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 ( 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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