By G. Fischer
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Additional resources for Complex Analytic Geometry
Zn) ] = R. It becomes continuous if we provide 0s•163 x Z) w i t h the Fr~chet topo- logy of compact c o n v e r g e n c e and R with the Krull topology. Since I' c R is closed in the Krull topology and h = ~ h (i) i=O we conclude h ~ I' This finally implies I = I'. ,H r 6 0 s ( U ) [z 1 . . If I' c 0uxcn denotes coherence of I and I' generating I. We may U of s in S and representatives ,Zn]. t h e s h e a f o f i d e a l s g e n e r a t e d b y H1 , . . , H r t h e implies that there is an open neighbourhood Z of o in C n such that I = I' on U x Z, if we shrink U sufficiently.
E. = YA(XxA) if Zr Zn(XxA) and Y' is minimal with respect to this property, X is any closed subspace such that = Yn(XxA), then Y' is a closed subspace of Z. d) AnY' is a n a l y t i c a l l y rare in Y'. e) AnY is a n a l y t i c a l l y rare i n Y if and only if Y' = Y. We call Y' the closure of Y\A in X and denote it by closx(YxA). Proof. a) is clear since c) follows I c I[A]. immediately from the d e f i n i t i o n of the gap sheaf. b) Put B := IYIxA. A s s e r t i o n c) implies B c Conversely, UA(Y\A) if p E AxB, = ~.
The scalar m u l t i p l i c a t i o n (t,z I .... e. over S. A closed complex S, if it is invariant under if there is a commutative diagram subspace the scalar 45 C x S • Cn J C with x - > S • s J - -~-'-> a holomorphic If Xr X map ~' S x C n is a c o n e s ~ induces ~n, (which then (z I .... ,Zn) ~ is the for any restriction fixed k E C* of ~). ,kZn) automorphisms ~X: For X ~ S • cn ~ the d e f i n i t i o n lization ! and ~X: S x s X ~ of an " a b s t r a c t as a s u b s p a c e of X. cone" X over S x cn we r e f e r If U c S is o p e n we c o n s i d e r to any p o l y n o m i a l S and a possible rea- .
Complex Analytic Geometry by G. Fischer