Lemma 42.69.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume $\dim _\delta (\text{Supp}(\mathcal{F})) \leq k + 1$. Then the element

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] \in K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$

lies in the subgroup $B_ k(X)$ of Lemma 42.69.3 and maps to the element $c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1}$ via the map $B_ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$.

Proof. Let

$0 \to \mathcal{K} \to \mathcal{F} \to \mathcal{F}' \to 0$

be the short exact sequence constructed in Divisors, Lemma 31.4.6. This in particular means that $\mathcal{F}'$ has no embedded associated points. Since the support of $\mathcal{K}$ is nowhere dense in the support of $\mathcal{F}$ we see that $\dim _\delta (\text{Supp}(\mathcal{K})) \leq k$. We may re-apply Divisors, Lemma 31.4.6 starting with $\mathcal{K}$ to get a short exact sequence

$0 \to \mathcal{K}'' \to \mathcal{K} \to \mathcal{K}' \to 0$

where now $\dim _\delta (\text{Supp}(\mathcal{K}'')) < k$ and $\mathcal{K}'$ has no embedded associated points. Suppose we can prove the lemma for the coherent sheaves $\mathcal{F}'$ and $\mathcal{K}'$. Then we see from the equations

$[\mathcal{F}]_{k + 1} = [\mathcal{F}']_{k + 1} + [\mathcal{K}']_{k + 1} + [\mathcal{K}'']_{k + 1}$

(use Lemma 42.10.4),

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] = [\mathcal{F}' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}'] + [\mathcal{K}' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'] + [\mathcal{K}'' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'']$

(use the $\otimes \mathcal{L}$ is exact) and the trivial vanishing of $[\mathcal{K}'']_{k + 1}$ and $[\mathcal{K}'' \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{K}'']$ in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$ that the result holds for $\mathcal{F}$. What this means is that we may assume that the sheaf $\mathcal{F}$ has no embedded associated points.

Assume $X$, $\mathcal{F}$ as in the lemma, and assume in addition that $\mathcal{F}$ has no embedded associated points. Consider the sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ X$, the corresponding closed subscheme $i : Z \to X$ and the coherent $\mathcal{O}_ Z$-module $\mathcal{G}$ constructed in Divisors, Lemma 31.4.7. Recall that $Z$ is a locally Noetherian scheme without embedded points, $\mathcal{G}$ is a coherent sheaf without embedded associated points, with $\text{Supp}(\mathcal{G}) = Z$ and such that $i_*\mathcal{G} = \mathcal{F}$. Moreover, set $\mathcal{N} = \mathcal{L}|_ Z$.

By Divisors, Lemma 31.25.4 the invertible sheaf $\mathcal{N}$ has a regular meromorphic section $s$ over $Z$. Let us denote $\mathcal{J} \subset \mathcal{O}_ Z$ the sheaf of denominators of $s$. By Lemma 42.69.6 there exist short exact sequences

$\begin{matrix} 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow {1} & \mathcal{G} & \to & \mathcal{Q}_1 & \to & 0 \\ 0 & \to & \mathcal{J}\mathcal{G} & \xrightarrow {s} & \mathcal{G} \otimes _{\mathcal{O}_ Z} \mathcal{N} & \to & \mathcal{Q}_2 & \to & 0 \end{matrix}$

such that $\dim _\delta (\text{Supp}(\mathcal{Q}_ i)) \leq k$ and such that the cycle $[\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k$ is a representative of $c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1}$. We see (using the fact that $i_*(\mathcal{G} \otimes \mathcal{N}) = \mathcal{F} \otimes \mathcal{L}$ by the projection formula, see Cohomology, Lemma 20.52.2) that

$[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}] = [i_*\mathcal{Q}_2] - [i_*\mathcal{Q}_1]$

in $K_0(\textit{Coh}_{\leq k + 1}(X)/\textit{Coh}_{\leq k - 1}(X))$. This already shows that $[\mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}] - [\mathcal{F}]$ is an element of $B_ k(X)$. Moreover we have

\begin{eqnarray*} [i_*\mathcal{Q}_2]_ k - [i_*\mathcal{Q}_1]_ k & = & i_*\left( [\mathcal{Q}_2]_ k - [\mathcal{Q}_1]_ k \right) \\ & = & i_*\left(c_1(\mathcal{N}) \cap [\mathcal{G}]_{k + 1} \right) \\ & = & c_1(\mathcal{L}) \cap i_*[\mathcal{G}]_{k + 1} \\ & = & c_1(\mathcal{L}) \cap [\mathcal{F}]_{k + 1} \end{eqnarray*}

by the above and Lemmas 42.26.4 and 42.12.4. And this agree with the image of the element under $B_ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(X)$ by definition. Hence the lemma is proved. $\square$

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