The Stacks project

81.3 Cycles

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 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 $Z_ k(X)$.

In other words, a $k$-cycle on $X$ is a locally finite formal $\mathbf{Z}$-linear combination of integral closed subspaces (Remark 81.2.3) of $\delta $-dimension $k$. Addition of $k$-cycles $\alpha = \sum n_ Z[Z]$ and $\beta = \sum m_ Z[Z]$ is given by

\[ \alpha + \beta = \sum (n_ Z + m_ Z)[Z], \]

i.e., by adding the coefficients.

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