The Stacks project

Lemma 6.33.3. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $(\mathcal{F}_ i, \varphi _{ij})$ be a glueing data of sheaves of abelian groups, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules for some sheaf of rings $\mathcal{O}$ on $X$. Then the construction in the proof of Lemma 6.33.2 above leads to a sheaf of abelian groups, resp. sheaf of algebraic structures, resp. sheaf of $\mathcal{O}$-modules.

Proof. This is true because in the construction the set of sections $\mathcal{F}(W)$ over an open $W$ is given as the equalizer of the maps

\[ \xymatrix{ \prod _{i \in I} \mathcal{F}_ i(W \cap U_ i) \ar@<1ex>[r] \ar@<-1ex>[r] & \prod _{i, j\in I} \mathcal{F}_ i(W \cap U_ i \cap U_ j) } \]

And in each of the cases envisioned this equalizer gives an object in the relevant category whose underlying set is the object considered in the cited lemma. $\square$


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