I’ve been trying to re-understand some of the proofs in commutative and homological algebra. I never really had a good feeling for spectral sequences, but they seemed to crop up in purely theoretical proofs quite frequently. (Of course, they crop up in computations quite frequently, too.) After learning about derived categories it became possible to re-interpret many of these proofs. That’s what I’d like to do in this post.
Here is a toy example of a result, which does not use spectral sequences in its usual proof, but which can be interpreted in terms of the derived category.
Proposition 1 Let be a local noetherian ring with residue field . Then a finitely generated -module such that is free.
Let’s try to understand the usual proof in terms of the derived category. Throughout, this will mean the bounded-below derived category of -modules: in other words, this is the category of bounded-below complexes of projectives and homotopy classes of maps. Any module can be identified with an object of by choosing a projective resolution.
So, suppose satisfies . Another way of saying this is that the derived tensor product
has no homology in negative degrees (it is in degree zero). Choose a free -module with a map which induces an isomorphism . Then we have that
by hypothesis. In particular, if is the cofiber (in ) of , then .
We’d like to conclude from this that is actually zero, or that : this will imply the desired freeness. Here, we have:
Lemma 2 (Derived Nakayama) Let have finitely generated homology. Suppose . Then . (more…)