Summary: Let $X$ be a separated scheme of finite type over an algebraically closed field $k$ and let $m$ be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod $m$ Suslin homology and the first mod $m$ tame étale cohomology of $X$. We show that the induced homomorphism from the mod $m$ Suslin homology to the abelianized tame fundamental group of $X$ mod $m$ is surjective. It is an isomorphism of finite abelian groups if $(m, char(k)) = 1$, and for general $m$ if resolution of singularities holds over $k$.

14F35, 14F43, 14C25

Suslin homology, higher dimensional class field theory, tame fundamental group