Algebra and geometry in several complex variables by Palamodov V.

By Palamodov V.

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Extra resources for Algebra and geometry in several complex variables

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It is called the inverse image (or pull back) of the sheaf π. Examples 1. Let X be a topological space; for any open U ⊂ X we consider the space C (U) of continuous functions f : U → C. For V ⊂ U the restriction mapping C (U) → C (V) : f → f |V is defined. This is a contravariant functor from T op (X) to the category of C-vector spaces. This functor is a sheaf, denoted C (X). 2. Let M be a complex analytic manifold. The sheaf of (germs) holomorphic functions is defined on M ; we use the notation O (M ) .

2) i

Then there exists an integer l and a N¨other differential operator q : O n → [On /p]l for I. Corollary 3 Let I be an arbitrary ideal in O n , I = I1 ∩ ... ,ps be the associated prime ideals. , s. 2 For a proof we take a primary decomposition of I and apply the above Theorem to each component. This theory is generalized for arbitrary modules of finite type. Theorem 4 Let M be an arbitrary O n -module of finite type, N is a submodule N = N1 ∩ ... ,ps are associated prime ideals. For each j there exists an integer lj and an O n -differential operator qj : M → [O n /p]lj of N¨other type such that ∩ Ker qj = N .

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