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Let $X$ be a noetherian space. We say a subset $Z$ of $X$ is constructible in $X$, if it is a finite union of locally closed subsets of $X$.

There is the following theorem of Chevalley(we are not supposed to prove it in this thread).

Theorem of Chevalley Let $X$ be a scheme. Let $Y$ be a noetherian scheme. Let $f\colon X \rightarrow Y$ be a morphism of finite type. Then $f(Z)$ is constructible in $Y$ for every constructible subset $Z$ of $X$.

Hartshorne Exercise II. 3.19 (a) is as follows. Show that the above thorem can be reduced to the following proposition.

Let $X, Y$ be affine and integral noetherian schemes. Let $f\colon X \rightarrow Y$ be a dominant morphism of finite type. Then $f(X)$ is constructible in $Y$.

How do we show this?

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    I think one can take scheme-theoretic image with reduced induced structure, this makes the irruducible component to be a integral scheme...2012-12-16

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