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

1. 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).

2. A $k$-cycle on $X$ is a cycle

$\alpha = \sum n_ Z [Z]$

where $n_ Z \not= 0 \Rightarrow \dim _\delta (Z) = k$.

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

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