Lemma 42.40.3 (02UI)—The Stacks project

Lemma 42.40.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Suppose that ${\mathcal E}$ sits in an exact sequence

\[ 0 \to {\mathcal E}_1 \to {\mathcal E} \to {\mathcal E}_2 \to 0 \]

of finite locally free sheaves $\mathcal{E}_ i$ of rank $r_ i$. The total Chern classes satisfy

\[ c({\mathcal E}) = c({\mathcal E}_1) c({\mathcal E}_2) \]

in $A^*(X)$.

Proof. By Lemma 42.35.3 we may assume that $X$ is integral and we have to show the identity when capping against $[X]$. By induction on $r_1$. The case $r_1 = 1$ is Lemma 42.40.2. Assume $r_1 > 1$. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ denote the projective space bundle associated to $\mathcal{E}_1$. Note that

$\pi ^* : \mathop{\mathrm{CH}}\nolimits _*(X) \to \mathop{\mathrm{CH}}\nolimits _*(P)$ is injective, and
$\pi ^*\mathcal{E}_1$ sits in a short exact sequence $0 \to \mathcal{F} \to \pi ^*\mathcal{E}_1 \to \mathcal{L} \to 0$ where $\mathcal{L}$ is invertible.

The first assertion follows from the projective space bundle formula and the second follows from the definition of a projective space bundle. (In fact $\mathcal{L} = \mathcal{O}_ P(1)$.) Let $Q = \pi ^*\mathcal{E}/\mathcal{F}$, which sits in an exact sequence $0 \to \mathcal{L} \to Q \to \pi ^*\mathcal{E}_2 \to 0$. By induction we have

\begin{eqnarray*} c(\pi ^*\mathcal{E}) \cap [P] & = & c(\mathcal{F}) \cap c(\pi ^*\mathcal{E}/\mathcal{F}) \cap [P] \\ & = & c(\mathcal{F}) \cap c(\mathcal{L}) \cap c(\pi ^*\mathcal{E}_2) \cap [P] \\ & = & c(\pi ^*\mathcal{E}_1) \cap c(\pi ^*\mathcal{E}_2) \cap [P] \end{eqnarray*}

Since $[P] = \pi ^*[X]$ we win by Lemma 42.38.5. $\square$

Comments (2)

Comment #7538 by Hao Peng on July 30, 2022 at 00:26

I think we can have a better proof for this one if we use double induction on rank of $\mathcal E_1$ and $\mathcal E_2$ , we can use splitting principle to reduce the invertible sheaf case, which is treated before.

Comment #7664 by Stacks Project on August 15, 2022 at 11:42

I am leaving this as is.

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