Cohomological Methods in Algebraic Geometry

We begin this paper with a discussion of n-dimensional algebraic Vector bundles. Vector bundles of rank n may be identified with locally free Ox-modules of rank n. Several sections are dedicated to reviewing the basic notions of scheme-theory. We show how to construct the projectization of arbitrary rings. This construction corresponds to the construction of the tautological line bundle over an arbitrary paracompact space encountered in classical topology. Next we construct the projectization P(E) of a quasi-coherent sheaf E on a scheme X. This construction corresponds to the projectization of an arbitrary vector bundle encountered in algebraic topology. Next we review some basic facts of Chow rings. We use the Chow ring as well the bundle projectization construction to define the Chern classes. Next we give a brief overview of cohomological constructions arising from an interplay between algebraic topology and algebraic geometry through categorical constructions.