Let $K$ be a field with infinitely many elements(I assume $\mathbb{C}$), $A = K[x_1,\dots,x_n]/I$ a finitely generated K–algebra. Then after a general linear coordinate change(I dont get this line), there exists a number $r\le n$, and an inclusion $K[x_1,\dots,x_r]\subseteq A$, such that A is a finitely generated $K[x_1,\dots,x_r]$–module. If moreover,$I\neq(0)$, then $r < n$. If $V = V (I)$, we also say that the projection on the first r coordinates $π : V \rightarrow K^r$ is a Noether normalization of $V$. Could any one explain me the geometric motivation behind the fact?I will be pleased if some one explain me the fact over $\mathbb{C}[x_1,x_2]$
Noether Normalization
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algebraic-geometry
commutative-algebra
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0Dear Makuasi, Have you looked at [this MO question and answers](http://mathoverflow.net/questions/42275/choosing-the-algebraic-independent-elements-in-noethers-normalization-lemma/42363#42363)? Regards, – 2012-05-17
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0Dear Matt E, I did not understand properly any of the explanation. – 2012-05-18
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0Dear Makuasi, If you didn't understand any of the explanation, then perhaps you should ask about some more basic points of geometry than Noether normalization. The concepts being used in that discussion are basic ones such as projection from a point to a hyperplane and so on. Do you not know about these? Regards, – 2012-05-19
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0No dear sir, then what are you recomending to study first? – 2012-05-19