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I don't understand the proof of Noether Normalization Lemma in "Algebraic Geometry and Arithmetic Curves" . Liu considers first the case $k[X_1,X_0]/I$, then $k[X_1,X_1]/I=k[X_1]/I$, then again $k[X_1,X_2]/I$. It feels strange that induction proves the theorem for two variables, then for one variable, then again for two variable and the rest cases. Is there a mistake in subscripts or does $k[X_1,X_0]/I=k/I$?

It looks like a mistake or then I have misunderstood the notation of $k[X_1,X_0]$ as the proof given in follows from induction on number of generators.

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You're looking at Proposition 1.9? I don't see the objects that you're describing. – Dylan Moreland Jul 6 '12 at 21:33
Yes. He takes $A=k[X_1,\ldots,X_n]/I$. If $n=0$ we have $A=k[X_1,X_0]/I$. If $n=1$ we have $A=k[X_1]/I$. If $n\geq 2$ then we have $A=k[X_1,\ldots,X_n]/I$. – Jaakko Seppälä Jul 6 '12 at 21:44
If $n=0$ we have $A=k$. The notation $1,2,\dots,n$ means all the integers greater than 1 and less than or equal to $n$. Hence when $N=0$ you have the empty set. – JSchlather Jul 6 '12 at 21:46
$n=0$ is $A=k/I$, I think. – Dylan Moreland Jul 6 '12 at 21:47
Maybe one of the commenters should post an answer (maybe community wiki) in order to insure this question does not stay unanswered. – M Turgeon Jul 6 '12 at 23:13

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