Lemma 18.30.9. In Situation 18.30.5 assume (1), (2), and (3) hold. Let $\mathcal{O}$ be a sheaf of rings. Assume given a map

$\bigoplus \nolimits _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i}$

with $U_ i$ and $V_ j$ in $\mathcal{B}$, and coverings $\{ U_{ik} \to U_ i\} _{k \in K_ i}$ with $U_{ik} \in \mathcal{B}$. Then there exist finite subsets $K'_ i \subset K_ i$ and a finite set $L$ of $W_ l \in \mathcal{B}$ and a commutative diagram

$\xymatrix{ \bigoplus _{l \in L} j_{W_ l!}\mathcal{O}_{W_ l} \ar[d] \ar[r] & \bigoplus _{i = 1, \ldots , n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} }$

inducing an isomorphism on cokernels of the horizontal maps.

Proof. Since $U_ i$ is quasi-compact, we may choose finite subsets $K'_ i \subset K_ i$ as in Lemma 18.30.2. Then since $\bigoplus _{i = 1, \ldots , n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \to \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i}$ is surjective, we can find coverings $\{ V_{jm} \to V_ j\} _{m \in M_ j}$ with $V_{jm} \in \mathcal{B}$ such that we can find a commutative diagram

$\xymatrix{ \bigoplus _{j = 1, \ldots , m} \bigoplus _{m \in M_ j} j_{V_{jm}!}\mathcal{O}_{V_{jm}} \ar[d] \ar[r] & \bigoplus _{i = 1, \ldots n} \bigoplus _{k \in K'_ i} j_{U_{ik}!}\mathcal{O}_{U_{ik}} \ar[d] \\ \bigoplus _{j = 1, \ldots , m} j_{V_ j!}\mathcal{O}_{V_ j} \ar[r] & \bigoplus _{i = 1, \ldots , n} j_{U_ i!}\mathcal{O}_{U_ i} }$

Since $V_ j$ is quasi-compact, we can choose finite subsets $M'_ j \subset M_ j$ as in Lemma 18.30.2. Set

$L = \left(\coprod \nolimits _{i = 1, \ldots , n} K'_ i \times K'_ i \right) \coprod \left(\coprod \nolimits _{j = 1, \ldots , m} M'_ j\right)$

and for $l = (k, k') \in K'_ i \times K'_ i \subset L$ set $W_ l = U_{ik} \times _{U_ i} U_{ik'}$ and for $l = m \in M'_ j \subset L$ set $W_ l = V_{jm}$. Since we have the exact sequences of Lemma 18.30.2 for the families $\{ U_{ik} \to U_ i\} _{k \in K'_ i}$ we conclude that we get a diagram as in the statement of the lemma (details omitted), except that it is not yet clear that $W_ l \in \mathcal{B}$. However, since $W_ l$ is quasi-compact for all $l \in L$ we do another application of Lemma 18.30.2 and find finite families of maps $\{ W_{lt} \to W_ l\} _{t \in T_ l}$ with $W_{lt} \in \mathcal{B}$ such that $\bigoplus j_{W_{lt}!}\mathcal{O}_{W_{lt}} \to j_{W_ l!}\mathcal{O}_{W_ l}$ is surjective. Then we replace $L$ by $\coprod _{l \in L} T_ l$ and everything is clear. $\square$

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