Editing Chern class (section)

===Algebraic geometry===

In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:

*For complex varieties the Chern classes can take values in ordinary cohomology, as above.
*For varieties over general fields, the Chern classes can take values in cohomology theories such as [[etale cohomology]] or [[l-adic cohomology]].
*For varieties ''V'' over general fields the Chern classes can also take values in homomorphisms of [[Chow group]]s CH(V): for example, the first Chern class of a line bundle over a variety ''V'' is a homomorphism from CH(''V'') to CH(''V'') reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the [[cap product]].