In the study of local theory for holomorphic varieties, the Local Parametrization Theorem states that in $\mathbb{C}^n$,for any irreducible germ of holomorphic variety $V$ at 0, there exist a nonsingular linear change of coordinate under which the natural projection $\pi : \mathbb{C}^d \times \mathbb{C}^{n-d} \to \mathbb{C}^d$ is a finite branched holomorphic covering of $\mathbb{C}^d$.

On the other hand, the Noether normalization Theorem, applied in the context of reduced holomorphic algebra, states that for every reduced holomorphic algebra $A$ there exist an integer $d$ and a finite injective algebra homomorphism $\phi : _d\mathcal{O}_0 \to A$.

It seems to me that the two theorems are telling us the exact same thing, but one is stated in a more geometric language while the other in a algebraic one. Is this true?

If so, why do we have two separate theorems? Were they developed independently?

If they are not equivalent. How can I see the difference?

  • 1
    $\begingroup$ Isn't the Noether normalization Thm more general? $\endgroup$ May 23, 2012 at 6:20
  • $\begingroup$ The normalization theorem does certainly apply to more general situations where $A$ can be any finitely generated modules over any field $k$, and it seems that the local parametrization theorem is a special case of it (although I have trouble seeing how to prove it). If that's the case, doesn't it make the whole local parametrization theorem redundant? Especially considering that it was known as far back as the 1920s? $\endgroup$
    – ssquidd
    May 23, 2012 at 22:39


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