The Stacks project

Example 42.19.5. Here is a “strange” example. Suppose that $S$ is the spectrum of a field $k$ with $\delta$ as in Example 42.7.2. Suppose that $X = C_1 \cup C_2 \cup \ldots$ is an infinite union of curves $C_ j \cong \mathbf{P}^1_ k$ glued together in the following way: The point $\infty \in C_ j$ is glued transversally to the point $0 \in C_{j + 1}$ for $j = 1, 2, 3, \ldots$. Take the point $0 \in C_1$. This gives a zero cycle $[0] \in Z_0(X)$. The “strangeness” in this situation is that actually $[0] \sim _{rat} 0$! Namely we can choose the rational function $f_ j \in R(C_ j)$ to be the function which has a simple zero at $0$ and a simple pole at $\infty$ and no other zeros or poles. Then we see that the sum $\sum (i_ j)_*\text{div}(f_ j)$ is exactly the $0$-cycle $[0]$. In fact it turns out that $\mathop{\mathrm{CH}}\nolimits _0(X) = 0$ in this example. If you find this too bizarre, then you can just make sure your spaces are always quasi-compact (so $X$ does not even exist for you).