The Stacks project

Lemma 17.17.2. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology of $X$ and assume that each $U \in \mathcal{B}$ is quasi-compact. Then every sheaf of sets on $X$ is a filtered colimit of sheaves of the form

17.17.2.1
\begin{equation} \label{modules-equation-towards-constructible-sets} \text{Coequalizer}\left( \xymatrix{ \coprod \nolimits _{b = 1, \ldots , m} j_{V_ b!}\underline{S_ b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{a = 1, \ldots , n} j_{U_ a!}\underline{S_ a} } \right) \end{equation}

with $U_ a$ and $V_ b$ in $\mathcal{B}$ and $S_ a$ and $S_ b$ finite sets.

Proof. By Lemma 17.17.1 every sheaf of sets $\mathcal{F}$ is the target of a surjection whose source $\mathcal{F}_0$ is a coproduct of sheaves the form $j_{U!}\underline{S}$ with $U \in \mathcal{B}$ and $S$ finite. 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

\[ \xymatrix{ \coprod \nolimits _{b \in B} j_{V_ b!}\underline{S_ b} \ar@<1ex>[r] \ar@<-1ex>[r] & \coprod \nolimits _{a \in A} j_{U_ a!}\underline{S_ a} } \]

for some index sets $A$, $B$ and $V_ b$ and $U_ a$ in $\mathcal{B}$ and $S_ a$ and $S_ b$ finite. For every finite subset $B' \subset B$ there is a finite subset $A' \subset A$ such that the coproduct over $b \in B'$ maps into the coproduct over $a \in A'$ via both maps. Namely, we can view the right hand side as a filtered colimit with injective transition maps. Hence taking sections over the quasi-compact opens $V_ b$, $b \in B'$ commutes with this coproduct, see Sheaves, Lemma 6.29.1. Thus our sheaf is the colimit of the cokernels of these maps between finite coproducts. $\square$


Comments (2)

Comment #3642 by Brian Conrad on

Near the start of the proof, replace "source is a coprod " with "source is a coproduct", 2nd line after the displayed expression replace "coprod" with "coproduct", and on 3rd to last line replace with .


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