The Stacks project

Remark 42.19.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Suppose we have infinite collections $\alpha _ i, \beta _ i \in Z_ k(X)$, $i \in I$ of $k$-cycles on $X$. Suppose that the supports of $\alpha _ i$ and $\beta _ i$ form locally finite collections of closed subsets of $X$ so that $\sum \alpha _ i$ and $\sum \beta _ i$ are defined as cycles. Moreover, assume that $\alpha _ i \sim _{rat} \beta _ i$ for each $i$. Then it is not clear that $\sum \alpha _ i \sim _{rat} \sum \beta _ i$. Namely, the problem is that the rational equivalences may be given by locally finite families $\{ W_{i, j}, f_{i, j} \in R(W_{i, j})^*\} _{j \in J_ i}$ but the union $\{ W_{i, j}\} _{i \in I, j\in J_ i}$ may not be locally finite.

In many cases in practice, one has a locally finite family of closed subsets $\{ T_ i\} _{i \in I}$ such that $\alpha _ i, \beta _ i$ are supported on $T_ i$ and such that $\alpha _ i = \beta _ i$ in $\mathop{\mathrm{CH}}\nolimits _ k(T_ i)$, in other words, the families $\{ W_{i, j}, f_{i, j} \in R(W_{i, j})^*\} _{j \in J_ i}$ consist of subschemes $W_{i, j} \subset T_ i$. In this case it is true that $\sum \alpha _ i \sim _{rat} \sum \beta _ i$ on $X$, simply because the family $\{ W_{i, j}\} _{i \in I, j\in J_ i}$ is automatically locally finite in this case.

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