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  "body": "\n## Statement\n\nLet $Y$ be a [[Noetherian scheme|ND6XEY68P4]] and let $f \\colon X \\to Y$ be a [[morphism|JXRKH6P1Z4]] [[of finite type|AHTE5DVX61]].\nA subset of a Noetherian scheme is *constructible* if it is a finite union of sets $U \\cap Z$.\nHere each $U$ is open and each $Z$ is closed in the [[Zariski topology|8SQK7RTB63]].\nThen $f(C)$ is a constructible subset of $Y$ for every constructible subset $C$ of $X$.\nIn particular, the image $f(X)$ is constructible.\n\n## Remarks\n\nThe proof reduces to the case where $Y = \\operatorname{Spec} A$ is affine and integral and $X$ is affine.\nThe key step shows that if $f(X)$ is dense in $Y$, then $f(X)$ contains a nonempty open subset of $Y$.\nThis follows from generic freeness or from Noether normalization over the fraction field of $A$.\nNoetherian induction on the closed complement then finishes the proof.\nFor example, over a field $k$, the morphism $\\mathbb{A}^2_k \\to \\mathbb{A}^2_k$ given by $(x, y) \\mapsto (x, xy)$ has image $\\set{u \\ne 0} \\cup \\set{(0, 0)}$.\nThis image is constructible but neither open nor closed, so constructible sets are the right class for images.\nWithout the finite type hypothesis the image can fail to be constructible.\nFor instance, $\\operatorname{Spec} \\Q \\to \\operatorname{Spec} \\Z$ has image the generic point alone, which is not constructible.\n\n## Explanations\n\n### L1 human\n\nThe image of an algebraic map need not be open or closed.\nChevalley's theorem says that it is never much worse than that.\nIt is always built from finitely many pieces, each the part of a closed set that lies in an open set.\nIn the language of logic, the projection of a set defined by polynomial equations and inequations is again of that form.\n\n### L2 human\n\nOver an algebraically closed field, the theorem is the geometric form of quantifier elimination for algebraically closed fields.\nIt is used to show that properties of the fibres of a morphism hold on constructible sets of the base.\nTogether with generic flatness, it gives the usual stratifications of the base of a family into finitely many pieces over which the family is well behaved.\nA general version holds for morphisms of finite presentation between arbitrary schemes, with a suitable definition of constructible sets.\n",
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