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By R. O. Wells Jr. (auth.)

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7, we let X = C and let ß be the holomorphic functions on C. 10). Then we have the following exact sequence of sheaves: 0- d - ß- ß/d- O. We note that (0/ dt - {C,0, if x = 0 if x =1= O. 9: Let X be a connected Hausdorff space and let a, b be two distinct points in X. Let Z denote the constant sheaf of integers on X and d denote the subsheaf of Z wh ich vanishes at a and b. Then O-d-Z-Z/d-O is exact and (Z / dt - {Z, 0, if x = a or x = b if x =1= a and x =1= b. tNote that "0" here is the identity element in an abelian group.

Resolutions 0/ Sheaves Sec. 4, since the presheaf of quotients of sections of Z by sections of d violates Axiom S2' Following the terminology of homological algebra for modules, we make the following definitions where sheaf means sheaf of abelian groups or sheaf of modules. 2). Aresolution of a sheaf 5' is an exact sequenee of sheaves of the form 0--5' __ 5'0 __ 5'1 __ .. • , which we also denote symbolically by 0--5' - - 5'*, the maps being understood. We shall see later that various types of information for a given sheaf 5' can be obtained from knowledge of agiven resolution.

Namely, since drp = 0, we have that there is aUE 8~-' (using germs at x), so that du = rp, where r = p + q is the total degree of rp. Thus we see that if we write u = Ur-I,D + ... D. \ on a complex manifold X. 1 term (see Bigolin [I]). This is analogous to the resolution of e by 8°,* and has a similar usefulness. 51 Cohom%gy Theory Sec. 3 3. Cohomology Theory In this section we want to present abrief development of sheaf co homology theory.

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