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Zn . Then, as noted above, n M ˜ is actually generelements zm+1 , . . , zn . By the theorem of the primitive element, n M ˜ by a single element which may be chosen to be a linear combination of ated over m M zm+1 , . . , zn . Another linear change of variables effecting only these coordinates can be used to transform this element into zm+1 . Such a change of variables does not change the fact that P is regular in the variables zm+1 , . . 6. This completes the proof. Let P be a prime ideal of n H.

Let π −1 (p) = {q1 , · · · , qk } and suppose we have chosen for each i a neighborhood Ui of qi in V0 such that Ui ∩Uj = ∅ for i = j. If A is a neighborhood of p with compact closure ¯ − Ui is a compact subset of V0 . The collection of sets of this form is in W0 , then π −1 (A) closed under finite intersection and so, if they are all non-empty, then there is a point q in their intersection. Then, necessarily, π(q) = p and, hence, q be one of the qi . This is not possible since q is in the complement of each Ui .

Thus, In−1 satisfies condition (iii) as desired. Either m = n − 1 or we may now conclude as above that there is fm−1 ∈ In−1 = n−1 H ∩ I which is regular in zn−1 . Clearly we can repeat this procedure n − m times to achieve condition (i). This completes the proof. We say that a germ π : V → W of a holomorphic map between two germs of varieties is finite if π −1 (0) = (0). 22 Theorem. A germ π : V → W of a holomorphic mapping between two germs of holomorphic varieties is finite if and only if for each irreducible component Vi of V , the image π(Vi ) is the germ of a holomorphic subvariety of W and π : Vi → π(Vi ) is a finite branched holomorphic cover.

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### Notes on several complex variables by Taylor J.L.

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