Lemma 115.24.4 (02TF)—The Stacks project

Lemma 115.24.4. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral and $\dim _\delta (X) = n$. Let $D$, $D'$ be effective Cartier divisors on $X$. Then

\[ D \cdot [D']_{n - 1} = D' \cdot [D]_{n - 1} \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$.

First proof of Lemma 115.24.4. First, let us prove this in case $X$ is quasi-compact. In this case, apply Lemma 115.23.11 to $X$ and the two element set $\{ D, D'\} $ of effective Cartier divisors. Thus we get a proper morphism $b : X' \to X$, a finite collection of effective Cartier divisors $D'_ j \subset X'$ intersecting pairwise in codimension $\geq 2$, with $b^{-1}(D) = \sum n_ j D'_ j$, and $b^{-1}(D') = \sum m_ j D'_ j$. Note that $b_*[b^{-1}(D)]_{n - 1} = [D]_{n - 1}$ in $Z_{n - 1}(X)$ and similarly for $D'$, see Lemma 115.23.5. Hence, by Chow Homology, Lemma 42.26.4 we have

\[ D \cdot [D']_{n - 1} = b_*\left(b^{-1}(D) \cdot [b^{-1}(D')]_{n - 1}\right) \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$ and similarly for the other term. Hence the lemma follows from the equality $b^{-1}(D) \cdot [b^{-1}(D')]_{n - 1} = b^{-1}(D') \cdot [b^{-1}(D)]_{n - 1}$ in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X')$ of Lemma 115.24.3.

Note that in the proof above, each referenced lemma works also in the general case (when $X$ is not assumed quasi-compact). The only minor change in the general case is that the morphism $b : U' \to U$ we get from applying Lemma 115.23.11 has as its target an open $U \subset X$ whose complement has codimension $\geq 3$. Hence by Chow Homology, Lemma 42.19.3 we see that $\mathop{\mathrm{CH}}\nolimits _{n - 2}(U) = \mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$ and after replacing $X$ by $U$ the rest of the proof goes through unchanged. $\square$

Second proof of Lemma 115.24.4. Let $\mathcal{I} = \mathcal{O}_ X(-D)$ and $\mathcal{I}' = \mathcal{O}_ X(-D')$ be the invertible ideal sheaves of $D$ and $D'$. We denote $\mathcal{I}_{D'} = \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{O}_{D'}$ and $\mathcal{I}'_ D = \mathcal{I}' \otimes _{\mathcal{O}_ X} \mathcal{O}_ D$. We can restrict the inclusion map $\mathcal{I} \to \mathcal{O}_ X$ to $D'$ to get a map

\[ \varphi : \mathcal{I}_{D'} \longrightarrow \mathcal{O}_{D'} \]

and similarly

\[ \psi : \mathcal{I}'_ D \longrightarrow \mathcal{O}_ D \]

It is clear that

\[ \mathop{\mathrm{Coker}}(\varphi ) \cong \mathcal{O}_{D \cap D'} \cong \mathop{\mathrm{Coker}}(\psi ) \]

and

\[ \mathop{\mathrm{Ker}}(\varphi ) \cong \frac{\mathcal{I} \cap \mathcal{I}'}{\mathcal{I}\mathcal{I}'} \cong \mathop{\mathrm{Ker}}(\psi ). \]

Hence we see that

\[ \gamma = [\mathcal{I}_{D'}] - [\mathcal{O}_{D'}] = [\mathcal{I}'_ D] - [\mathcal{O}_ D] \]

in $K_0(\textit{Coh}_{\leq n - 1}(X))$. On the other hand it is clear that

\[ [\mathcal{I}'_ D]_{n - 1} = [D]_{n - 1}, \quad [\mathcal{I}_{D'}]_{n - 1} = [D']_{n - 1}. \]

and that

\[ \mathcal{O}_ X(D') \otimes \mathcal{I}'_ D = \mathcal{O}_ D, \quad \mathcal{O}_ X(D) \otimes \mathcal{I}_{D'} = \mathcal{O}_{D'}. \]

By Chow Homology, Lemma 42.69.7 (applied two times) this means that the element $\gamma $ is an element of $B_{n - 2}(X)$, and maps to both $c_1(\mathcal{O}_ X(D')) \cap [D]_{n - 1}$ and to $c_1(\mathcal{O}_ X(D)) \cap [D']_{n - 1}$ and we win (since the map $B_{n - 2}(X) \to \mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$ is well defined – which is the key to this proof). $\square$

The Stacks project

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