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*This is my semi active blog, each article is a by-product of me attempting to understand some mathematics. Most articles, hence, are just rehashes of existing literature and most original contributions are either through mistakes which I apologise for in advance, or through non-traditional reordering of topics to suit my taste. It can however provide a starting point to some of the content, or an alternate perspective if you're already familiar with the material.*

June 7, 2023 (geometry)

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 desiresâ€“ 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.

July 6, 2021 (geometry)

*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.