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.

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