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 concept is a common instrument for dealing with questions which contain neighborhood options and worldwide patching. "La thought de faisceau s'introduit parce qu'il s'agit de passer de donnees 'locales' a l'etude de proprietes 'globales'" [CAR], p. 622. The equipment of sheaf conception are algebraic. The idea of a sheaf used to be first brought in 1946 through J. LERAY in a brief be aware Eanneau d'homologie d'une illustration, C. R. Acad. Sci. 222, 1366-68. after all sheaves had happened implicitly a lot previous in arithmetic. The "Monogene analytische Functionen", which ok. WEIERSTRASS glued jointly from "Func- tionselemente durch analytische Fortsetzung", are easily the attached elements of the sheaf of germs of holomorphic features on a RIEMANN surface*'; and the "ideaux de domaines indetermines", simple within the paintings of okay. OKA on account that 1948 (cf. [OKA], p. eighty four, 107), are only sheaves of beliefs of germs of holomorphic capabilities. hugely unique contributions to arithmetic usually are not preferred at the start. thankfully H. CARTAN instantly discovered the nice 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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Dimcr£'(1P,,) = n + 1. 4. Holomorphically Convex Spaces and Stein Spaces. All complex spaces considered in this and the following paragraphs are assumed to have a countable topology. A (not necessarily reduced) complex space X is called holomorphically convex, if for every infinite discrete and closed subset M of X there exists a holomorphic function h on X such that the set of values of [h] on M is unbounded. A holomorphically convex space X is called a STEIN space if every compact complex subspace of X is finite.

On X and every STEIN covering U of X we can define 36 1. Complex Spaces we call Hq(X,9') the q-th cohomology module of X with coefficients in the coherent sheaf 9'. 0 Every short exact sequence o~9" ~9' ~9''' ~o of coherent sheaves on an arbitrary complex space X gives rise to the long exact cohomology sequence o ~ HO (U, 9") ~ HO (U, 9') ~ HO (U, 9''') ~ HI (U, 9") ~ .... If X is a STEIN space we conclude that the induced sequence O~9"(X)~9'(X)~9'(X")~o of global sections is exact. e. J(X)-sequence 9"(X)~9'(X)~9'''(X) is exact.

D-coherent. )D are isomorphic to the G:::-algebra of convergent power series in n variables. )D-sheaves, however - and this is what really counts - coherence is preserved. The Extension Principle will be applied again and again in this book. 5. Analytic Image Sheaves. )x)--+(Y, llJy) be a holomorphic map. )x-module Y the image sheaf f*(Y) is well-defined and, in a 18 1. Complex Spaces canonical way, an f*(IPx)-sheaf. iy-module is called the analytic image sheaf of [I' (with respect to the map (f,J)).