The Stacks project

Lemma 114.23.1. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\{ i_ j : D_ j \to X \} _{j \in J}$ be a locally finite collection of effective Cartier divisors on $X$. Let $n_ j > 0$, $j\in J$. Set $D = \sum _{j \in J} n_ j D_ j$, and denote $i : D \to X$ the inclusion morphism. Let $\alpha \in Z_{k + 1}(X)$. Then

\[ p : \coprod \nolimits _{j \in J} D_ j \longrightarrow D \]

is proper and

\[ i^*\alpha = p_*\left(\sum n_ j i_ j^*\alpha \right) \]

in $\mathop{\mathrm{CH}}\nolimits _ k(D)$.

Proof. The proof of this lemma is made a bit longer than expected by a subtlety concerning infinite sums of rational equivalences. In the quasi-compact case the family $D_ j$ is finite and the result is altogether easy and a straightforward consequence of Chow Homology, Lemma 42.24.2 and Divisors, Lemma 31.27.5 and the definitions.

The morphism $p$ is proper since the family $\{ D_ j\} _{j \in J}$ is locally finite. Write $\alpha = \sum _{a \in A} m_ a [W_ a]$ with $W_ a \subset X$ an integral closed subscheme of $\delta $-dimension $k + 1$. Denote $i_ a : W_ a \to X$ the closed immersion. We assume that $m_ a \not= 0$ for all $a \in A$ such that $\{ W_ a\} _{a \in A}$ is locally finite on $X$.

Observe that by Chow Homology, Definition 42.29.1 the class $i^*\alpha $ is the class of a cycle $\sum m_ a\beta _ a$ for certain $\beta _ a \in Z_ k(W_ a \cap D)$. Namely, if $W_ a \not\subset D$ then $\beta _ a = [D \cap W_ a]_ k$ and if $W_ a \subset D$, then $\beta _ a$ is a cycle representing $c_1(\mathcal{O}_ X(D)) \cap [W_ a]$.

For each $a \in A$ write $J = J_{a, 1} \amalg J_{a, 2} \amalg J_{a, 3}$ where

  1. $j \in J_{a, 1}$ if and only if $W_ a \cap D_ j = \emptyset $,

  2. $j \in J_{a, 2}$ if and only if $W_ a \not= W_ a \cap D_1 \not= \emptyset $, and

  3. $j \in J_{a, 3}$ if and only if $W_ a \subset D_ j$.

Since the family $\{ D_ j\} $ is locally finite we see that $J_{a, 3}$ is a finite set. For every $a \in A$ and $j \in J$ we choose a cycle $\beta _{a, j} \in Z_ k(W_ a \cap D_ j)$ as follows

  1. if $j \in J_{a, 1}$ we set $\beta _{a, j} = 0$,

  2. if $j \in J_{a, 2}$ we set $\beta _{a, j} = [D_ j \cap W_ a]_ k$, and

  3. if $j \in J_{a, 3}$ we choose $\beta _{a, j} \in Z_ k(W_ a)$ representing $c_1(i_ a^*\mathcal{O}_ X(D_ j)) \cap [W_ j]$.

We claim that

\[ \beta _ a \sim _{rat} \sum \nolimits _{j \in J} n_ j \beta _{a, j} \]

in $\mathop{\mathrm{CH}}\nolimits _ k(W_ a \cap D)$.

Case I: $W_ a \not\subset D$. In this case $J_{a, 3} = \emptyset $. Thus it suffices to show that $[D \cap W_ a]_ k = \sum n_ j [D_ j \cap W_ a]_ k$ as cycles. This is Lemma 114.22.10.

Case II: $W_ a \subset D$. In this case $\beta _ a$ is a cycle representing $c_1(i_ a^*\mathcal{O}_ X(D)) \cap [W_ a]$. Write $D = D_{a, 1} + D_{a, 2} + D_{a, 3}$ with $D_{a, s} = \sum _{j \in J_{a, s}} n_ jD_ j$. By Divisors, Lemma 31.27.5 we have

\begin{eqnarray*} c_1(i_ a^*\mathcal{O}_ X(D)) \cap [W_ a] & = & c_1(i_ a^*\mathcal{O}_ X(D_{a, 1})) \cap [W_ a] + c_1(i_ a^*\mathcal{O}_ X(D_{a, 2})) \cap [W_ a] \\ & & + c_1(i_ a^*\mathcal{O}_ X(D_{a, 3})) \cap [W_ a]. \end{eqnarray*}

It is clear that the first term of the sum is zero. Since $J_{a, 3}$ is finite we see that the last term agrees with $\sum \nolimits _{j \in J_{a, 3}} n_ jc_1(i_ a^*\mathcal{L}_ j) \cap [W_ a]$, see Divisors, Lemma 31.27.5. This is represented by $\sum _{j \in J_{a, 3}} n_ j \beta _{a, j}$. Finally, by Case I we see that the middle term is represented by the cycle $\sum \nolimits _{j \in J_{a, 2}} n_ j[D_ j \cap W_ a]_ k = \sum _{j \in J_{a, 2}} n_ j\beta _{a, j}$. Whence the claim in this case.

At this point we are ready to finish the proof of the lemma. Namely, we have $i^*D \sim _{rat} \sum m_ a\beta _ a$ by our choice of $\beta _ a$. For each $a$ we have $\beta _ a \sim _{rat} \sum _ j \beta _{a, j}$ with the rational equivalence taking place on $D \cap W_ a$. Since the collection of closed subschemes $D \cap W_ a$ is locally finite on $D$, we see that also $\sum m_ a \beta _ a \sim _{rat} \sum _{a, j} m_ a\beta _{a, j}$ on $D$! (See Chow Homology, Remark 42.19.6.) Ok, and now it is clear that $\sum _ a m_ a\beta _{a, j}$ (viewed as a cycle on $D_ j$) represents $i_ j^*\alpha $ and hence $\sum _{a, j} m_ a\beta _{a, j}$ represents $p_* \sum _ j i_ j^*\alpha $ and we win. $\square$


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 02TC. Beware of the difference between the letter 'O' and the digit '0'.