Grothendieck proved any algebraic vector bundle on \(\mathbb{P}^1\) splits into a direct sum of line bundles, and consequently any coherent sheaf on the variety is a direct sum of line bundles and torsion sheaves. This behaviour, and its ramifications on the derived category, exhibit how \(\mathbb{P}^1\) is very close to a principal ideal domain. The results above can be proven with as much or as little machinery as one desires– indeed one can take the point of view that this classification of coherent sheaves is a mere consequence of existence of Jordan normal forms of matrices. In this article however we look at a more cohomological proof, showing how Serre duality shines in these situations.

Among the first ideas one encounters in the study of homotopy groups is to consider covering spaces– things that locally look like products with some discrete topological space \(\Delta\) but can have vastly different global properties than a simple product space. In fact the number of non-trivial covering spaces is a measure of the topological complexity of the space, in a sense made precise by the Galois correspondence of fundamental groups. It is then natural to ask what happens if one replaces \(\Delta\) with more a complicated topological space \(F\): what results is a richer analogue of a covering space called a fiber bundle, or a vector bundle if \(F\) is a vector space.