Lemma 43.17.1. Let $X$ be a nonsingular variety. Let $a, b \in \mathbf{P}^1$ be distinct closed points. Let $k \geq 0$.

If $W \subset X \times \mathbf{P}^1$ is a closed subvariety of dimension $k + 1$ which intersects $X \times a$ properly, then

$[W_ a]_ k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and

$[W_ a]_ k = \text{pr}_{X, *}(W \cdot X \times a)$ as cycles on $X$.

Let $\alpha $ be a $(k + 1)$-cycle on $X \times \mathbf{P}^1$ which intersects $X \times a$ and $X \times b$ properly. Then $pr_{X,*}( \alpha \cdot X \times a - \alpha \cdot X \times b)$ is rationally equivalent to zero.

Conversely, any $k$-cycle which is rationally equivalent to $0$ is of this form.

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