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  "body": "\n## Definition\n\nLet $S = \\bigoplus_{d \\ge 0} S_d$ be a commutative [[graded ring|8KB4SQGC0C]], and let $S_+ = \\bigoplus_{d \u003e 0} S_d$.\nAs a set, $\\operatorname{Proj} S$ is the set of [[homogeneous|6ZBNGS0P98]] [[prime ideals|00MHHZ9SNE]] $\\mathfrak{p}$ of $S$ with $S_+ \\not\\subseteq \\mathfrak{p}$.\nIts closed sets are the sets $V_+(\\mathfrak{a}) = \\set{\\mathfrak{p} \\in \\operatorname{Proj} S : \\mathfrak{a} \\subseteq \\mathfrak{p}}$ for homogeneous ideals $\\mathfrak{a}$.\nFor a homogeneous $f \\in S_+$, let $D_+(f) = \\set{\\mathfrak{p} \\in \\operatorname{Proj} S : f \\notin \\mathfrak{p}}$.\nLet $S_{(f)}$ be the ring of elements of degree $0$ in the [[localization|4C1AE84N5N]] $S_f$, that is, the fractions $a / f^n$ with $a$ homogeneous of degree $n \\deg f$.\n\nThe structure sheaf $\\mathcal{O}$ on $\\operatorname{Proj} S$ is the unique sheaf of rings such that, for every homogeneous $f \\in S_+$:\n\n1. $\\mathcal{O}(D_+(f)) = S_{(f)}$;\n2. $(D_+(f), \\mathcal{O}|_{D_+(f)})$ is isomorphic to the [[affine scheme|B3V3XJ2W8G]] $\\operatorname{Spec} S_{(f)}$, by $\\mathfrak{p} \\mapsto \\mathfrak{p} S_f \\cap S_{(f)}$.\n\nThe sets $D_+(f)$ cover $\\operatorname{Proj} S$, so $(\\operatorname{Proj} S, \\mathcal{O})$ is a [[scheme|5A35TYZD02]].\n\n## Remarks\n\nExample: for a ring $A$, projective $n$-space over $A$ is $\\mathbb{P}^n_A = \\operatorname{Proj} A[x_0, \\ldots, x_n]$, with $\\deg x_i = 1$.\nIt is covered by the $n+1$ affine pieces $D_+(x_i) \\cong \\operatorname{Spec} A[x_0/x_i, \\ldots, x_n/x_i]$.\nNon-example: for $S = k[x_0, \\ldots, x_n]$ over a field $k$, the ideal $S_+ = (x_0, \\ldots, x_n)$ is a homogeneous prime ideal.\nIt is not a point of $\\operatorname{Proj} S$, since it contains $S_+$; it would correspond to the excluded origin of $k^{n+1}$.\nFor a homogeneous polynomial $F$ in $x_0, \\ldots, x_n$, the scheme $\\operatorname{Proj} k[x_0, \\ldots, x_n]/(F)$ is the projective hypersurface $F = 0$.\n",
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