Lemma 18.30.7 (093C)—The Stacks project

Lemma 18.30.7. In Situation 18.30.5 assume (1) and (2) hold.

Every sheaf of sets is a filtered colimit of sheaves of the form

18.30.7.1
\begin{equation} \label{sites-modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{j = 1, \ldots , m} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i = 1, \ldots , n} h_{U_ i}^\# } \right) \end{equation}

with $U_ i$ and $V_ j$ in $\mathcal{B}$.
If $\mathcal{O}$ is a sheaf of rings, then every $\mathcal{O}$-module is a filtered colimit of sheaves of the form

18.30.7.2
\begin{equation} \label{sites-modules-equation-towards-constructible} \mathop{\mathrm{Coker}}\left( \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} \right) \end{equation}

with $U_ i$ and $V_ j$ in $\mathcal{B}$.

Proof. Proof of (1). By Lemma 18.30.6 every sheaf of sets $\mathcal{F}$ is the target of a surjection whose source is a coprod $\mathcal{F}_0$ of sheaves the form $h_{U}^\# $ with $U \in \mathcal{B}$. Applying this to $\mathcal{F}_0 \times _\mathcal {F} \mathcal{F}_0$ we find that $\mathcal{F}$ is a coequalizer of a pair of maps

\[ \xymatrix{ \coprod \nolimits _{j \in J} h_{V_ j}^\# \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{i \in I} h_{U_ i}^\# } \]

for some index sets $I$, $J$ and $V_ j$ and $U_ i$ in $\mathcal{B}$. For every finite subset $J' \subset J$ there is a finite subset $I' \subset I$ such that the coproduct over $j \in J'$ maps into the coprod over $i \in I'$ via both maps, see Sites, Lemma 7.17.7. (Details omitted; hint: an infinite coproduct is the filtered colimit of the finite sub-coproducts.) Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts.

Proof of (2). By Lemma 18.30.6 every module is a quotient of a direct sum of modules of the form $j_{U!}\mathcal{O}_ U$ with $U \in \mathcal{B}$. Thus every module is a cokernel

\[ \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j \in J} j_{V_ j!}\mathcal{O}_{V_ j} \longrightarrow \bigoplus \nolimits _{i \in I} j_{U_ i!}\mathcal{O}_{U_ i} \right) \]

for some index sets $I$, $J$ and $V_ j$ and $U_ i$ in $\mathcal{B}$. For every finite subset $J' \subset J$ there is a finite subset $I' \subset I$ such that the direct sum over $j \in J'$ maps into the direct sum over $i \in I'$, see Lemma 18.30.4. Thus our module is the colimit of the cokernels of these maps between finite direct sums. $\square$

The Stacks project

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