Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ 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 $X_1,\ldots,X_n$ such that $M$ is finitely generated over $\operatorname{Sym}(W)$ 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?


3 Answers 3


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.

  • $\begingroup$ 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. $\endgroup$ Oct 21, 2009 at 12:19

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. DOI 10.1007/978-3-540-73542-7_3 is a page that links to the appropriate chapter. It is possible that a subscription is required to open it though.


A good reference seems

Adjamagbo, K.; Winiarski, T. Refined Noether normalization theorem and sharp degree bounds for dominating morphisms. Comm. Algebra 33 (2005), no. 7, 2387–2393. (Reviewer: David R. Finston)

They state the theorem as follows:

Theorem 1. Let $B=\mathbb K[x_1,\ldots,x_n]=\mathbb K[X_1,\ldots,X_n]/I$ be an absolutely integral affine algebra of dimension $r>0$ over an infinite field $\mathbb K$, where $I$ is an ideal of $\mathbb K[X_1,\ldots,X_n]$ and $d$ is the degree of its variety in $\mathbb A^n(\mathbb K)$. If $L$ ifs a generic linear automorphism of $\mathbb K^n$ with components $L_1,\ldots,L_n$, and $A=\mathbb K[L(x_1,\ldots,x_n),\ldots,L_r(x_1,\ldots,x_n)]\subseteq B$, then $B$ is a finite module over $A$, the fractions field of $B$ is an extension of the dractions field of $A$ with separable degree $d$, and the cardinality of each fiber of the canonical morphism from $\operatorname{Spec}(B)$ to $\operatorname{Spec}(A)$ is majorated by $d$.

Edit: Another good reference might be the red book of varieties and schemes, page 126 at the bottom, this is a remark concerning Proposition 6 in §7.


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