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  "body": "\n## Statement\n\nLet $A$ and $B$ be commutative rings with identity.\n\n1. The natural map $A \\to \\mathcal{O}_{\\operatorname{Spec} A}(\\operatorname{Spec} A)$ is a ring isomorphism.\n2. The map that sends a [[ring homomorphism|0VTH8KPCG2]] $\\varphi \\colon A \\to B$ to the induced [[morphism of schemes|JXRKH6P1Z4]] $\\operatorname{Spec} B \\to \\operatorname{Spec} A$ is a bijection\n$$\\Hom_{\\mathrm{Ring}}(A, B) \\to \\Hom_{\\mathrm{Sch}}(\\operatorname{Spec} B, \\operatorname{Spec} A).$$\nIts inverse sends a morphism $(f, f^\\#)$ to its map on global sections $A \\to B$.\n\nConsequently, the [[functor|28NQ0KRHD3]] $\\operatorname{Spec}$ is an [[equivalence|K7A04ZBD92]] from the [[opposite category|DAMK8XE1FR]] of commutative rings to the category of [[affine schemes|B3V3XJ2W8G]].\nA quasi-inverse is $X \\mapsto \\mathcal{O}_X(X)$.\n\n## Remarks\n\nPart 1 is a gluing argument: a global section is given locally by fractions $a_i / f_i^{n_i}$ on basic open sets $D(f_i)$ that cover $\\operatorname{Spec} A$.\nSince the $f_i$ then generate the unit ideal, these fractions glue to a single element of $A$.\nFor part 2, the stalk maps of a morphism must be local.\nThis forces $f(\\mathfrak{q})$ to be the preimage of $\\mathfrak{q}$ under the map on global sections.\nThen the sheaf map is determined on each $D(g)$ by localizing.\nExample: for a field $k$, the morphisms $\\operatorname{Spec} k \\to \\operatorname{Spec} k[x]$ over $k$ correspond to $k$-algebra maps $k[x] \\to k$, that is, to the elements $a \\in k$, the $k$-points of the line.\nThe locality condition is needed.\nLet $R$ be a discrete valuation ring with fraction field $K$.\nSend the point of $\\operatorname{Spec} K$ to the closed point of $\\operatorname{Spec} R$, with the inclusion $R \\to K$ on sections.\nThis is a morphism of ringed spaces, but it is not the morphism induced by $R \\to K$, which hits the generic point.\n\n## Explanations\n\n### L1 human\n\nAffine schemes and commutative rings carry the same information, with arrows turned around.\nA map of spaces $\\operatorname{Spec} B \\to \\operatorname{Spec} A$ is the same thing as a map of rings $A \\to B$, which pulls functions back.\nSo every question about affine schemes can be translated into algebra, and every ring can be thought of as a space.\n\n### L2 human\n\nThis is the reason the [[spectrum of a ring|FB45BHHA2J]] is the right local model for algebraic geometry.\nIt extends to all schemes: for any scheme $X$, morphisms $X \\to \\operatorname{Spec} A$ correspond to ring maps $A \\to \\mathcal{O}_X(X)$.\nIt is used constantly.\nFor example, it shows that the fiber product of affine schemes is the spectrum of a tensor product.\nIt also lets one describe a scheme through its functor of points on rings.\n",
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