Lemma 62.5.9 (0H4X)—The Stacks project

Lemma 62.5.9. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $r \geq 0$ be an integer. Let $g : S' \to S$ be a morphism of schemes and $X' = S' \times _ S X$ . Assume that for every $s \in S$ there exists a point $s' \in S'$ with $g(s') = s$ and such that $\kappa (s')/\kappa (s)$ is a separable extension of fields. Then

For families $\alpha _1$ and $\alpha _2$ of $r$ -cycles on fibres of $X/S$ if $g^*\alpha _1 = g^*\alpha _2$ , then $\alpha _1 = \alpha _2$ .
Given a family $\alpha '$ of $r$ -cycles on fibres of $X'/S'$ if $\text{pr}_1^*\alpha ' = \text{pr}_2^*\alpha '$ as families of $r$ -cycles on fibres of $(S' \times _ S S') \times _ S X / (S' \times _ S S')$ , then there is a unique family $\alpha$ of $r$ -cycles on fibres of $X/S$ such that $g^*\alpha = \alpha '$ .

Proof. Part (1) follows from the injectivity of the base change map discussed in Section 62.3. (This argument works as long as $S' \to S$ is surjective.)

Let $\alpha '$ be as in (2). Denote $\alpha '' = \text{pr}_1^*\alpha ' = \text{pr}_2^*\alpha '$ the common value.

Let $(X/S)^{(r)}$ be the set of $x \in X$ with $\text{trdeg}_{\kappa (f(x))}(\kappa (x)) = r$ and similarly define $(X'/S')^{(r)}$ and $(X''/S'')^{(r)}$ Taking coefficients, we may think of $\alpha '$ and $\alpha ''$ as functions $\alpha ' : (X'/S')^{(r)} \to \mathbf{Z}$ and $\alpha '' : (X''/S'')^{(r)} \to \mathbf{Z}$ . Given a function

$\varphi : (X/S)^{(r)} \to \mathbf{Z}$

we define $g^*\varphi : (X'/S')^{(r)} \to \mathbf{Z}$ by analogy with our base change operation. Namely, say $x' \in (X'/S')^{(r)}$ maps to $x \in X$ , $s' \in S'$ , and $s \in Z$ . Denote $Z' \subset X'_{s'}$ and $Z \subset X_ s$ the integral closed subschemes with generic points $x'$ and $x$ . Note that $\dim (Z') = r$ . If $\dim (Z) < r$ , then we set $(g^*\varphi )(x') = 0$ . If $\dim (Z) = r$ , then $Z'$ is an irreducible component of $Z_{s'}$ and hence has a multiplicity $m_{Z', Z_{s'}}$ . Call this $m(x', g)$ . Then we define

$(g^*\varphi )(x') = m(x', g) \varphi (x)$

Note that the coefficients $m(x', g)$ are always positive integers (see for example Lemma 62.3.1). We similarly have base change maps

$\text{pr}_1^*, \text{pr}_2^* : \text{Map}((X'/S')^{(r)}, \mathbf{Z}) \longrightarrow \text{Map}((X''/S'')^{(r)}, \mathbf{Z})$

It follows from the associativity of base change that we have $\text{pr}_1^* \circ g^* = \text{pr}_2^* \circ g^*$ (small detail omitted). To be explicitly, in terms of the maps of sets this equality just means that for $x'' \in (X''/S'')^{(r)}$ we have

$m(x'', \text{pr}_1) m(\text{pr}_1(x''), g) = m(x'', \text{pr}_2) m(\text{pr}_2(x''), g)$

provided that $\text{pr}_1(x'')$ and $\text{pr}_2(x'')$ are in $(X''/S'')^{(r)}$ . By Lemma 62.5.8 and an elementary argument ¹ using the previous displayed equation, it follows that there exists a unique map

$\alpha : (X/S)^{(r)} \to \mathbf{Q}$

such that $g^*\alpha = \alpha '$ . To finish the proof it suffices to show that $\alpha$ has integer values (small detail omitted: one needs to see that $\alpha$ determines a locally finite sum on each fibre which follows from the corresponding fact for $\alpha '$ ). Given any $x \in (X/S)^{(r)}$ with image $s \in S$ we can pick a point $s' \in S'$ such that $\kappa (s')/\kappa (s)$ is separable. Then we may choose $x' \in (X'/S')^{(r)}$ mapping to $s$ and $x$ and we see that $m(x', g) = 1$ because $Z_{s'}$ is reduced in this case. Whence $\alpha (x) = \alpha '(x')$ is an integer. $\square$

[1] Given

$x \in (X/S)^{(r)}$ pick

$x' \in (X'/S')^{(r)}$ mapping to

$x$ and set

$\alpha (x) = \alpha '(x')/m(x', g)$ . This is well defined by the formula and the lemma.

The Stacks project

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