# Hartshorne Exercise II. 3.19 (c)

Hartshorne Exercise II. 3.19 (c) is as follows. Prove the following theorem of Chevalley by using Exercise II. 3.19 (a) and (b) and noetherian induction on $Y$. How do we prove this?

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 $X$.

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This is the third part of this question you have asked on this site only to write an answer to it yourself. I would prefer to see all users make such posts on their personal blogs, rather than making this site a sort of public notebook. If many users did this sort of thing, the site would be much more difficult to use. For this reason I have downvoted this question. –  Carl Mummert Dec 19 '12 at 2:01
!Makoto Kato: It is not this site's policy; the policies of the math SE are determined by its users. We are free to decide as a community that we do not welcome a certain type of post even if it is welcome on other SE sites. –  Carl Mummert Dec 19 '12 at 2:33
To be clear, it is not answering your own question that is the issue. It is using the site as a kind of notebook to record your work. If a user occasionally answers their own question, especially after getting a hint, that is wonderful. But the purpose of the site is not for asking questions that the asker is already able to solve; I feel that distorts the meaning of "question". –  Carl Mummert Dec 19 '12 at 2:38
I have explained everything I wish to explain, and there is nothing else that I wish to add at this time. –  Carl Mummert Dec 20 '12 at 17:20
Insanity: Doing the same thing over and over again and expecting different results. -Albert Einstein –  robjohn Dec 21 '12 at 0:50

Hopefully, you are not expecting someone to post a complete proof of this question (later edit: because this is a well known theorem whose proof you can look up). Check section $8.4$ of Foundations of Algebraic Geometry by Ravi Vakil (notes available here: http://math.stanford.edu/~vakil/216blog/). He develops the machinery to prove this theorem in this section and has many exercises (in particular, I think he uses Noetherian induction). If you are looking for a hint, he gives you one and asks you to make the argument precise.