This is the analogue of Chow Homology, Section 42.8
Since we are not assuming our spaces are quasi-compact we have to be a little careful when defining cycles. We have to allow infinite sums because a rational function may have infinitely many poles for example. In any case, if X is quasi-compact then a cycle is a finite sum as usual.
Definition 82.3.1. In Situation 82.2.1 let X/B be good. Let k \in \mathbf{Z}.
A cycle on X is a formal sum
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).
A k-cycle on X is a cycle
where n_ Z \not= 0 \Rightarrow \dim _\delta (Z) = k.
The abelian group of all k-cycles on X is denoted Z_ k(X).
In other words, a k-cycle on X is a locally finite formal \mathbf{Z}-linear combination of integral closed subspaces (Remark 82.2.3) of \delta -dimension k. Addition of k-cycles \alpha = \sum n_ Z[Z] and \beta = \sum m_ Z[Z] is given by
i.e., by adding the coefficients.
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