TY - JOUR
T1 - The Étale Homology and the Cycle Maps in Adic Coefficients
AU - Li , Ting
JO - Communications in Mathematical Research 
VL - 1
SP - 68
EP - 87
PY - 2021
DA - 2021/05
SN - 29
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/cmr/19030.html
KW - $ℓ$-adic cohomology, cycle map, derived category.
AB - <p style="text-align: justify;">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.</p>