By Friedrich Hirzebruch

In recent times new topological tools, specifically the speculation of sheaves based by way of J. LERAY, were utilized effectively to algebraic geometry and to the idea of features of a number of advanced variables. H. CARTAN and J. -P. SERRE have proven how primary theorems on holomorphically whole manifolds (STEIN manifolds) will be for mulated when it comes to sheaf thought. those theorems indicate many proof of functionality conception as the domain names of holomorphy are holomorphically whole. they could even be utilized to algebraic geometry as the supplement of a hyperplane part of an algebraic manifold is holo morphically entire. J. -P. SERRE has got vital effects on algebraic manifolds by means of those and different equipment. lately a lot of his effects were proved for algebraic kinds outlined over a box of arbitrary attribute. ok. KODAIRA and D. C. SPENCER have additionally utilized sheaf idea to algebraic geometry with nice good fortune. Their tools range from these of SERRE in that they use options from differential geometry (harmonic integrals and so on. ) yet don't make any use of the idea of STEIN manifolds. M. F. ATIYAH and W. V. D. HODGE have dealt effectively with difficulties on integrals of the second one sort on algebraic manifolds with assistance from sheaf conception. i used to be capable of interact with okay. KODAIRA and D. C. SPENCER in the course of a remain on the Institute for complex research at Princeton from 1952 to 1954.

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2. (FR1) and sequence of a 36 be given, with s 6 S. (Z,Y,N,s,f,g). Then (u, idN) the first, Complete Complete fu s to a triangle to a triangle (W,X,N,t,fu,h). is a map of two sides of the second triangle so there is a map v: W >Z into g i v i n g a m o r p h i s m of triangles. V Z t/u idN X >Y U Now sv = ut, so it remains = 0 to prove t 6 S. s 6 S, we have H(Ti(N)) for all i 6 ~ sequence of the first triangle. sequence of the second triangle, Indeed, since by the long exact Applying this to the long exact we H(Ti(t)) find is an i s o m o r p h i s m for all i 6 ~.

ZP+I(x ") > I p+l to a map easily that the resulting map isomorphism, as required. fp+l: f: xP+l X" are in IP/im I p-I A' is Extend the natural map > I p+I. 7. let I be the (additive) of A. A be an abelian category, and subcategory of injective objects Then the natural functor K+(I) is fully faithful. > D+(A) (Note that the results of section 3 carry over to additive subcategories of abelian categories. , if every admits an injection into an inJective object) is an equivalence of categories. Proof.

Of Lemma call the h o m o t o p y (k,t): morphism > Y" as above. cone of s. 4, and so is h o m o t o p i c operator T(I') 9 Y" i Then we have the e q u a t i o n V = (idi. , O) = (k,t) Separating the components, we d Z + d I (k,t) find Let Z" is acyclic, satisfies to zero. the Let us 42 id I = dk + kd + ts and dt - td = O . Thus t: Y" homotopic > I" to 1). 6. Lemma Let (i) element of is a m o r p h i s m Let P is a h o m o t o p y A be an a b e l i a n category. A E v e r y object of Ob A X" Assume (ii) and assume admits an i n j e c t i o n I" of o b j e c t s of o > Y >x P that into an X 6 P, then x~ > xI is an exact sequence, Then every X" I" of objects Let A' of A'.