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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