### Unstable motivic homotopy categories in Nisnevich and cdh-topologies, by Vladimir Voevodsky

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 built out of smooth schemes
with respect to the Nisnevich topology plays a distinguished role but
in some cases it is very 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. In order to do it we show that the standard cd-topologies on
the category of Noetherian schemes, including the cdh-topology,
satisfy certain conditions which allows one to use the generalized
version of the Brown-Gersten approach to the homotopy theory of
simplicial sheaves.

Vladimir Voevodsky <vladimir@ias.edu>