Definition 42.8.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$.

A

*cycle on $X$*is a formal sum\[ \alpha = \sum n_ Z [Z] \]where the sum is over integral closed subschemes $Z \subset X$, each $n_ Z \in \mathbf{Z}$, and the collection $\{ Z; n_ Z \not= 0\} $ is locally finite (Topology, Definition 5.28.4).

A

*$k$-cycle*on $X$ is a cycle\[ \alpha = \sum n_ Z [Z] \]where $n_ Z \not= 0 \Rightarrow \dim _\delta (Z) = k$.

The abelian group of all $k$-cycles on $X$ is denoted $Z_ k(X)$.

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