Hopefully there will be something here...
Excisive functors and homology theories have a close connection. I demonstrated their relation together with spectra and how the failure of the identity functor being excisive gives rise to the stable homotopy groups.
Differential forms on a smooth manifold is a powerful tool from the perspectives of algebra and geometry. We may mimic the construction over to varieties to obtain a similar theory. Interestingly, while the cohomology groups they give are isomorphic by a theorem of Grothendieck, the module of differential forms are not isomorphic even if we feed it the ring of smooth functions.
I took a lot of notes during my undergraduate years, you can see them in my github.