The Étale Homology and the Cycle Maps in Adic Coefficients

Abstract

In this article, we define the $ℓ$-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups: proper push-forward, flat pull-back, base change, cap-product, etc. In particular, on singular varieties, this kind of $ℓ$-adic homology behaves much better than the classical $ℓ$-adic cohomology. As an application, we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields. And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.