The Stacks project

Lemma 42.54.6. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Let $Y \to X$ be a morphism which is locally of finite type. Given integers $r$, $n$ assume

  1. $\mathcal{N}$ is locally free of rank $r$,

  2. every irreducible component of $Y$ has $\delta $-dimension $n$,

  3. $\dim _\delta (Z \times _ X Y) \leq n - r$, and

  4. for $\xi \in Z \times _ X Y$ with $\delta (\xi ) = n - r$ the local ring $\mathcal{O}_{Y, \xi }$ is Cohen-Macaulay.

Then $c(Z \to X, \mathcal{N}) \cap [Y]_ n = [Z \times _ X Y]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$.

Proof. The statement makes sense as $Z \times _ X Y$ is a closed subscheme of $Y$. Because $\mathcal{N}$ has rank $r$ we know that $c(Z \to X, \mathcal{N}) \cap [Y]_ n$ is in $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$. Since $\dim _\delta (Z \cap Y) \leq n - r$ the chow group $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$ is freely generated by the cycle classes of the irreducible components $W \subset Z \times _ X Y$ of $\delta $-dimension $n - r$. Let $\xi \in W$ be the generic point. By assumption (2) we see that $\dim (\mathcal{O}_{Y, \xi }) = r$. On the other hand, since $\mathcal{N}$ has rank $r$ and since $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is surjective, we see that the ideal sheaf of $Z$ is locally cut out by $r$ equations. Hence the quasi-coherent ideal sheaf $\mathcal{I} \subset \mathcal{O}_ Y$ of $Z \times _ X Y$ in $Y$ is locally generated by $r$ elements. Since $\mathcal{O}_{Y, \xi }$ is Cohen-Macaulay of dimension $r$ and since $\mathcal{I}_\xi $ is an ideal of definition (as $\xi $ is a generic point of $Z \times _ X Y$) it follows that $\mathcal{I}_\xi $ is generated by a regular sequence (Algebra, Lemma 10.104.2). By Divisors, Lemma 31.20.8 we see that $\mathcal{I}$ is generated by a regular sequence over an open neighbourhood $V \subset Y$ of $\xi $. By our description of $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$ it suffices to show that $c(Z \to X, \mathcal{N}) \cap [V]_ n = [Z \times _ X V]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X V)$. This follows from Lemma 42.54.5 because the excess normal sheaf is $0$ over $V$. $\square$

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