By G. Fischer

ISBN-10: 3540078576

ISBN-13: 9783540078579

ISBN-10: 354038121X

ISBN-13: 9783540381211

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**Extra resources for Complex Analytic Geometry**

**Example text**

A s s u m e there O ~ f (Ker(0(U) If a ( Ap is a n o n - z e r o - d i v i s o r , then is an open U a X and ~ 0(U\A)). Then we define I:= Ann(Ouf] If x ( U\A, c 0 U. then fx = 0 and I x = 0X, x. Hence supp(0u/l ) c AnU. On the other hand, Consequently Ip c o n s i s t s Remark. plex (see X' m A . by using We want if there to prove result For this p u r p o s e space, subset A of a reduced can be o b t a i n e d a generalization ]). that fp } O. is no i r r e d u c i b l e primary deoomposition their subset.

FORSTER). closed, By KRULLs intersection theorem G c Fp is hence := pp-I (G) c F(X) M is closed and a Fr~chet by f]'''''fr" bourhoods Mj space. If V o z V I ~... Denote by G c F the subsheaf generated is a fundamental system of Stein neigh- of p, we define := {f E F(X): fx E G x for every x E Vj} = N x~Vj px]G x for j = 0,1,2,... J c M is a closed subspace. ,a r E 0x(V j) such that f = a]f] + ... + arf r on Vj. ,a r ~ 0x(U) f = ~aif Since there M= i on U. is a n ~ N with V n c U, we have f ~ Mn, and hence U j=O M..

24 Example. Consider the holomorphic map @: C ~ C, x ~ x 2. e. the simple point with the ideal gene- function y. e. a double point. In particular, duced. This is a very c h a r a c t e r i s t i c spaces can occur in a natural way. Corollary. is the subspace de-1(y,) is not re- example how non reduced complex Let X be a complex space with (open or closed) complex sub- spaces X I and X 2. Then there is a unique complex subspace X I n X 2 of X I and X 2 such that the square of canonical X2 injections >X is cartesian.

### Complex Analytic Geometry by G. Fischer

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