### Relative cycles and Chow sheaves, by Andrei Suslin and Vladimir Voevodsky

For a scheme X of finite type over a Noetherian scheme S we define a group of
relative equidimensional cycles. We show that it is contravariantly
functorial with respect to base change and thus provides a presheaf on the
category of Noetherian schemes over S. Moreover this presheaf turns out to
be a sheaf in a Grothendieck topology called the cdh-topology. The main goal
of the paper is to study these Chow sheaves. In the particular case of X
being a projective variety over a field of characteristic zero the Chow sheaf
of effective cycles of dimension d is representable by the corresponding Chow
variety. Even in this case though it turns out to be more convenient to work
with sheaves than with varieties. In particular we construct certain short
exact sequenecs of Chow sheaves which in the case of varieties over a field
lead to localization long exact sequences in algebraic cycle homology and
which do not have any obvious analog for Chow varieties.

Andrei Suslin <suslin@lomi.spb.su>

Vladimir Voevodsky <vladimir@math.harvard.edu>