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  "body": "\n## Definition\n\nLet $I$ be an index set, and for each $i \\in I$ let $X_i$ be a [[scheme|5A35TYZD02]].\nFor all $i, j \\in I$, let $U_{ij} \\subseteq X_i$ be an open subscheme with $U_{ii} = X_i$.\nLet $\\varphi_{ij} \\colon U_{ij} \\to U_{ji}$ be isomorphisms of schemes such that:\n\n1. $\\varphi_{ii}$ is the identity of $X_i$ for every $i$.\n2. $\\varphi_{ij}(U_{ij} \\cap U_{ik}) = U_{ji} \\cap U_{jk}$ for all $i, j, k$.\n3. $\\varphi_{ik} = \\varphi_{jk} \\circ \\varphi_{ij}$ on $U_{ij} \\cap U_{ik}$ for all $i, j, k$ (the cocycle condition).\n\nA *gluing* of these data is a scheme $X$ with [[open immersions|GFC5YKSN5H]] $\\psi_i \\colon X_i \\to X$ such that:\n\n1. the open sets $\\psi_i(X_i)$ cover $X$;\n2. $\\psi_i(U_{ij}) = \\psi_i(X_i) \\cap \\psi_j(X_j)$ for all $i, j$;\n3. $\\psi_i = \\psi_j \\circ \\varphi_{ij}$ on $U_{ij}$ for all $i, j$.\n\nSuch a gluing always exists, and it is unique up to a unique isomorphism compatible with the maps $\\psi_i$.\n\n## Remarks\n\nExample: let $k$ be a field, $X_1 = \\operatorname{Spec} k[t]$, $X_2 = \\operatorname{Spec} k[s]$, $U_{12} = D(t)$ and $U_{21} = D(s)$.\nLet $\\varphi_{12}$ be the isomorphism given by the ring map $k[s, s^{-1}] \\to k[t, t^{-1}]$, $s \\mapsto t^{-1}$, and $\\varphi_{21} = \\varphi_{12}^{-1}$.\nWith only two indices the cocycle condition holds, and the gluing is the projective line $\\mathbb{P}^1_k$.\nIf 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.\nNon-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 $2 \\ne 0$.\nThe composite $\\varphi_{21} \\circ \\varphi_{12}$ is given by $t \\mapsto 2t$, so it is not $\\varphi_{11} = \\id$, and clause 3 fails for $i = k = 1$, $j = 2$.\n",
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