 2.4. We fix an arbitrary integer n ≥ 0, and consider the following set of open immersions which have a closed complement of codimension at least n + 1The letter B stands for birational.
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