The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 6.33.2. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. Given any glueing data $(\mathcal{F}_ i, \varphi _{ij})$ for sheaves of sets with respect to the covering $X = \bigcup U_ i$ there exists a sheaf of sets $\mathcal{F}$ on $X$ together with isomorphisms

\[ \varphi _ i : \mathcal{F}|_{U_ i} \to \mathcal{F}_ i \]

such that the diagrams

\[ \xymatrix{ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]_{\varphi _ i} \ar[d]_{\text{id}} & \mathcal{F}_ i|_{U_ i \cap U_ j} \ar[d]^{\varphi _{ij}} \\ \mathcal{F}|_{U_ i \cap U_ j} \ar[r]^{\varphi _ j} & \mathcal{F}_ j|_{U_ i \cap U_ j} } \]

are commutative.

Proof. First proof. In this proof we give a formula for the set of sections of $\mathcal{F}$ over an open $W \subset X$. Namely, we define

\[ \mathcal{F}(W) = \{ (s_ i)_{i \in I} \mid s_ i \in \mathcal{F}_ i(W \cap U_ i), \varphi _{ij}(s_ i|_{W \cap U_ i \cap U_ j}) = s_ j|_{W \cap U_ i \cap U_ j} \} . \]

Restriction mappings for $W' \subset W$ are defined by the restricting each of the $s_ i$ to $W' \cap U_ i$. The sheaf condition for $\mathcal{F}$ follows immediately from the sheaf condition for each of the $\mathcal{F}_ i$.

We still have to prove that $\mathcal{F}|_{U_ i}$ maps isomorphically to $\mathcal{F}_ i$. Let $W \subset U_ i$. In this case the condition in the definition of $\mathcal{F}(W)$ implies that $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$. And the commutativity of the diagrams in the definition of a glueing data assures that we may start with any section $s \in \mathcal{F}_ i(W)$ and obtain a compatible collection by setting $s_ i = s$ and $s_ j = \varphi _{ij}(s_ i|_{W \cap U_ j})$.

Second proof (sketch). Let $\mathcal{B}$ be the set of opens $U \subset X$ such that $U \subset U_ i$ for somje $i \in I$. Then $\mathcal{B}$ is a base for the topology on $X$. For $U \in \mathcal{B}$ we pick $i \in I$ with $U \subset U_ i$ and we set $\mathcal{F}(U) = \mathcal{F}_ i(U)$. Using the isomorphisms $\varphi _{ij}$ we see that this prescription is “independent of the choice of $i$”. Using the restriction mappings of $\mathcal{F}_ i$ we find that $\mathcal{F}$ is a sheaf on $\mathcal{B}$. Finally, use Lemma 6.30.6 to extend $\mathcal{F}$ to a unique sheaf $\mathcal{F}$ on $X$. $\square$


Comments (2)

Comment #3423 by Samir Canning on

Maybe a slightly shorter proof can be given as follows: define a base for the topology on consisting of all the open sets of each and then define on the base in the obvious way. Then just use tag 009N to get a (unique) sheaf on .


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