Volume 29, Issue 1
The Étale Homology and the Cycle Maps in Adic Coefficients

Commun. Math. Res., 29 (2013), pp. 68-87.

Published online: 2021-05

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• 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.

• Keywords

$ℓ$-adic cohomology, cycle map, derived category.

14F20, 14C25

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@Article{CMR-29-68, author = {Ting and Li and and 18926 and and Ting Li}, title = {The Étale Homology and the Cycle Maps in Adic Coefficients}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {29}, number = {1}, pages = {68--87}, 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.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19030.html} }
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 -

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.

Ting Li. (2021). The Étale Homology and the Cycle Maps in Adic Coefficients. Communications in Mathematical Research . 29 (1). 68-87. doi:
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