----- > [!proposition] Proposition. ([[scheme gluing]]) > Let $\{ X_{i} \}$ be a family of [[scheme|schemes]] (possibly infinite) and suppose for each $i \neq j$ we are given an [[subscheme|open subscheme]] $U_{ij} \subset X_{i}$. Suppose also given for each $i \neq j$ an [[morphism of locally ringed spaces|isomorphism]] of [[scheme|schemes]] $\varphi_{ij}:U_{ij} \to U_{ji}$, such that the [[cocycle conditions]] hold: > 1. For each $i,j$, $\varphi_{ij}=\varphi_{ji}^{-1}$; > 2. For each $i,j,k$, $\varphi_{ij}$ restricts to a map $U_{ij} \cap U_{ik} \xrightarrow{\sim} U_{ji} \cap U_{jk}$, i.e. $\varphi_{ij}(U_{ij} \cap U_{ik})=U_{ji} \cap U_{jk}$, and $\varphi_{ik}=\varphi_{jk} \circ \varphi_{ij}$ on $U_{ij} \cap U_{ik}$. > > Then there is a [[scheme]] $\widetilde{X}$, together with [[morphism of locally ringed spaces|morphisms]] $\psi_{i}:X_{i} \to \widetilde{X}$ for each $i$, such that > 1. (Open embeddings) $\psi_{i}$ is an [[isomorphism]] of $X_{i}$ with an [[subscheme|open subscheme]] of $\widetilde{X}$; > 2. (Covering) The $\psi_{i}(X_{i})$ [[cover]] $\widetilde{X}$; > 3. ($U_{ij}$s prescribe the overlaps) $\psi_{i}(U_{ij})=\psi_{i}(X_{i}) \cap \psi_{j}(X_{j})$; > 4. (Transitions) $\psi_{i}=\psi_{j} \circ \varphi_{ij}$ on $U_{ij}$. > > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQANEAX1PU1z5CKcqWLU6TVuwCqAfWBYAVtx58QGbHgJEATGIkMWbRCHnAlWVdwkwoAc3hFQAMwBOEALZIyIHBCR9SWN2AB1Qhjc0AAssBWVVGkZ6ACMYRgAFAW1hEDcse2icNVcPb0RffyRRYOlTcLRsOSwSkHcvQJoqxBqjOpAGpqUQJNT0rK0hdnzC4ptuIA > \begin{tikzcd} > & X & \\ > U_{ij} \arrow[rr, "\varphi_{ij}"'] \arrow[ru, "\psi_i"] & & U_{ji} \arrow[lu, "\psi_j"'] > \end{tikzcd} > \end{document} > ``` > > [!proof]- Proof. ([[scheme gluing]]) > ~ ----- #### ![[CleanShot 2025-02-17 at [email protected]]] See also: [[sheaf gluing]]. **Topological considerations.** First we examine what $X$ should look like as a [[topological space]], and what the $\psi_{i}$ should be as [[homeomorphism|homeomorphisms]] of the $X_{i}$ onto their images. The intuitive thing to do is to put $\widetilde{X}:= \frac{\coprod_{i} X_{i}}{\sim}$ for the [[equivalence relation]] $\sim$ generated by[^2] $(i,x_{i} \in X_{i}) \sim (j, x_{j} \in X_{j}) \iff x_{i} \in U_{ij} \text{ and } \varphi_{ij}(x_{i})=x_{j}.$ Then $\psi_{i}:X_{i} \to X$ will be defined as the composition $X_{i} \hookrightarrow \coprod_{i}X_{i} \twoheadrightarrow \widetilde{X}$ of the canonical injection and projection. $\widetilde{X}:=\im \psi_{i}$ consists of classes of the form $\widetilde{X}:=\im \psi_{i}=\{ [(i, x_{i})]: x_{i} \in X_{i} \}.$ There is an inverse $\phi_{i}: \widetilde{X}_{i} \to X_{i}$ given by $\phi_{i}([\ell, x_{\ell}]):=\varphi_{\ell i}(x_{\ell}) = x_{i}$, where $\varphi_{ii}$ is understood to be the [[identity map]]: $\phi([(i, x_{i})])=x_{i}$. Note that $\phi_{i}$ is [[continuous]] because $\psi_{i}$ is an [[open map]].[^1] So $\psi_{i}$ is indeed a [[homeomorphism]] onto its image. At this stage, we can show that condition (4) holds for the topological maps involved (later we will show it also holds for the [[morphism of (pre)sheaves|sheaf morphisms]] involved): letting $u_{ij} \in U_{ij}$, we have $\begin{align} \psi_{j} \circ \varphi_{ij}(u_{ij}) & = [j, \varphi_{ij}(u_{ij})] \\ &= [i, u_{ij}] \\ & = \psi_{i}(u_{ij}). \end{align}$ As the conditions (2) and (3) that we are required to show are purely [[topological space|topological]] questions concerning the [[continuous]] maps $\psi_{i}$ we can verify them now too. (2) is clear because any $\widetilde{x} \in \widetilde{X}$ lifts under the projection (a [[surjection]]) to $(i, x_{i})$ for some $i$ and $x_{i} \in X_{i}$, and thus $\psi(x_{i})=\widetilde{x}$. For (3), the inclusion $\psi_{i}(U_{ij})\subset \widetilde{X}_{i} \cap \widetilde{X}_{j}$ is immediate from the fact that $\psi_{i}=\psi_{j} \circ \varphi_{ij}$ as topological maps. For the reverse inclusion, if $\widetilde{x} \in \widetilde{X}_{j} \cap \widetilde{X}_{j}$, then $\widetilde{x}=[i, x_{i}]$ for some $x_{i} \in X_{i}$ but also $\widetilde{x}=[j, x_{j}]$ for some $x_{j} \in X_{j}$, meaning that $(i, x_{i}) \sim (j, x_{j})$ and in particular that $x_{i} \in U_{ij}$ from the very definition of $\sim$. Hence $\widetilde{x}=\psi_{i}(x_{i}) \in \psi(U_{ij})$. **Schematic considerations.