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By Palamodov V.

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Extra info for Algebra and geometry in several complex variables

Example text

It can be formulated as follows: f belongs to the ideal Iz in the formal algebra Fz generated by elements of I. The last ideal is called localization of I at z. 1 Differential operators in modules Definition. Let A be a commutative algebra, E, F be A-modules, l ≥ 0 is an integer; a differential operator q : E → F of order ≤ l is a linear operator such that for an arbitrary a ∈ A . (ad a) q = aq − qa is a differential of order ≤ l − 1. A differential operator of order 0 is, by definition, an A-morphism.

Claim that the set πj (Z ∩ V) is a hypersurface in Vj = Dj ×B if Dj and B are 9 . sufficiently small. 7). By Weierstrass Lemma 1 we can replace f by a pp p with respect to a suitable coordinate system. Apply the previous Theorem; one of the factors belongs to I (Z) because of this ideal is prime. We can assume that p is irreducible. This pp generates the ideal Ij . Indeed, if g ∈ Ij , applying Weierstrass Lemma (ii) we reduce g to a pp q. The resultant R (p, q) belongs to Id and does not vanish identically.

We apply the same arguments to Q, Q and so on. 3) with m1 = ... , Ps . Now we consider the case k > 1. Let B be a suitable nbd of a for the function D = Skj (P ) Now we call ∆ = {D = 0} the discriminant set of P. Choose a point b ∈ B\∆; there exists a root, say α1 of multiplicity exactly k of the polynomial P (z1 , b) . Choose a curve γ ⊂ B\∆ that joins b and a point z and construct analytic continuation of α1 along γ. , βl all the roots of P (z1 , z ) that can be obtained in this way. 5) and Q by taking the product of binomials z1 − βj over all other roots in the point z .

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