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The Stacks project

Lemma 18.30.11. In Situation 18.30.5 assume (1), (2), and (3) hold. Let \mathcal{O} be a sheaf of rings. Let \mathcal{A} \subset \textit{Mod}(\mathcal{O}) be the full subcategory of modules isomorphic to a cokernel as in (18.30.7.2). If the kernel of every map of \mathcal{O}-modules of the form

\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}, is in \mathcal{A}, then \mathcal{A} is weak Serre subcategory of \textit{Mod}(\mathcal{O}).

Proof. We will use the criterion of Homology, Lemma 12.10.3. By the results of Lemmas 18.30.8 and 18.30.10 it suffices to see that the kernel of a map \mathcal{F} \to \mathcal{G} between objects of \mathcal{A} is in \mathcal{A}. To prove this choose presentations

\bigoplus A_{V_ j} \to \bigoplus A_{U_ i} \to \mathcal{F} \to 0 \quad \text{and}\quad \bigoplus A_{T_ j} \to \bigoplus A_{W_ i} \to \mathcal{G} \to 0

In this proof the direct sums are always finite, and we write A_ U = j_{U!}\mathcal{O}_ U for U \in \mathcal{B}. Using Lemmas 18.30.1 and 18.30.9 and arguing as in the proof of Lemma 18.30.10 we may assume that the map \mathcal{F} \to \mathcal{G} lifts to a map of presentations

\xymatrix{ \bigoplus A_{V_ j} \ar[r] \ar[d] & \bigoplus A_{U_ i} \ar[r] \ar[d] & \mathcal{F} \ar[r] \ar[d] & 0 \\ \bigoplus A_{T_ j} \ar[r] & \bigoplus A_{W_ i} \ar[r] & \mathcal{G} \ar[r] & 0 }

Then we see that

\mathop{\mathrm{Ker}}(\mathcal{F} \to \mathcal{G}) = \mathop{\mathrm{Coker}}\left(\bigoplus A_{V_ j} \to \mathop{\mathrm{Ker}}\left( \bigoplus A_{T_ j} \oplus \bigoplus A_{U_ i} \to \bigoplus A_{W_ i}\right)\right)

and the lemma follows from the assumption and Lemma 18.30.8. \square


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