Sheaves

Introduction In the last post we studied homology theories as abstract functors from stable $\infty$-categories to abelian categories. We showed that for every stable $\infty$-category $\mathcal{C}$ there is a universal homology theory which all others factor through, namely the Yoneda embedding into the Freyd envelope, $$ y\colon \mathcal{C}\longrightarrow A(\mathcal{C}). $$The fact that this is universal means that for any homology theory $H\colon \mathcal{C}\longrightarrow \mathcal{A}$ there is an essentially unique factorization ...

The first two posts ([1],[2]) I ever did on this blog - now over a year ago - were posts about algebraic geometry. In particular we explored the geometric implications of some of the algebraic results I was learning in my commutative algebra class. Last summer I also wrote a post about sheaves, and left it off by claiming to soon write about schemes. If you scroll through the blog we have covered a bunch of different topics, but the blog post on schemes, seems to have fallen through the cracks. Today we will rectify this situation. I have my algebraic geometry exam this week, so this is both an explainer-post, and a “making sure I understand the course material”-post. These types of posts have in fact become common on this blog, but hopefully that is ok. ...

This summer I’m participating in Ravi Vakils pseudocourse on algebraic geometry, AGITTOC. Hence this summer serves as a wonderful opportunity to learn and write about cool mathematics. For long I have wanted to dive deeper into this abstract topic after just dipping my toes in during my bachelor thesis, and now it is time. Ravi though us in the first lecture that we shouldn’t study abstract objects without a cause, i.e. we need to ask ourselves why we want to learn about the objects, or the mathematics that lies ahead. I want to study algebraic geometry because I really like algebraic topology, and a lot of the concepts and notions of algebraic topology are abstracted in algebraic geometry, and several concepts gets a new viewpoint or gets some new tools to use to study them. I thought I would start off by discussing one of the fundamental objects of study in algebraic geometry, namely sheaves. These objects abstract, concretize and formalize several other mathematical notions, some of which we already know. One of these in particular is sheaf cohomology which can be viewed to generalize both singular, deRham and Cech cohomology. We are going to look into this cohomology theory in a later post. ...