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  "body": "\n## Definition\n\nLet $f \\colon X \\to Y$ be a [[morphism of schemes|JXRKH6P1Z4]] and let $y \\in Y$.\nLet $\\kappa(y)$ be the [[residue field|G51E8MW7J1]] of $Y$ at $y$.\nLet $\\iota_y \\colon \\operatorname{Spec} \\kappa(y) \\to Y$ be the canonical morphism with image $y$.\nThe *fiber* of $f$ over $y$ is the [[fiber product|1DZMT4S9FB]]\n$$X_y = X \\times_Y \\operatorname{Spec} \\kappa(y),$$\ntaken with respect to $f$ and $\\iota_y$.\nIt is a scheme over $\\kappa(y)$.\n\n## Remarks\n\nThe projection $X_y \\to X$ is a homeomorphism of the underlying space of $X_y$ onto $f^{-1}(y)$.\nExample: let $f \\colon \\operatorname{Spec} \\Z[i] \\to \\operatorname{Spec} \\Z$ and $y = (p)$ for a prime $p$.\nThen $X_y = \\operatorname{Spec}(\\Z[i] \\otimes_\\Z \\F_p) = \\operatorname{Spec} \\F_p[x]/(x^2 + 1)$.\nFor $p = 5$ this is $\\operatorname{Spec} \\F_5[x]/((x-2)(x+2))$, two points.\nFor $y = (0)$ the fiber is $\\operatorname{Spec} \\Q(i)$, the generic fiber.\nNon-example: the fiber is in general not the reduced point structure on $f^{-1}(y)$.\nLet $k$ be a field with $2 \\ne 0$, let $f \\colon \\operatorname{Spec} k[t] \\to \\operatorname{Spec} k[x]$ be given by $x \\mapsto t^2$, and let $y = (x)$.\nThen $X_y = \\operatorname{Spec} k[t]/(t^2)$, which is not reduced, while the single point of $f^{-1}(y)$ with its reduced structure is $\\operatorname{Spec} k$.\n",
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