### Bivariant cycle cohomology, by E. M. Friedlander and Vladimir Voevodsky

In this paper we construct a bivariant (co-)homology theory on the category of
algebraic varieties over a field which admits resolution of
singularities. We show that this theory has the standard good properties,
such as localization long exact sequences (with respect to the covariant
argument) and Mayer-Vietories long exact sequences (with respect to the
contravariant argument). We also prove the duality theorem which asserts
that for smooth varieties the corresponding covariant and contravariant
theories are isomorphic. We further use it to
define four "motivic (co-)homology" theories (motivic homology,
Borel-Moore motivic homology, motivic cohomlogy and motivic cohomology
with compact supports) which are related by duality isomorphisms in the
smooth case.

E. M. Friedlander <eric@math.nwu.edu>

Vladimir Voevodsky <vladimir@math.harvard.edu>