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 conception is a common instrument for dealing with questions which contain neighborhood options and worldwide patching. "La proposal 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 concept are algebraic. The proposal of a sheaf used to be first brought in 1946 via J. LERAY in a brief observe Eanneau d'homologie d'une illustration, C. R. Acad. Sci. 222, 1366-68. in fact sheaves had happened implicitly a lot past 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 capabilities on a RIEMANN surface*'; and the "ideaux de domaines indetermines", simple within the paintings of okay. OKA due to the fact 1948 (cf. [OKA], p. eighty four, 107), are only sheaves of beliefs of germs of holomorphic services. hugely unique contributions to arithmetic are not favored at the start. thankfully H. CARTAN instantly discovered the good significance of LERAY'S new summary suggestion of a sheaf. within the polycopied notes of his Semina ire on the E. N. S.

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Proof: We proceed by induction on n. The case n=O is clear, hence we assume n>O. Let gE(9o, g:J=O, be arbitrary; we choose coordinates Zl' ""zn_l' Wsuch that g has finite order b in w. In order to show that (90 is noetherian it is enough to show that the residue ring (9o/(9og is always noetherian. 1 we have an (9~ module isomorphism (9o/(9og~(9't. By induction hypothesis (9~ is a noetherian ring. Hence (9o/(9og is a noetherian (9~-module and therefore a noetherian ring. In order to show that (90 is factorial we use GAUSS' lemma which says that a polynomial ring R [T] over a factorial ring R is factorial again.

A STEIN space. Furthermore we note: Let f: X -+ Y be a holomorphic map, let U resp. V be holomorphically convex open sets in X resp. Y. Then the intersection U nf-1(V) is a holo- 34 1. Complex Spaces morphically convex open set in X. If in addition V is a tersection V Ilf-1(V) is a STEIN space. STEIN space the in- Proof: We only have to verify the convexity assertion. Take an infinite discrete and closed subset M of V Ilf-1(V). It is an exercise in set topology to see that there exists an infinite subset M' of M such that M' is either discrete and closed in V or that f(M') is an infinite discrete and closed set in V.

Fn(x)); it never occurred to anyone to mention explicitly the attached "lifting" sheaf map l: (J)cr;n-+ f*«(J)x)' Since the days when nonreduced spaces came into being one was forced to proceed with more care. Clearly every holomorphic map (f, f): X -+