Lemma 42.31.6. In the situation of Lemma 42.31.5 assume $Y$ is locally of finite type over $(S, \delta )$ as in Situation 42.7.1. Then we have $i_1^*p^*\alpha = p_1^*i^*\alpha $ in $\mathop{\mathrm{CH}}\nolimits _{k - 1}(D_1)$ for all $\alpha \in \mathop{\mathrm{CH}}\nolimits _ k(Y)$.

**Proof.**
Let $W \subset Y$ be an integral closed subscheme of $\delta $-dimension $k$. We distinguish two cases.

Assume $W \subset D$. Then $i^*[W] = c_1(\mathcal{L}_1) \cap [W] + c_1(\mathcal{L}_2) \cap [W]$ in $\mathop{\mathrm{CH}}\nolimits _{k - 1}(D)$ by our definition of gysin homomorphisms and the additivity of Lemma 42.24.2. Hence $p_1^*i^*[W] = p_1^*(c_1(\mathcal{L}_1) \cap [W]) + p_1^*(c_1(\mathcal{L}_2) \cap [W])$. On the other hand, we have $p^*[W] = [p^{-1}(W)]_{k + 1}$ by construction of flat pullback. And $p^{-1}(W) = W_1 \cup W_2$ (scheme theoretically) where $W_ i = p_ i^{-1}(W)$ is a line bundle over $W$ by the lemma (since formation of the diagram commutes with base change). Then $[p^{-1}(W)]_{k + 1} = [W_1] + [W_2]$ as $W_ i$ are integral closed subschemes of $L$ of $\delta $-dimension $k + 1$. Hence

by construction of gysin homomorphisms, the definition of flat pullback (for the second equality), and compatibility of $c_1 \cap -$ with flat pullback (Lemma 42.25.2). Since $W_1 \cap W_2$ is the zero section of the line bundle $W_1 \to W$ we see from Lemma 42.31.4 that $[W_1 \cap W_2]_ k = p_1^*(c_1(\mathcal{L}_2) \cap [W])$. Note that here we use the fact that $D_1$ is the line bundle which is the relative spectrum of the inverse of $\mathcal{L}_2$. Thus we get the same thing as before.

Assume $W \not\subset D$. In this case, both $i_1^*p^*[W]$ and $p_1^*i^*[W]$ are represented by the $k - 1$ cycle associated to the scheme theoretic inverse image of $W$ in $D_1$. $\square$

## 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Your email address will not be published.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.

## Comments (2)

Comment #4640 by awllower on

Comment #4789 by Johan on