By Hans Grauert

...Je mehr ich tiber die Principien der Functionentheorie nachdenke - und ich thue dies unablassig -, urn so fester wird meine Uberzeugung, dass diese auf dem Fundamente algebraischer Wahrheiten aufgebaut werden muss (WEIERSTRASS, Glaubensbekenntnis 1875, Math. Werke II, p. 235). 1. Sheaf thought is a common instrument for dealing with questions which contain neighborhood ideas and international patching. "La concept de faisceau s'introduit parce qu'il s'agit de passer de donnees 'locales' a l'etude de proprietes 'globales'" [CAR], p. 622. The tools of sheaf idea are algebraic. The inspiration of a sheaf was once first brought in 1946 by means of J. LERAY in a quick notice Eanneau d'homologie d'une illustration, C. R. Acad. Sci. 222, 1366-68. after all sheaves had happened implicitly a lot prior in arithmetic. The "Monogene analytische Functionen", which ok. WEIERSTRASS glued jointly from "Func- tionselemente durch analytische Fortsetzung", are easily the hooked up parts of the sheaf of germs of holomorphic services on a RIEMANN surface*'; and the "ideaux de domaines indetermines", simple within the paintings of okay. OKA for the reason that 1948 (cf. [OKA], p. eighty four, 107), are only sheaves of beliefs of germs of holomorphic services. hugely unique contributions to arithmetic aren't liked at the start. thankfully H. CARTAN instantly discovered the good significance of LERAY'S new summary idea of a sheaf. within the polycopied notes of his Semina ire on the E. N. S.

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**Example text**

It remains to show! e . i'x=gx' XEX. }x,x which coincide on the generators Zp ... , zn of m«(J)cr;n,o)' This implies Ix=gx'*) Surjectivity: Take fI>"" fn E(J) X (X). First assume that X is isomorphic to a model space in a domain D in

I)(f-l(V)). I)). :l'. :l'. If, besides f: X -t Y, we have a second holomorphic map g: Y -tZ, we know (cf. /' the equation (gof)*(cp)=g*(f*(cp)) holds. }x. } x; we have X' = f -1 (Y') and f induces a holomorphic map X' -t Y' commuting with injections. X' is called the complex inverse image space of Y'. g. the fiber f- 1 (0) of the map f:

D Holomorphic maps induce product maps in an obvious manner: If f: X -+ X', g: Y-+ Y' are any holomorphic maps there exists a unique holomorphic product map f x g: X x Y-+ X' X Y' such that the diagram ! \ XxY X~X' fxg \ I )X'xY' Y~Y' is commutative. If f and g are biholomorphic so is generalization of this last statement is easy to prove: f x g. The following f: X -+ X' and g: Y-+ Y' are closed embeddings then f x g: X x Y-+ Y' is a closed embedding. § resp. / is the ideal in {9X' resp. (Dy. defining the subspace f(X) resp.