6.33 Glueing sheaves
In this section we glue sheaves defined on the members of a covering of $X$. We first deal with maps.
Lemma 6.33.1. Let $X$ be a topological space. Let $X = \bigcup U_ i$ be an open covering. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of sets on $X$. Given a collection
\[ \varphi _ i : \mathcal{F}|_{U_ i} \longrightarrow \mathcal{G}|_{U_ i} \]
of maps of sheaves such that for all $i, j \in I$ the maps $\varphi _ i, \varphi _ j$ restrict to the same map $\mathcal{F}|_{U_ i \cap U_ j} \to \mathcal{G}|_{U_ i \cap U_ j}$ then there exists a unique map of sheaves
\[ \varphi : \mathcal{F} \longrightarrow \mathcal{G} \]
whose restriction to each $U_ i$ agrees with $\varphi _ i$.
Proof.
For each open subset $U \subset X$ define
\[ \varphi _ U : \mathcal{F}(U) \to \mathcal{G}(U), \quad s \mapsto \varphi _ U(s) \]
where $\varphi _ U(s)$ is the unique section verifying
\[ (\varphi _ U(s))|_{U \cap U_ i} = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}). \]
Existence and uniqueness of such a section follows from the sheaf axioms due to the fact that
\begin{align*} ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{U \cap U_ i \cap U_ j} & = (\varphi _ i)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = (\varphi _ j)_{U \cap U_ i \cap U_ j}(s|_{U \cap U_ i \cap U_ j})\\ & = ((\varphi _ j)_{U \cap U_ j}(s|_{U \cap U_ j}))|_{U \cap U_ i \cap U_ j}. \end{align*}
This family of maps gives us indeed a map of sheaves: Let $V \subset U \subset X$ be open subsets then
\[ (\varphi _ U(s))|_ V = \varphi _ V(s|_ V) \]
since for each $i \in I$ the following holds
\begin{align*} (\varphi _ U(s))|_{V \cap U_ i} & = ((\varphi _ U(s))|_{U \cap U_ i})|_{V \cap U_ i}\\ & = ((\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i}))|_{V \cap U_ i}\\ & = (\varphi _ i)_{V \cap U_ i}(s|_{V \cap U_ i})\\ & = \varphi _ V(s_{V})|_{V \cap U_ i}. \end{align*}
Furthermore, its restriction to each $U_ i$ agrees with $\varphi _ i$ since given $U \subset X$ open subset and $s \in \mathcal{F}(U \cap U_ i)$ then
\begin{align*} \varphi _{U \cap U_ i}(s) & = \varphi _{U \cap U_ i}(s)|_{U \cap U_ i}\\ & = (\varphi _ i)_{U \cap U_ i}(s|_{U \cap U_ i})\\ & = (\varphi _ i)_{U \cap U_ i}(s). \end{align*}
$\square$
The previous lemma implies that given two sheaves $\mathcal{F}$, $\mathcal{G}$ on the topological space $X$ the rule
\[ U \longmapsto \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (U)}( \mathcal{F}|_ U, \mathcal{G}|_ U) \]
defines a sheaf. This is a kind of internal hom sheaf. It is seldom used in the setting of sheaves of sets, and more usually in the setting of sheaves of modules, see Modules, Section 17.22.
Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. For each $i \in I$ let $\mathcal{F}_ i$ be a sheaf of sets on $U_ i$. For each pair $i, j \in I$, let
\[ \varphi _{ij} : \mathcal{F}_ i|_{U_ i \cap U_ j} \longrightarrow \mathcal{F}_ j|_{U_ i \cap U_ j} \]
be an isomorphism of sheaves of sets. Assume in addition that for every triple of indices $i, j, k \in I$ the following diagram is commutative
\[ \xymatrix{ \mathcal{F}_ i|_{U_ i \cap U_ j \cap U_ k} \ar[rr]_{\varphi _{ik}} \ar[rd]_{\varphi _{ij}} & & \mathcal{F}_ k|_{U_ i \cap U_ j \cap U_ k} \\ & \mathcal{F}_ j|_{U_ i \cap U_ j \cap U_ k} \ar[ru]_{\varphi _{jk}} } \]
We will call such a collection of data $(\mathcal{F}_ i, \varphi _{ij})$ a glueing data for sheaves of sets with respect to the covering $X = \bigcup U_ i$.
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 some $i \in I$. 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 $X$.
$\square$
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.
$\square$
Lemma 6.33.4. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. The functor which associates to a sheaf of sets $\mathcal{F}$ the following collection of glueing data
\[ (\mathcal{F}|_{U_ i}, (\mathcal{F}|_{U_ i})|_{U_ i \cap U_ j} \to (\mathcal{F}|_{U_ j})|_{U_ i \cap U_ j} ) \]
with respect to the covering $X = \bigcup U_ i$ defines an equivalence of categories between $\mathop{\mathit{Sh}}\nolimits (X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules.
Proof.
The functor is fully faithful by Lemma 6.33.1 and essentially surjective (via an explicitly given quasi-inverse functor) by Lemma 6.33.2.
$\square$
This lemma means that if the sheaf $\mathcal{F}$ was constructed from the glueing data $(\mathcal{F}_ i, \varphi _{ij})$ and if $\mathcal{G}$ is a sheaf on $X$, then a morphism $f : \mathcal{F} \to \mathcal{G}$ is given by a collection of morphisms of sheaves
\[ f_ i : \mathcal{F}_ i \longrightarrow \mathcal{G}|_{U_ i} \]
compatible with the glueing maps $\varphi _{ij}$. Similarly, to give a morphism of sheaves $g : \mathcal{G} \to \mathcal{F}$ is the same as giving a collection of morphisms of sheaves
\[ g_ i : \mathcal{G}|_{U_ i} \longrightarrow \mathcal{F}_ i \]
compatible with the glueing maps $\varphi _{ij}$.
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