The Stacks project

42.44 Chern classes and sections

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

\[ s \in \Gamma (X', f^*\mathcal{E}) \]

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

\[ c_ r(\mathcal{E}) \cap [X']_ n = [Z(s)]_{n - r} \]

in $\mathop{\mathrm{CH}}\nolimits _*(X')$.

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

\[ 0 \to \mathcal{E}' \to \pi ^*\mathcal{E} \to \mathcal{O}_ P(1) \to 0 \]

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

\[ [Z(t)]_{n + r - 2} = c_1(\mathcal{O}_ P(1)) \cap [P]_{n + r - 1} \]

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

\begin{align*} c_ r(\pi ^*\mathcal{E}) \cap [P]_{n + r - 1} & = c_{r - 1}(\mathcal{E}') \cap c_1(\mathcal{O}_ P(1)) \cap [P]_{n + r - 1} \\ & = c_{r - 1}(\mathcal{E}') \cap [Z(t)]_{n + r - 2} \\ & = [Z(s')]_{n - 1} \\ & = [Z(\pi ^*s)]_{n - 1} \end{align*}

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

\[ 0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0 \]

be a short exact sequence of finite locally free $\mathcal{O}_ X$-modules. Consider the closed embedding

\[ i : N' = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}((\mathcal{N}')^\vee )) \longrightarrow N = \underline{\mathop{\mathrm{Spec}}}_ X(\text{Sym}(\mathcal{N}^\vee )) \]

For $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(X)$ we have

\[ i_*(p')^*\alpha = p^*(c_{top}(\mathcal{E}) \cap \alpha ) \]

where $p' : N' \to X$ and $p : N \to X$ are the structure morphisms.

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

\[ s \in \Gamma (N, p^*\mathcal{E}) = \Gamma (X, p_*p^*\mathcal{E}) = \Gamma (X, \mathcal{E} \otimes _{\mathcal{O}_ X} \text{Sym}(\mathcal{N}^\vee ) \supset \Gamma (X, \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, \mathcal{E})) \]

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

  1. $p^*[Y] = [p^{-1}(Y)]$ since $p^{-1}(Y)$ is integral of $\delta $-dimension $n + r$,

  2. $(p')^*[Y] = [(p')^{-1}(Y)]$ since $(p')^{-1}(Y)$ is integral of $\delta $-dimension $n + r'$,

  3. 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

\[ (p')^*[Y] = c_{r - r'}(p^*\mathcal{E}) \cap p^*[Y] \quad \text{in}\quad \mathop{\mathrm{CH}}\nolimits _*(N) \]

by Lemma 42.44.1. This proves the lemma. $\square$

Comments (2)

Comment #9271 by AprilGrimoire on

I'm kind of confused by the setup here. In the setup we have is a dim function on , then what does mean? Can I understand it as the usual dimension notion?

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