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  "body": "\n## Summary\n\nThe Proj construction makes a [[scheme|5A35TYZD02]] $\\operatorname{Proj} S$ out of a [[graded ring|8KB4SQGC0C]] $S$.\nIts points are the homogeneous prime ideals that do not contain all elements of positive degree.\nIt plays for projective geometry the role that $\\operatorname{Spec}$ plays for affine geometry: $\\operatorname{Proj}$ of a polynomial ring is projective space.\n\n## Explanations\n\n### L1 human\n\nA point of projective space is given by coordinates $[x_0 : \\cdots : x_n]$ that matter only up to a common factor.\nSo only homogeneous polynomials, whose zero sets do not depend on the factor, make sense on projective space.\nThe Proj construction turns this into algebra: it uses the homogeneous prime ideals of a graded ring.\nIt is covered by affine pieces, one for each homogeneous element $f$ of positive degree, on which $f \\ne 0$.\nFor $S = k[x_0, \\ldots, x_n]$ and $f = x_0$, that piece is the affine space with coordinates $x_1/x_0, \\ldots, x_n/x_0$.\n",
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