The Stacks project

Lemma 81.27.3. In Situation 81.2.1 let $X/B$ be good. Let $\mathcal{E}$ be a finite locally free sheaf of rank $r$ on $X$. Let

\[ p : E = \underline{\mathop{\mathrm{Spec}}}(\text{Sym}^*(\mathcal{E})) \longrightarrow X \]

be the associated vector bundle over $X$. Then $p^* : \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _{k + r}(E)$ is an isomorphism for all $k$.

Proof. (For the case of linebundles, see Lemma 81.25.2.) For surjectivity see Lemma 81.25.1. Let $(\pi : P \to X, \mathcal{O}_ P(1))$ be the projective space bundle associated to the finite locally free sheaf $\mathcal{E} \oplus \mathcal{O}_ X$. Let $s \in \Gamma (P, \mathcal{O}_ P(1))$ correspond to the global section $(0, 1) \in \Gamma (X, \mathcal{E} \oplus \mathcal{O}_ X)$. Let $D = Z(s) \subset P$. Note that $(\pi |_ D : D \to X , \mathcal{O}_ P(1)|_ D)$ is the projective space bundle associated to $\mathcal{E}$. We denote $\pi _ D = \pi |_ D$ and $\mathcal{O}_ D(1) = \mathcal{O}_ P(1)|_ D$. Moreover, $D$ is an effective Cartier divisor on $P$. Hence $\mathcal{O}_ P(D) = \mathcal{O}_ P(1)$ (see Divisors on Spaces, Lemma 70.7.8). Also there is an isomorphism $E \cong P \setminus D$. Denote $j : E \to P$ the corresponding open immersion. For injectivity we use that the kernel of

\[ j^* : \mathop{\mathrm{CH}}\nolimits _{k + r}(P) \longrightarrow \mathop{\mathrm{CH}}\nolimits _{k + r}(E) \]

are the cycles supported in the effective Cartier divisor $D$, see Lemma 81.15.2. So if $p^*\alpha = 0$, then $\pi ^*\alpha = i_*\beta $ for some $\beta \in \mathop{\mathrm{CH}}\nolimits _{k + r}(D)$. By Lemma 81.27.2 we may write

\[ \beta = \pi _ D^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^{r - 1} \cap \pi _ D^* \beta _{r - 1}. \]

for some $\beta _ i \in \mathop{\mathrm{CH}}\nolimits _{k + i}(X)$. By Lemmas 81.24.1 and 81.19.4 this implies

\[ \pi ^*\alpha = i_*\beta = c_1(\mathcal{O}_ P(1)) \cap \pi ^*\beta _0 + \ldots + c_1(\mathcal{O}_ D(1))^ r \cap \pi ^*\beta _{r - 1}. \]

Since the rank of $\mathcal{E} \oplus \mathcal{O}_ X$ is $r + 1$ this contradicts Lemma 81.19.4 unless all $\alpha $ and all $\beta _ i$ are zero. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0ERW. Beware of the difference between the letter 'O' and the digit '0'.