Unstable motivic homotopy categories in Nisnevich and cdh-topologies

By Vladimir Voevodsky. 2008

88 citations recorded by the source corpus.

Abstract The motivic homotopy categories can be defined with respect to different topologies and different underlying categories of schemes. For a number of reasons (mainly because of the Gluing Theorem) the motivic homotopy category of smooth schemes with respect to the Nisnevich topology plays a distinguished role but in some cases it is desirable to be able to work with all schemes instead of the smooth ones. In this paper we prove that, under the resolution of singularities assumption, the unstable motivic homotopy category of all schemes over a field with respect to the cdh-topology is almost equivalent to the unstable motivic homotopy category of smooth schemes over the same field with respect to the Nisnevich topology.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8d3d70993a892e508ed5ecad57b9aa6180d3b8b9

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Galois equivariance and stable motivic homotopy theory (2014, unknown): “It follows from [51] that |Σ∞ P1 X•+| → ΣP1X+ is a stable equivalence in SHk.”
Motivic Eilenberg-MacLane spaces (2008, background): “Proof: By [39] the class of local equivalences in the cdh-topology on ∆(Sch/k)#+ is cl∆̄((WNis)+ ∪ (Wlcd)+) where Wlcd is defined in the same way as WNis but with respect to the lower distinguished squares (i.”
Homotopy theory of simplicial sheaves in completely decomposable topologies (2008, methodology): “A closely related density structure on the category of Noetherian schemes is considered in [11].”