# Algebraic Geometry II: Cohomology of Algebraic Varieties. by V. I. Danilov (auth.), I. R. Shafarevich (eds.)

Y be a continuous map. The direct image functor f*, from the category of Abelian sheaves on X to the category of Abelian sheaves on Y, is additive and left exact (but not necessarily exact). The corresponding derived functors, denoted by Rn J*, are defined with the help of flabby resolutions. Fora sheaf Fon X, the sheaf Rn f*(F) is, in fact, a sheaf on Y associated to a presheaf h particular, if f maps X to a point, then Rn f*(F) coincides essentially with the cohomology Hn(X, F).

Projections. As in the case of the Chow ring, one can verify that K (IP'n x Y) is generated over K(Y) by the classes of sheaves O(m). (By the way, the long exact sequence of Example 1 from Sect. 2 yields the relation among the O(m)'s in K(IP'n)). So, it is suffice to verify the formula for O(m), which can be clone by explicit calculations, as in Example 4 of Sect. 4. Embeddings. By the deformation to the normal bundle, everything is reduced to the case when Xis embedded as the zero section of a vector bundle V(E).

The Koszul Complex. Complexes of modules, like ( *), arise in many problems. We will recall a few facts; for details, see (Fulton-Lang (1985), Griffiths-Harris (1978), Grothendieck (1968b)). For simplicity, we restriet ourselves to the case M = A. Then the complex ( *) is a tensor product of the "elementary" two term complexes concentrated in dimensions 0 and 1. ''') A-. tnA c:::: A~A . Then IC(f 00 ) is the inductive limit of IC(r) as n---. oo. Since cohomology commute with lim, it is helpful to examine the complex K·(r).