2.3 Algebraic cycles

In this section, \(X\) is a separated scheme of finite type over a field \(k\).

Suppose \(k\) is a nonnegative integer. We set \(X_{(k)}\) equal to the set of closed subvarieties \(V\subseteq X\) with \(\dim V = k\). Then \(X_{(k)}\) is naturally isomorphic (as a set) to the set of all points \(x\in X\) with \(\dim \overline\{ x\} = k\). The equivalence sends a point \(x\in X\) to its closure and it sends a closed subvariety \(V\subseteq X\) to its generic point.

We set \(Z_k X\) equal to the free abelian group on the set \(X_{(k)}\). This is group of dimension \(k\) algebraic cycles on \(X\).