Cohomologia do espaço projetivo e a caracterização dos fibrados sobre p1

Abstract

This work’s main objective is to prove a theorem by Grothendieck which characterizes vector bundles over P1. The theorem states that if E is a vector bundle over P1, than the associated sheaf E is of type O(a1)⊕O(a2)...⊕O(ar), with a i ∈ Z and this decomposition is unique.We will follow the road used by TEIXIDOR (Massachusetts 2002). In order to be able to do that, we’ll visit some results on coherent sheaves and cohomology of the projective space. On the first chapter, some commutative algebra results are introduced and used as we move foward to prove a lemma by Grothendieck which heps us to prove a theorem about finiteness of coherent sheaves. On the second, we develop the initial part of coherent sheaves theory and show that on a complete variaty over a field k, the space of global sections of a coherent sheaf has finite dimension. On the third part we talk about sheaf cohomology aiming to study the cohomology of the projective space via ˇCech cohomology. In particular, for sheaves of type O(n), n ∈ Z and coherent sheaves when O X (1) is a very ample sheaf. In the last chapter we show that every vector bundle corresponds to a locally free sheaf, we introduce the functor e and Ext and prove the main theorem.