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Suppose that M is a finitely generated module over A=k[X_1,...,X_n] of Krull dimension m with k an infinite field. Then one version of Noether normalisation says there is an m-dimensional k-subspace W of the k-vector space spanned by X_1,...,X_n such that M is finitely generated over Sym(W) considered as a subring of A.

As is surely well-known, in fact one can show that the set of m-dimensional k-vector spaces W that work is open in the appropriate Grassmannian. My question is where is there a reference for this fact in the literature?

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up vote 3 down vote accepted

Ok, I might be missing something (I often am) but I believe that this does it. Scroll up to page 452, line 3.9. The book is Effective Methods in Algebraic Geometry by Rossi and Spangher.

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I think that does what I want. Thank you. It would be nice to have a reference that isn't the middle of a proof of something else entirely though. If anyone has any other suggestions they are very welcome. –  Simon Wadsley Oct 21 '09 at 12:19
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In case anyone else has the same question and discovers this page I have just found a more explicit reference for this result: Remark 3.4.4 of A Singular introduction to commutative algebra by Greuel and Pfister. http://www.springerlink.com/content/u62645311l0h2256/ is a page that links to a pdf of the appropriate chapter. It is possibe that a subscription is required to open it though.

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