Lemma 43.17.1 (0B07)—The Stacks project

Table of contents
Part 3: Topics in Scheme Theory
Chapter 43: Intersection Theory
Section 43.17: Intersection product using Tor formula
Lemma 43.17.1 (cite)

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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
1. $[W_ a]_ k = W \cdot X \times a$ as cycles on $X \times \mathbf{P}^1$, and
2. $[W_ a]_ k = \text{pr}_{X, *}(W \cdot X \times a)$ 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.

Proof. First we observe that $X \times a$ is an effective Cartier divisor in $X \times \mathbf{P}^1$ and that $W_ a$ is the scheme theoretic intersection of $W$ with $X \times a$. Hence the equality in (1)(a) is immediate from the definitions and the calculation of intersection multiplicity in case of a Cartier divisor given in Lemma 43.16.4. Part (1)(b) holds because $W_ a \to X \times \mathbf{P}^1 \to X$ maps isomorphically onto its image which is how we viewed $W_ a$ as a closed subscheme of $X$ in Section 43.8. Parts (2) and (3) are formal consequences of part (1) and the definitions. $\square$

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