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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]].\nFor a point $x \\in X$, let $y = f(x)$, and let $f^\\# \\colon \\mathcal{O}_{Y,y} \\to \\mathcal{O}_{X,x}$ be the induced map of [[stalks|JTAKAPTT8H]].\nThese stalks are [[local rings|MZ02BKPHJA]], and $f^\\#$ makes $\\mathcal{O}_{X,x}$ a module over $\\mathcal{O}_{Y,y}$.\nThe morphism $f$ is *flat at $x$* if $\\mathcal{O}_{X,x}$ is a [[flat module|73RYYEVDKA]] over $\\mathcal{O}_{Y,y}$.\nThe morphism $f$ is *flat* if it is flat at every point of $X$.\n\n## Remarks\n\nFor a ring $A$, the morphism $\\mathbb{A}^n_A \\to \\operatorname{Spec} A$ is flat.\nEach local ring of $\\mathbb{A}^n_A$ is a localization of $A[x_1, \\dots, x_n]$, which is a free $A$-module, and localizations of flat modules are flat.\nFor a field $k$, the closed immersion $\\operatorname{Spec} k \\to \\operatorname{Spec} k[t]$ given by $t \\mapsto 0$ is not flat.\nIts stalk map is $k[t]_{(t)} \\to k$, and $k$ is not flat over $k[t]_{(t)}$, since multiplication by $t$ is injective on $k[t]_{(t)}$ but becomes zero on $k$.\nFor a morphism of affine schemes $\\operatorname{Spec} B \\to \\operatorname{Spec} A$, flatness is equivalent to flatness of $B$ as an $A$-module.\n",
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