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  "body": "\n## Definition\n\nLet $X$ and $Y$ be [[schemes|5A35TYZD02]].\nA *morphism of schemes* $X \\to Y$ is a morphism of [[locally ringed spaces|B7FPT8JTXR]], that 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, so for each open $V \\subseteq Y$ it gives a ring map $\\mathcal{O}_Y(V) \\to \\mathcal{O}_X(f^{-1}(V))$, compatible with restrictions.\n3. For every $x \\in X$, the induced map of stalks $f^\\#_x \\colon \\mathcal{O}_{Y,f(x)} \\to \\mathcal{O}_{X,x}$ is a [[local homomorphism|P8YTMGEN9R]].\n\nThe composite of $(f, f^\\#) \\colon X \\to Y$ and $(g, g^\\#) \\colon Y \\to Z$ is $(g \\circ f, g_*(f^\\#) \\circ g^\\#)$.\n\n## Remarks\n\nExample: a ring homomorphism $\\varphi \\colon A \\to B$ gives a morphism $\\operatorname{Spec} B \\to \\operatorname{Spec} A$ with $f(\\mathfrak{q}) = \\varphi^{-1}(\\mathfrak{q})$.\nOn stalks it is the map $A_{\\varphi^{-1}(\\mathfrak q)} \\to B_{\\mathfrak q}$, which sends the maximal ideal into $\\mathfrak{q} B_{\\mathfrak q}$, so it is local.\nNon-example: let $R$ be a discrete valuation ring with fraction field $K$.\nLet $f$ send the single point of $\\operatorname{Spec} K$ to the closed point of $\\operatorname{Spec} R$, and let $f^\\#$ come from the inclusion $R \\to K$.\nThis is a morphism of ringed spaces, but it fails clause 3.\nThe stalk map $R \\to K$ is not local, since it sends the maximal ideal of $R$ into the units of $K$.\n",
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