By Palamodov V.
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Extra resources for Algebra and geometry in several complex variables
Definition. A A-differential operator q : E → F is called operator of N¨other type, if for any element a ∈ A there exists a A-morphism b : F → F such that qa = bq. If q is of N¨other type then S = Ker q is a submodule of E : if e ∈ Ker q, a ∈ A, then q (ae) = bq (e) = 0. The operator q is called N¨other operator for S. Problem 2. o. q : O0n → Cl . , n. Problem 3. Let A be an Artin C-algebra. Show that any linear bijection q : A → Cl is a A-operator of N¨other type. This fact is generalized as follows: Theorem 2 Let I be a primary ideal in O n associated to a prime ideal p.
3) Both ideals (I, a) and I, bk are strictly larger than I and our assertion will imply that I is reducible. 3) we show that any element c of the right side belongs to I. We have c = i + ubk for some i ∈ I and u ∈ A. 2). This implies ib + ubk+1 = cb ∈ I, ubk+1 ∈ I consequently u ∈ I : bk+1 = I : bk ! d. 3 Corollary 4 An arbitrary ideal in a N¨otherian algebra is equal to intersection of primary ideals. Proof. 1 we prove that I can be written as intersection of irreducible ideals I = I1 ∩ ... 4) Each ideal Ir is primary according to the previous Theorem.
Zn . We wish to choose a dense linear free system in I. e. of nonnegative integers). The monomials are linearly free and the span is equal to the subalgebra of polynomials. It is dense in F in the sense that an arbitrary series a is equal to a polynomial up to an element of mk for arbitrary k. e. for any two different elements we have either i j or i ≺ j. Note the property: if i, j, k ∈ Nn are arbitrary, then i j is equivalent to i + k j + k. Now for an arbitrary i ∈ Nn the subspace F(i) ⊂F of series that contains only monomials z j for j i, is an ideal in F.