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  "body": "\n## Summary\n\nThe fiber of a [[morphism of schemes|JXRKH6P1Z4]] $f \\colon X \\to Y$ over a point $y$ is a scheme $X_y$ over the [[residue field|G51E8MW7J1]] $\\kappa(y)$.\nIts underlying space is the preimage $f^{-1}(y)$, but it also carries a scheme structure, which may be non-reduced.\nA morphism is often thought of as a family of schemes, its fibers, parametrized by the points of $Y$.\n\n## Explanations\n\n### L1 human\n\nThe fiber over $y$ is \"everything in $X$ that lies above $y$\", made into a scheme of its own.\nTake $\\operatorname{Spec} \\Z[i] \\to \\operatorname{Spec} \\Z$.\nOver the prime $5$ the fiber has two points, because $x^2 + 1$ factors modulo $5$.\nOver the prime $3$ it has one point with residue field the field of $9$ elements.\nOver the prime $2$ it has one point, but the fiber is $\\operatorname{Spec} \\F_2[x]/((x+1)^2)$, which has a nonzero nilpotent: this records that $2$ ramifies.\n",
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