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  "body": "\n## Definition\n\nLet $\\mathrm{Sch}$ be the category of [[schemes|5A35TYZD02]] and [[morphisms of schemes|JXRKH6P1Z4]], and let $X$ be a scheme.\nThe *functor of points* of $X$ is the [[functor|28NQ0KRHD3]]\n$$h_X \\colon \\mathrm{Sch}^{\\mathrm{op}} \\to \\mathrm{Set}, \\quad h_X(T) = \\Hom_{\\mathrm{Sch}}(T, X),$$\non the [[opposite category|DAMK8XE1FR]] of $\\mathrm{Sch}$.\nFor a morphism $u \\colon T' \\to T$, the map $h_X(u)$ sends $a \\colon T \\to X$ to $a \\circ u$.\nAn element of $h_X(T)$ is a *$T$-point* of $X$; for a ring $A$, one writes $X(A) = h_X(\\operatorname{Spec} A)$.\n\n## Remarks\n\nBy the [[Yoneda lemma|D6DF1ARPVD]], the rule $X \\mapsto h_X$ is fully faithful, so a scheme is determined up to unique isomorphism by its functor of points.\nIt suffices to know $X(A)$ for all commutative rings $A$.\nExample: for $X = \\mathbb{A}^n_\\Z = \\operatorname{Spec} \\Z[x_1, \\ldots, x_n]$, a morphism $\\operatorname{Spec} A \\to X$ is a ring map $\\Z[x_1, \\ldots, x_n] \\to A$.\nSuch a map is given by the images of the $x_i$, so $X(A) = A^n$.\nNon-example: the constant functor $F$ with $F(T) = \\set{0, 1}$ for all $T$ is not of the form $h_X$.\nFor the empty scheme $\\emptyset$, the set $h_X(\\emptyset)$ has exactly one element for every $X$, while $F(\\emptyset)$ has two.\nA functor isomorphic to some $h_X$ is called [[representable|KCCHH87H6E]], and $X$ then represents it.\n",
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