# Noether normalization over $\mathbb{Z}$

I would like to know what is a correct analogue of Noether normalization theorem for rings finitely generated over $\mathbb Z$. Obviously, Noether normalization can not hold "literately" in this case since, for example the ring $\mathbb Z_2[X]$ does not contain a polynomial subring with coefficients in $\mathbb Z$ over which it is finite.

I am asking this question to better understand the second part of the answer of Qing Liu to the question given here: http://mathoverflow.net/questions/57515/one-point-in-the-post-of-terence-tao-on-ax-grothendieck-theorem

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## 1 Answer

Take a look at this: http://www.math.lsa.umich.edu/~hochster/615W10/supNoeth.pdf. It proves the generalized version of Noether Normalization, which is what you need (or rather what Qing Liu uses in his answer). In general I think Mel Hochster's notes are really good.

The above link was in the following answer http://mathoverflow.net/questions/42276, which was a link in Qing Liu's answer that you mention in your question.

Sorry, I should mention what the general version of Noether Normalization is that Hochster proves in his notes:

Let $D$ be a domain, and $R$ a finitely generated $D$ algebra. There exists a nonzero $f \in D$, and a finite injective ring map $D_f[X_1,\dots,X_n] \hookrightarrow R_f$. Here the $X_i$ are indeterminates.

Note how the above version implies Noether Normalization over a field. Although, if you know some basic scheme theory, I feel like Qing Liu's answer involving constructible sets is equally enlightening.

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Dear Rankeya, thank you for the answer and for affirming that Hochester's notes are good:) ! This is important information. – agleaner Oct 14 '12 at 12:43
Also, I have one more question. Would you advise some (not too scary) place where to read the proof of Chevalet theorem on constructive sheaves? – agleaner Oct 14 '12 at 12:59
If you meant Chevalley's Theorem on constructible sets, then Ravi Vakil's notes, "Foundations of Algebraic Geometry", available on his website has a nice section on constructible sheaves and Chevalley's theorem. I believe he proves the theorem in section 8.4 of his notes. Note, however, that Ravi leaves many things as exercises, which depending on your background might be time consuming. Also, when you want to know about any topic in AG (even CA), I recommend the Stacks Project. It has a wonderful new search feature, which allows you to go straight to what you want. – Rankeya Oct 14 '12 at 14:15
More importantly, it has a section on constructible sets, and Chevalley's theorem. Most of these sources might appear scary the first time you use them. I was scared the first time I saw the Stacks Project. If you don't let your fears get to you, I guarantee that you will be rewarded and learn some beautiful math. – Rankeya Oct 14 '12 at 14:15
It is interesting, I never heard about Stacks Project... I see it has over 3000 pages :)... Will try to see if I will be able to use it. – agleaner Oct 14 '12 at 14:29
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