Homology of schemes, II, by Vladimir Voevodsky

May, 1994. Revised version. In this paper we study a class of functors called pretheories on the category of smooth schemes over a perfect field. Morally, a pretheory is a contravariant functor equipped with transfers with respect to relative divisors with compact support on smooth relative curves. Among the examples of such functors are K-cohomology, bivariant cycle homology of Friedlander and Gabber ("Cycle spaces and intersection theory", in the book "Topological Methods in Modern Mathematics") and in some sense algebraic singular homology (see "Singular homology of abstract algebraic varieties", by Suslin and Voevodsky). Our main result states that Zariski cohomology with coefficients in a homotopy invariant pretheory is again a homotopy invariant pretheory. Together with some other properties of pretheories this result allows us to prove the Mayer-Vietoris property for algebraic singular homology and in the case when there is resolution of singularities to prove the localization property for algebraic cycle homology. One of the most important technical tools which allows us to deal effectively with pretheories is the Nisnevich topology on algebraic varieties.

Vladimir Voevodsky <vladimir@math.harvard.edu>