Cycle class
= Cycle class
{title2=$\operatorname{cl}(Z)$}
The cycle class of an <algebraic cycle> is its image under the <cycle class map>. For a codimension-$r$ cycle on a smooth complex projective variety, it belongs to $H^{2r}(X,\mathbb Z)$ and is the <Poincare dual> of its fundamental homology class. The map factors through the <Chow group>, but a <Chow class> retains finer information than its cohomological cycle class.