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  "body": "\n## Definition\n\nLet $(X, \\mathcal{O}_X)$ be a [[ringed space|KNDXDE7K8M]], that is, a topological space $X$ with a [[sheaf|TB6TM6R4MD]] $\\mathcal{O}_X$ of commutative rings on $X$.\nIt is a *locally ringed space* if for every point $x \\in X$ the [[stalk|JTAKAPTT8H]] $\\mathcal{O}_{X,x}$ is a [[local ring|MZ02BKPHJA]].\nThe maximal ideal of $\\mathcal{O}_{X,x}$ is written $\\mathfrak{m}_x$.\n\nLet $(X, \\mathcal{O}_X)$ and $(Y, \\mathcal{O}_Y)$ be locally ringed spaces.\nA *morphism of locally ringed spaces* $(X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ is a pair $(f, f^\\#)$ such that:\n\n1. $f \\colon X \\to Y$ is a [[continuous map|0VGFHTM75Y]].\n2. $f^\\# \\colon \\mathcal{O}_Y \\to f_* \\mathcal{O}_X$ is a morphism of sheaves of rings on $Y$.\n3. For every $x \\in X$, the induced ring map $f^\\#_x \\colon \\mathcal{O}_{Y, f(x)} \\to \\mathcal{O}_{X,x}$ is a [[local homomorphism|P8YTMGEN9R]], that is, $(f^\\#_x)^{-1}(\\mathfrak{m}_x) = \\mathfrak{m}_{f(x)}$.\n\n## Remarks\n\nExample: let $M$ be a smooth manifold and let $\\mathcal{O}_M(U)$ be the ring of smooth real functions on the open set $U$.\nA germ $g$ at $p$ is invertible in $\\mathcal{O}_{M,p}$ exactly when $g(p) \\ne 0$.\nSo the non-invertible germs form the ideal $\\set{g : g(p) = 0}$, and $\\mathcal{O}_{M,p}$ is local.\nNon-example: the one-point space with the ring $\\Z$ as its sheaf is a ringed space.\nIts only stalk $\\Z$ is not local, since $(2)$ and $(3)$ are two different maximal ideals.\nClause 3 is not automatic: a morphism of ringed spaces between locally ringed spaces need not satisfy it.\nIt is the clause that makes morphisms respect \"vanishing at a point\".\n",
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