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  "body": "\n## Definition\n\nLet $Y$ be a scheme and let $n \\ge 0$ be an integer.\nLet $\\mathbb{P}^n_{\\Z} = \\operatorname{Proj} \\Z[x_0, \\dots, x_n]$ be [[projective space|ZT9A6JK8QF]] over $\\Z$, given by the [[Proj construction|A39XQHCVNW]].\nLet $\\mathbb{P}^n_Y = \\mathbb{P}^n_{\\Z} \\times_{\\operatorname{Spec} \\Z} Y$ be the [[fiber product|1DZMT4S9FB]], with projection $p \\colon \\mathbb{P}^n_Y \\to Y$.\nA [[morphism of schemes|JXRKH6P1Z4]] $f \\colon X \\to Y$ is *projective* if $f = p \\circ i$ for some $n \\ge 0$ and some [[closed immersion|WWRWYSJDET]] $i \\colon X \\to \\mathbb{P}^n_Y$.\n\n## Remarks\n\nFor a field $k$ and a homogeneous polynomial $F \\in k[x_0, x_1, x_2]$ of positive degree, the curve $X = V(F) \\subseteq \\mathbb{P}^2_k$ is projective over $\\operatorname{Spec} k$.\nIndeed, $X \\to \\mathbb{P}^2_k$ is a closed immersion and $\\mathbb{P}^2_k \\to \\operatorname{Spec} k$ is the projection $p$.\nThe affine line $\\mathbb{A}^1_k \\to \\operatorname{Spec} k$ is not projective.\nA projective morphism is [[proper|9537RKBZPP]], and $\\mathbb{A}^1_k$ is not proper over $k$.\nThis is the definition in Hartshorne's textbook.\nIn EGA the definition is more general: $X$ is a closed subscheme of the projective bundle of a quasi-coherent $\\mathcal{O}_Y$-module of finite type.\nThe two definitions agree when $Y$ is affine.\n",
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