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  "body": "\n## Summary\n\nGluing builds a new [[scheme|5A35TYZD02]] from a family of schemes by identifying open pieces of them along isomorphisms.\nThe isomorphisms must agree on triple overlaps, a condition called the cocycle condition.\nGluing is how most schemes beyond affine ones are made, for example projective space.\nIt is also used to prove that constructions such as fiber products exist for all schemes.\n\n## Explanations\n\n### L1 human\n\nImagine making a sphere from two discs by sewing them along their edges.\nGluing of schemes does the same with algebraic pieces: one says which open parts of the pieces are to be identified, and by which maps.\nThe projective line comes from two copies of the affine line, with coordinates $t$ and $s$.\nThe parts where $t \\ne 0$ and $s \\ne 0$ are identified by the rule $s = 1/t$.\nThe point $t = 0$ and the point $s = 0$ stay separate, so the result is a line with one extra point at infinity.\n",
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