/sci/ - Science & Math

>>12577813
It is a functor between module categories. The ring homomorphism that induces it makes the codomain ring an algebra over the domain ring (not domain like an integral domain). The Hom would be a right adjoint to the restriction functor which is the one I was talking about in the old post, so you get a triple extension/induction - restriction - coextension/coinduction, which is two adjoint pairs. I think they should all be used in AG quite a lot.

I learned my CT from Awodey and Rotman's Introduction to Homological Algebra, and then just here and there while working on stuff. You will need homological algebra with all the derived stuff, so why not take a look? It will go through adjunctions and limits and that kind of things in a mostly algebraic setting, but you may also have examples of what things would be like for sets and other categories than just modules. His book also describes various types of rings fro a homological viewpoint. For example, let's say [math]R[/math] is von Neumann regular. Two equivalent ways to define it would be:
(1) For every [math]r\in R[/math], there is some [math]x\in R[/math] such that [math]rxr = r[/math].
(2) Every module over the ring is flat.
The first condition is nice if you want to check whether a field would be vNR, but the second one tells you how its modules do homological things. I recommend.
>I want to try thinking in both ways whenever I can so I'll be able to complement when I get stuck in one side, but actually doing is a whole another story... I hope I'll do my best!
That's a nice thing when you have a dual formalism. You will do your best! Victory or Valhalla!