Definition 81.3.1. In Situation 81.2.1 let $X/B$ be good. 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 subspaces $Z \subset X$, each $n_ Z \in \mathbf{Z}$, and $\{ |Z|; n_ Z \not= 0\}$ is a locally finite collection of subsets of $|X|$ (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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