By Hans Grauert

ISBN-10: 3642695825

ISBN-13: 9783642695827

ISBN-10: 3642695841

ISBN-13: 9783642695841

...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 idea is a basic device for dealing with questions which contain neighborhood recommendations 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 thought are algebraic. The suggestion 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. 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 attached elements of the sheaf of germs of holomorphic services on a RIEMANN surface*'; and the "ideaux de domaines indetermines", easy within the paintings of ok. OKA seeing that 1948 (cf. [OKA], p. eighty four, 107), are only sheaves of beliefs of germs of holomorphic features. hugely unique contributions to arithmetic are not preferred initially. thankfully H. CARTAN instantly discovered the good value of LERAY'S new summary thought of a sheaf. within the polycopied notes of his Semina ire on the E. N. S.

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**Example text**

Z~_l' W we have ex> g(O, w)= L Pj(c l b W, ••. , cn _ l W, w)= Pb(C l , ••• , Cn_l' l)w b+ .... Since pb=l=O, we may choose cl"",Cn_ l in such a way that Pb(C l " " , cn_ l ' 1) =1= O. Then g(O, w) =1= 0 and hence gi(O, w) =1= 0 Jor all i = 1, ... , t in the coordinates z~, ""Z~_l' W. 2. § 2. Algebraic Structure of (9 (:n, 0 W. RUCKERT, a student of W. KRULL, realized in 1931 that algebraic methods are very helpful to simplify local function theory. In his now classical paper Zum Eliminationsproblem der Potenzreihenideale, submitted in 1931 and published in 1933, he writes (cf.

Since Ja is generated by U 1 0(1, ... ,uk0(1, we conclude '$(Ja)=o. Analogously we have I/I(J,)=O. 3 yields a unique holomorphic map (h, h): Z - T such that (r/I, t/i)=(loh, ;;;h). Now (3) and (2) yield io f =io(poh) and jog= jo(qoh). 3, we conclude f=poh, g=qoh. The uniqueness of (h,h) follows from the uniqueness of r/I and I. }) is a canonical product. 4. Existence of Canonical Products. General Case. For simplicity we first deal with a semiglobal case. }y). }y). Proof: The sets Wi'=U;xY, iel, form an open covering of Xx¥.

Direct Products and Graphs 25 We now fix some notations which will be used in the next two paragraphs. For given complex spaces (X, (9x) and (Y, (9y) we always denote by p resp. q the continuous projection X x Y ~ X resp. X x Y -+ Y. A complex space (X x y, (9) is called a canonical (complex direct) product of (X, (9x) and (Y, (9y) if there exist sheaf maps p: (9x-+p*({9), q: (9y-+q*({9) such that (X x y, (9) together with the holomorphic projections (p, jJ) and (q, q) is a complex direct product of (X, (9x) and (Y, (9y).

### Coherent Analytic Sheaves by Hans Grauert

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