Gluing of schemes

Builds on Scheme Morphism of schemes Open immersion

Definition

Let $I$ be an index set, and for each $i \in I$ let $X_i$ be a scheme. For all $i, j \in I$ , let $U_{ij} \subseteq X_i$ be an open subscheme with $U_{ii} = X_i$ . Let $\varphi_{ij} \colon U_{ij} \to U_{ji}$ be isomorphisms of schemes such that:

$\varphi_{ii}$ is the identity of $X_i$ for every $i$ .
$\varphi_{ij}(U_{ij} \cap U_{ik}) = U_{ji} \cap U_{jk}$ for all $i, j, k$ .
$\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$ on $U_{ij} \cap U_{ik}$ for all $i, j, k$ (the cocycle condition).

A gluing of these data is a scheme $X$ with open immersions $\psi_i \colon X_i \to X$ such that:

the open sets $\psi_i(X_i)$ cover $X$ ;
$\psi_i(U_{ij}) = \psi_i(X_i) \cap \psi_j(X_j)$ for all $i, j$ ;
$\psi_i = \psi_j \circ \varphi_{ij}$ on $U_{ij}$ for all $i, j$ .

Such a gluing always exists, and it is unique up to a unique isomorphism compatible with the maps $\psi_i$ .

Remarks

Example: let $k$ be a field, $X_1 = \operatorname{Spec} k[t]$ , $X_2 = \operatorname{Spec} k[s]$ , $U_{12} = D(t)$ and $U_{21} = D(s)$ . Let $\varphi_{12}$ be the isomorphism given by the ring map $k[s, s^{-1}] \to k[t, t^{-1}]$ , $s \mapsto t^{-1}$ , and $\varphi_{21} = \varphi_{12}^{-1}$ . With only two indices the cocycle condition holds, and the gluing is the projective line $\mathbb{P}^1_k$ . If instead $\varphi_{12}$ is given by $s \mapsto t$ on $D(t)$ , the gluing is the affine line with a doubled origin, which is a scheme but not a separated one. Non-example: with $\varphi_{12}$ as in the first example, take $\varphi_{21}$ to be given by $t \mapsto 2s^{-1}$ over a field $k$ with $2 \ne 0$ . The composite $\varphi_{21} \circ \varphi_{12}$ is given by $t \mapsto 2t$ , so it is not $\varphi_{11} = \id$ , and clause 3 fails for $i = k = 1$ , $j = 2$ .

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Morphism of schemes concept
Open immersion concept
Scheme concept
Gluing of schemes definition

Tag	`HWGM5P7KCK`
URI	`m2://HWGM5P7KCK`
Kind	definition
Writing	W0 stub
MSC	14A15
Defines	Gluing of schemes
Uses	Scheme, Morphism of schemes, Open immersion
Changed	2 November 2026

Gluing of schemes

Definition

Remarks

Reading path

Further reading

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