** Recall the notation $\widetilde{X}_{i}:=\psi_{i}(X_{i})$. How should the [[scheme|structure sheaf]] $\mathcal{O}_{\widetilde{X}}$ be defined (and in turn the sheaf morphisms $\psi_{i}^{\sharp}$)? Each $\psi_{i}$ [[pushforward sheaf|pushes]] $\mathcal{O}_{X_{i}}$ forward to a [[sheaf]] on $\widetilde{X}$, yielding a sheaf $\mathcal{O}_{\widetilde{X}_{i}}:=(\psi_{i})_{*}(\mathcal{O}_{X_{i}})$, $\mathcal{O}_{\widetilde{X}_{i}}(V)=\mathcal{O}_{X_{i}}\big( \psi ^{-1}(V) \big)$ for $V \subset X$ open. Intuitively, a section $\boldsymbol s \in \mathcal{O}_{\widetilde{X}}(V)$ should be built out of sections $s_{i} \in \mathcal{O}_{X_{i}}(V)$, $i \in I$, subject to conditions related to compatible overlap. We define $\mathcal{O}_{\widetilde{X}}(V):= \left\{ \boldsymbol s=(s_{i})_{i \in I} \in \prod_{i \in I} (\psi_{i})_{*} \mathcal{O}_{X_{i}}(V) : \textcolor{thistle}{ \varphi_{ij}^{\sharp}(s_{i})=s_{j}} \right\}$ (see ipad camera roll, maybe not quite precise enough, but very nearly correct) with the obvious choice of restriction maps: $\rho_{VW}^{\widetilde{X}}\big( (s_{i})_{i \in I} \big):=\big( \rho_{VW}^{\widetilde{X}_{i}}(s_{i}) \big)_{i \in I}$ for $W \subset V \subset X$. I believe that if we omitted the $\textcolor{thistle}{\text{thistle-colored}}$ condition, $\mathcal{O}_{\widetilde{X}}$ would still be a [[sheaf]]. Said condition *is* certainly needed to guarantee that $\psi_{i}^{\sharp}=(\psi_{j} \circ \varphi_{ij})^{\sharp}$ (so that requirement $(4)$ holds). Let us check these things now. **Locality.** Suppose $\{ V_{m} \}$ is an [[cover|open cover]] of $V$. Then $(s_{i})_{i \in I} |_{V_{m}}=0$ for all $m$ iff $(s_{i} |_{V_{m}})_{i \in I}=0$ for all $m$, iff $s_{i} |_{V_{m}}=0$ for all $m$, which happens iff $s_{i}=0$ for all $m$ since $(\psi_{i})_{*}\mathcal{O}_{X_{i}}$ is a [[sheaf]]. **Gluing.** Suppose $\{ V_{m} \}$ is an [[cover|open cover]] of $V$, and that we are given a family of local sections $\boldsymbol s ^{m} \in (\psi_{i})_{*}\mathcal{O}_{X_{i}}(V)$ such that $\boldsymbol s ^{m} |_{V_{m} \cap V_{m'}}=\boldsymbol s^{m'} |_{V_{m} \cap V_{m'}}$ for all $m,m'$. That is, $(s_{i}^{m} |_{V_{m} \cap V_{m'}})_{i \in I}=(s_{i}^{m'} |_{V_{m} \cap V_{m'}})_{i \in I}$ for all $m,m'$. Since each $(\psi_{i})_{*}\mathcal{O}_{X_{i}}$ is a [[sheaf]], these glue into some $\boldsymbol s=(s_{i})_{i \in I} \in \mathcal{O}_{\widetilde{X}}(V)$ as desired. We have $\varphi_{ij}^{\sharp}(s_{i})=s_{j}$ because $\varphi_{ij}^{\sharp}(s_{i}) |_{V_{m}}=\varphi_{ij}^{\sharp}(s_{i} |_{V_{m}})=\varphi_{ij}^{\sharp}(s_{i}^{m})=s_{j}^{m}=s_{j} |_{V_{m}}$ for all $m$; from locality the result then follows. We next show $(4)$. We already did the topological part, just need the sheaf morphism part, here it is, sloppily handwritten: ![[CleanShot 2025-02-17 at [email protected]]] Finally, we must show that $\widetilde{X}$ is a [[scheme]] and that $\psi_{i}$ identifies $X$ with an [[subscheme|open subscheme]] of $\widetilde{X}$. This should follow from the fact that it is covered by the $\psi_{i}(X_{i})$, where $X_{i}=\{ U_{\alpha} \}$ for open affines $U_{\alpha}$, and hence the $\psi_{i}(U_{\alpha})$ give us a covering of $\widetilde{X}$ by [[affine scheme|open affines]]. ---- [^1]: $\psi_{i}$ is a composition of the canonical injection and projection, which are both open maps. [^2]: Note that the same [[equivalence relation]] is given by $(i,x_{i} \in X_{i}) \sim (j, x_{j} \in X_{j}) \iff x_{i} \in U_{i} \text{ and } x_{i}=\varphi_{ji}(x_{j}),$ we could have also used this definition of $\sim$. #### 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 > ```