By G. Fischer
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Additional info for Complex Analytic Geometry
3 Assertion b) immediately Lemma Let ~: X ~ Y be a finite 3. implies: holomorphic map, F an 0X-mOdule and q s Y. 11. = Fp -[-[ p(-l(q) Given a holomorphic q := ~(p). 3) 0X, p is an 0y,q-module. quasi-finite Theorem. homomorphism 0y,q + 0X, p finite over 0y,q, We recall map ~: X ~ Y, fix a point p-~ X and put Via the canonical 0X, p is called if it is a finitely over 0y,q, a fundamental conditions i) 0X, p is finite result from local analytic map, iii) p is an isolated over 0y,q. point geometry (see p E X and q := ~(p), are equivalent: over 0y,q.
42. U 0:I k = U Ann Ik If p 6 X' take a generator a 6 0X, p of Ip. Since 0[Y]p = U Ann(a k) ~ 0X, p, the residue class of a in 0X,,p is a n o n - z e r o - d i v i s o r . of the analytic i n t e r s e c t i o n of Y and X' it generates By d e f i n i t i o n the ideal of Y' in X' at p and the a s s e r t i o n is proved. 46. Let X be a complex space. ,C k c X analytic c) C c X ~ sets form the the following conditions: C constructible. constructible ~ C I n ... n C k and C I U ... O C k con- structible.
There is a closed X the diagonal map 6 x i s a n em- c o m p l e x s u b s p a c e DX,-~ X x X a n d a f a c - torization X Dx~Xx X of 6X, where 6~ is biholomorphic. D X is called the diagonal in X x X. ,fk,p fi| generate Proof. are holomorphic generate 1| I , 9 ,fk| the maximal Since ~X,p: of D X at IXI is hausdorff, 0X• By construction ideal mX,p' of p such then 1| k the sheaf of ideals p ( X we consider in a neighbourhood the canonical (p,p) 16XI is a topological ring homomorphism = 0X,p ~ 0X,p ~ 0X,p" of 6 X we have ( X x X.
Complex Analytic Geometry by G. Fischer