----- > [!proposition] Proposition. ([[the gluing lemma for sheaf morphisms]]) > Let $X$ be a [[topological space]] and let $\mathcal{F}$, $\mathcal{G}$ be two [[sheaf|sheaves]] on $X$. Suppose that $\{ U_{i} \}_{i \in I}$ is an [[cover|open cover]] of $X$, and that we are given [[morphism of (pre)sheaves|sheaf morphisms]] $f^{i}:\mathcal{F} |_{U_{i}} \to \mathcal{G} |_{U_{i}}$ for each $i \in I$, such that $f^{i}$ and $f^{j}$ agree on $U_{i} \cap U_{j}$ for all $i,j\in I$. Then the $\{ f^{i} \}$ 'glue' into a unique global morphism $f:\mathcal{F} \to \mathcal{G}$ satisfying $f |_{U_{i}}=f^{i}$ for all $i$. > > Moreover, if $p \in U_{i}$ then the [[(pre)sheaf stalk|stalk map]] $f_{p}:\mathcal{F}_{p} \to \mathcal{G}_p$ is merely $f^{i}_{p}:(\mathcal{F} |_{U_{i}})_{p} \to (\mathcal{G} |_{U_{i}})_{p}$. ^proposition > [!proposition] Corollary. (Gluing morphisms of locally ringed spaces) > Let $(X, \mathcal{O}_{X})$ and $(Y, \mathcal{O}_{Y})$ be [[locally ringed space|locally ringed spaces]], and suppose there is an [[cover|open cover]] $\{ U_{i} \}$ of $X$ and [[morphism of locally ringed spaces|morphisms]] $(f_{i}, f_{i}^{\sharp}):(U_{i}, \mathcal{O}_{X} |_{U_{i}}) \to (Y,\mathcal{O}_{Y})$ which agree on all pairwise domain overlaps $U_{i}\cap U_{j}$. Then the $(f_{i}, f_{i}^{\sharp})$ glue together to give a single [[morphism of locally ringed spaces|morphism]] $(f_{}, f_{}^{\sharp}):(X, \mathcal{O}_{X}) \to (Y, \mathcal{O}_{Y})$ agreeing with the $(f_{i}, f_{i}^{\sharp})$ on the relevant restrictions to the $U_{i}$. > >*In particular, we have a gluing lemma for [[scheme|schemes]].* > ^proposition > [!proof] Proof of Corollary. > The [[continuous|topological maps]] $f_{i}$ glue, by [[the gluing lemma]] for [[topological space|topological spaces]]. And as [[morphism of (pre)sheaves|sheaf morphisms]] the $f_{i}^{\sharp}$ glue into a [[morphism of (pre)sheaves|sheaf morphism]] $f$. Thus, the $(f_{i}, f_{i}^{\sharp})$ glue as [[morphism of ringed spaces|morphisms of ringed spaces]]. > It just needs to be checked that $f$ is in fact a [[morphism of locally ringed spaces]], i.e., that we additionally have that the induced [[ring homomorphism]] on [[(pre)sheaf stalk|stalks]] $f_{p}^{\sharp}:\mathcal{O}_{Y}, f(p) \to \mathcal{O}_{X}, p$ is a [[homomorphism of local rings|local homomorphism]] for all $p$. ^proof > [!proof]+ Proof. ([[the gluing lemma for sheaf morphisms]]) > > Let $U \subset X$ be open. It is [[cover|openly covered]], by the relevant (portions of) $U_{i}$: $U=\bigcup_{i} U \cap U_{i}$. For simplicity of notation, we will again write $\{ U_{i} \}$ for the cover of $U$. > Given $s \in \mathcal{F}(U)$, how should $f_{U}(s)$ be defined? > *Claim: the sections $f^{i}_{U_{i}}(s |_{U_{i}}) \in \mathcal{G}(U_{i})$ agree on overlaps.* Indeed, using [[natural transformation|naturality]] of each $f^{i}$ as a [[morphism of (pre)sheaves|(pre)sheaf morphism]] and the assumption that the $f^{i}$ agree on overlaps, we have $f_{U_{i}}^{i}(s |_{U_{i}}) |_{U_{i} \cap U_{j}}= f^{i}_{U_{i} \cap U_{j}}(s |_{U_{i} \cap U_{j}})=f^{j}_{U_{i} \cap U_{j}}(s |_{U_{i} \cap U_{j}})=f_{U_{j}}^{j}(s |_{U_{j}}) |_{U_{i} \cap U_{j}}.$Then by the sheaf gluing axiom, there is a unique section $t \in \mathcal{G}(U)$ such that $t |_{U_{i}}=f_{U_{i}}^{i}(s |_{U_{i}})$ for all $i$. We define $f_{U}(s):= t$. > Now, on component $V \subset U_{i}$, $f |_{U_{i}}:\mathcal{F}|_{U_{i}} \to \mathcal{G} |_{U_{i}}$ is given by $(f |_{U_{i}})_{V}(s) =f_{V}(s)$ > Finish by checking the claim regarding [[(pre)sheaf stalk|stalks]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```