Lemma 42.23.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. The maps

$Z_ k(X) \longrightarrow K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)), \quad \sum n_ Z[Z] \mapsto \left[\bigoplus \nolimits _{n_ Z > 0} \mathcal{O}_ Z^{\oplus n_ Z}\right] - \left[\bigoplus \nolimits _{n_ Z < 0} \mathcal{O}_ Z^{\oplus -n_ Z}\right]$

and

$K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X)) \longrightarrow Z_ k(X),\quad \mathcal{F} \longmapsto [\mathcal{F}]_ k$

are mutually inverse isomorphisms.

Proof. Note that if $\sum n_ Z[Z]$ is in $Z_ k(X)$, then the direct sums $\bigoplus \nolimits _{n_ Z > 0} \mathcal{O}_ Z^{\oplus n_ Z}$ and $\bigoplus \nolimits _{n_ Z < 0} \mathcal{O}_ Z^{\oplus -n_ Z}$ are coherent sheaves on $X$ since the family $\{ Z \mid n_ Z > 0\}$ is locally finite on $X$. The map $\mathcal{F} \to [\mathcal{F}]_ k$ is additive on $\textit{Coh}_{\leq k}(X)$, see Lemma 42.10.4. And $[\mathcal{F}]_ k = 0$ if $\mathcal{F} \in \textit{Coh}_{\leq k - 1}(X)$. By part (1) of Homology, Lemma 12.11.3 this implies that the second map is well defined too. It is clear that the composition of the first map with the second map is the identity.

Conversely, say we start with a coherent sheaf $\mathcal{F}$ on $X$. Write $[\mathcal{F}]_ k = \sum _{i \in I} n_ i[Z_ i]$ with $n_ i > 0$ and $Z_ i \subset X$, $i \in I$ pairwise distinct integral closed subschemes of $\delta$-dimension $k$. We have to show that

$[\mathcal{F}] = [\bigoplus \nolimits _{i \in I} \mathcal{O}_{Z_ i}^{\oplus n_ i}]$

in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Denote $\xi _ i \in Z_ i$ the generic point. If we set

$\mathcal{F}' = \mathop{\mathrm{Ker}}(\mathcal{F} \to \bigoplus \xi _{i, *}\mathcal{F}_{\xi _ i})$

then $\mathcal{F}'$ is the maximal coherent submodule of $\mathcal{F}$ whose support has dimension $\leq k - 1$. In particular $\mathcal{F}$ and $\mathcal{F}/\mathcal{F}'$ have the same class in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Thus after replacing $\mathcal{F}$ by $\mathcal{F}/\mathcal{F}'$ we may and do assume that the kernel $\mathcal{F}'$ displayed above is zero.

For each $i \in I$ we choose a filtration

$\mathcal{F}_{\xi _ i} = \mathcal{F}_ i^0 \supset \mathcal{F}_ i^1 \supset \ldots \supset \mathcal{F}_ i^{n_ i} = 0$

such that the successive quotients are of dimension $1$ over the residue field at $\xi _ i$. This is possible as the length of $\mathcal{F}_{\xi _ i}$ over $\mathcal{O}_{X, \xi _ i}$ is $n_ i$. For $p > n_ i$ set $\mathcal{F}_ i^ p = 0$. For $p \geq 0$ we denote

$\mathcal{F}^ p = \mathop{\mathrm{Ker}}\left(\mathcal{F} \longrightarrow \bigoplus \xi _{i, *}(\mathcal{F}_{\xi _ i}/\mathcal{F}_ i^ p)\right)$

Then $\mathcal{F}^ p$ is coherent, $\mathcal{F}^0 = \mathcal{F}$, and $\mathcal{F}^ p/\mathcal{F}^{p + 1}$ is isomorphic to a free $\mathcal{O}_{Z_ i}$-module of rank $1$ (if $n_ i > p$) or $0$ (if $n_ i \leq p$) in an open neighbourhood of $\xi _ i$. Moreover, $\mathcal{F}' = \bigcap \mathcal{F}^ p = 0$. Since every quasi-compact open $U \subset X$ contains only a finite number of $\xi _ i$ we conclude that $\mathcal{F}^ p|_ U$ is zero for $p \gg 0$. Hence $\bigoplus _{p \geq 0} \mathcal{F}^ p$ is a coherent $\mathcal{O}_ X$-module. Consider the short exact sequences

$0 \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p/\mathcal{F}^{p + 1} \to 0$

and

$0 \to \bigoplus \nolimits _{p > 0} \mathcal{F}^ p \to \bigoplus \nolimits _{p \geq 0} \mathcal{F}^ p \to \mathcal{F} \to 0$

of coherent $\mathcal{O}_ X$-modules. This already shows that

$[\mathcal{F}] = [\bigoplus \mathcal{F}^ p/\mathcal{F}^{p + 1}]$

in $K_0(\textit{Coh}_{\leq k}(X)/\textit{Coh}_{\leq k - 1}(X))$. Next, for every $p \geq 0$ and $i \in I$ such that $n_ i > p$ we choose a nonzero ideal sheaf $\mathcal{I}_{i, p} \subset \mathcal{O}_{Z_ i}$ and a map $\mathcal{I}_{i, p} \to \mathcal{F}^ p/\mathcal{F}^{p + 1}$ on $X$ which is an isomorphism over the open neighbourhood of $\xi _ i$ mentioned above. This is possible by Cohomology of Schemes, Lemma 30.10.6. Then we consider the short exact sequence

$0 \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{I}_{i, p} \to \bigoplus \mathcal{F}^ p/\mathcal{F}^{p + 1} \to \mathcal{Q} \to 0$

and the short exact sequence

$0 \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{I}_{i, p} \to \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{O}_{Z_ i} \to \mathcal{Q}' \to 0$

Observe that both $\mathcal{Q}$ and $\mathcal{Q}'$ are zero in a neighbourhood of the points $\xi _ i$ and that they are supported on $\bigcup Z_ i$. Hence $\mathcal{Q}$ and $\mathcal{Q}'$ are in $\textit{Coh}_{\leq k - 1}(X)$. Since

$\bigoplus \nolimits _{i \in I} \mathcal{O}_{Z_ i}^{\oplus n_ i} \cong \bigoplus \nolimits _{p \geq 0, i \in I, n_ i > p} \mathcal{O}_{Z_ i}$

this concludes the proof. $\square$

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