Filed under: Category Theory, Homological Algebra | Tags: Abelian Categories, complexes, Derived Functors, Dummit & Foote, Functors, Weibel
While studying commutative algebra and module theory, you’ll eventually be introduced to projective and injective modules. Then, suddenly, people will start throwing around words like derived functors, chain complexes, Ext and Tor. It can all be pretty daunting, especially if homological algebra and category theory is only taught as a ‘side-dish’ on a need to know basis, supplementary to whatever material is typically meant to be covered.
On the other hand, you could also be studying Algebraic Topology, where chain complexes first came about. Either way, we’ll be approaching the topic in a more or less abstract manner (hopefully) appropriate for both situations.
So, where do you go when many homological algebra books don’t talk about derived functors until the second half of the book? I’d recommend either An Introduction to Homological Algebra by C.A. Weibel or, my personal preference, Section 17.1 of Abstract Algebra by Dummit & Foote, which most students already own. Both start out from (co)chain complexes and go straight to derived functors.
The only downside to using Dummit & Foote is that they do proofs using specific functors (
Here I’ll give a basic introduction on how to get from an understanding of commutative algebra with a little category theory (including: functors, objects, (mono/epi)morphisms, exact sequences) to understanding what a derived functor is and how it is derived from left/right exact functor. Giving proof outlines and/or citations as needed.
