Vector Bundles

In the next couple years I will need to understand the ins and outs of different cohomology theories and the spectra that represents them. Some of the most important of these (for my research) can be described using $MU$ — the complex cobordism spectrum. We briefly met this spectrum — or at least its cohomology theory — when we discussed formal group laws. There we explained briefly a theorem of Quillen, stating that the universal formal group law over the Lazard ring corresponds to complex cobordism cohomology. We did not cover what complex cobordism actually is, so that is the plan for this post. ...

A part of mathematics I really an starting to enjoy more is mathematics that explain or develop connections between geometry or topology, and algebra. The first two posts on this blog was focused on developing some geometrical insight to two lemmas from commutative algebra, namely Noether’s normalization lemma and Zariski’s lemma. There are many more such connections worth discussing and exploring, and today I want to focus on one of these “bridges” between geometry and algebra, namely Swan’s theorem. This theorem tells us how nice objects “over” another object in geometry relate to nice objects “over” another object in algebra. ...