The \'Etale Homology and The Cycle Maps in Adic Coefficients

TL;DR

This paper introduces a new l-adic homology theory for schemes that improves upon classical cohomology, providing a simpler way to construct cycle maps and establishing their key properties.

Contribution

It defines an l-adic homology with functorial properties similar to Chow groups, offering a more effective approach to cycle maps in algebraic geometry.

Findings

L-adic homology behaves better than classical l-adic cohomology on singular varieties.

Cycle maps constructed kill algebraic equivalences.

Cycle maps commute with Chern actions of locally free sheaves.

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 that the classical l-adic cohomology. As an application, we give an much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields of finite cohomology dimension. And we prove these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.