A brief section whose main result is that we may compute the top Chern class of a finite locally free module using the vanishing locus of a “regular section.
Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{E}$ be a finite locally free $\mathcal{O}_ X$-module. Let $f : X' \to X$ be locally of finite type. Let
be a global section of the pullback of $\mathcal{E}$ to $X'$. Let $Z(s) \subset X'$ be the zero scheme of $s$. More precisely, we define $Z(s)$ to be the closed subscheme whose quasi-coherent sheaf of ideals is the image of the map $s : f^*\mathcal{E}^\vee \to \mathcal{O}_{X'}$.
Lemma 42.44.1. In the situation described just above assume $\dim _\delta (X') = n$, that $f^*\mathcal{E}$ has constant rank $r$, that $\dim _\delta (Z(s)) \leq n - r$, and that for every generic point $\xi \in Z(s)$ with $\delta (\xi ) = n - r$ the ideal of $Z(s)$ in $\mathcal{O}_{X', \xi }$ is generated by a regular sequence of length $r$. Then in $\mathop{\mathrm{CH}}\nolimits _*(X')$.
\[ c_ r(\mathcal{E}) \cap [X']_ n = [Z(s)]_{n - r} \]
Proof. Since $c_ r(\mathcal{E})$ is a bivariant class (Lemma 42.38.7) we may assume $X = X'$ and we have to show that $c_ r(\mathcal{E}) \cap [X]_ n = [Z(s)]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _{n - r}(X)$. We will prove the lemma by induction on $r \geq 0$. (The case $r = 0$ is trivial.) The case $r = 1$ is handled by Lemma 42.25.4. Assume $r > 1$.
Let $\pi : P \to X$ be the projective space bundle associated to $\mathcal{E}$ and consider the short exact sequence
By the projective space bundle formula (Lemma 42.36.2) it suffices to prove the equality after pulling back by $\pi $. Observe that $\pi ^{-1}Z(s) = Z(\pi ^*s)$ has $\delta $-dimension $\leq n - 1$ and that the assumption on regular sequences at generic points of $\delta $-dimension $n - 1$ holds by flat pullback, see Algebra, Lemma 10.68.5. Let $t \in \Gamma (P, \mathcal{O}_ P(1))$ be the image of $\pi ^*s$. We claim
Assuming the claim we finish the proof as follows. The restriction $\pi ^*s|_{Z(t)}$ maps to zero in $\mathcal{O}_ P(1)|_{Z(t)}$ hence comes from a unique element $s' \in \Gamma (Z(t), \mathcal{E}'|_{Z(t)})$. Note that $Z(s') = Z(\pi ^*s)$ as closed subschemes of $P$. If $\xi \in Z(s')$ is a generic point with $\delta (\xi ) = n - 1$, then the ideal of $Z(s')$ in $\mathcal{O}_{Z(t), \xi }$ can be generated by a regular sequence of length $r - 1$: it is generated by $r - 1$ elements which are the images of $r - 1$ elements in $\mathcal{O}_{P, \xi }$ which together with a generator of the ideal of $Z(t)$ in $\mathcal{O}_{P, \xi }$ form a regular sequence of length $r$ in $\mathcal{O}_{P, \xi }$. Hence we can apply the induction hypothesis to $s'$ on $Z(t)$ to get $c_{r - 1}(\mathcal{E}') \cap [Z(t)]_{n + r - 2} = [Z(s')]_{n - 1}$. Combining all of the above we obtain
which is what we had to show.
Proof of the claim. This will follow from an application of the already used Lemma 42.25.4. We have $\pi ^{-1}(Z(s)) = Z(\pi ^*s) \subset Z(t)$. On the other hand, for $x \in X$ if $P_ x \subset Z(t)$, then $t|_{P_ x} = 0$ which implies that $s$ is zero in the fibre $\mathcal{E} \otimes \kappa (x)$, which implies $x \in Z(s)$. It follows that $\dim _\delta (Z(t)) \leq n + (r - 1) - 1$. Finally, let $\xi \in Z(t)$ be a generic point with $\delta (\xi ) = n + r - 2$. If $\xi $ is not the generic point of the fibre of $P \to X$ it is immediate that a local equation of $Z(t)$ is a nonzerodivisor in $\mathcal{O}_{P, \xi }$ (because we can check this on the fibre by Algebra, Lemma 10.99.2). If $\xi $ is the generic point of a fibre, then $x = \pi (\xi ) \in Z(s)$ and $\delta (x) = n + r - 2 - (r - 1) = n - 1$. This is a contradiction with $\dim _\delta (Z(s)) \leq n - r$ because $r > 1$ so this case doesn't happen. $\square$
Lemma 42.44.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let be a short exact sequence of finite locally free $\mathcal{O}_ X$-modules. Consider the closed embedding For $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we have where $p' : N' \to X$ and $p : N \to X$ are the structure morphisms.
\[ 0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0 \]
\[ i : N' = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}((\mathcal{N}')^\vee )) \longrightarrow N = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}(\mathcal{N}^\vee )) \]
\[ i_*(p')^*\alpha = p^*(c_{top}(\mathcal{E}) \cap \alpha ) \]
Proof. Here $c_{top}(\mathcal{E})$ is the bivariant class defined in Remark 42.38.11. By its very definition, in order to verify the formula, we may assume that $\mathcal{E}$ has constant rank. We may similarly assume $\mathcal{N}'$ and $\mathcal{N}$ have constant ranks, say $r'$ and $r$, so $\mathcal{E}$ has rank $r - r'$ and $c_{top}(\mathcal{E}) = c_{r - r'}(\mathcal{E})$. Observe that $p^*\mathcal{E}$ has a canonical section
corresponding to the surjection $\mathcal{N} \to \mathcal{E}$ given in the statement of the lemma. The vanishing scheme of this section is exactly $N' \subset N$. Let $Y \subset X$ be an integral closed subscheme of $\delta $-dimension $n$. Then we have
$p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of $\delta $-dimension $n + r$,
$(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of $\delta $-dimension $n + r'$,
the restriction of $s$ to $p^{-1}Y$ has vanishing scheme $(p')^{-1}Y$ and the closed immersion $(p')^{-1}Y \to p^{-1}Y$ is a regular immersion (locally cut out by a regular sequence).
We conclude that
by Lemma 42.44.1. This proves the lemma. $\square$
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