A Road to the Grothendieck Spectral Sequence:Derived Functors II

In the previous post, I gave a little glimpse into derived functors and somewhat motivated their construction. In this post, we’ll get our hands dirty with homological algebra to continue setting up the required machinery and go through many proofs.


In the previous post, I promised to continue the proof of a lemma which establishes a long exact sequence. Before giving the proof, let me mention a few facts which will be useful.

If we’re given a short exact sequence 0 \to I \to A \to B \to 0 in an abelian category where I is an injective object. From the map I \to I and the monomorphism I \hookrightarrow A, we can extend to a map A \to I such that the composition is the identity. Using the Splitting Lemma, one obtains a non-canonical splitting A \simeq I \oplus B. Applying F,

0 \to F(A) \to F(I) \oplus F(B) \to F(B).

The identity map B \to B factors through the projection map \pi:A \to B, the same holds true after applying F, in particular, the last map is surjective!

0 \to F(A) \to F(I) \oplus F(B) \to F(B) \to 0.

Proof of Lemma 1:

Step 1:A morphism between objects with injective resolution induces a chain map between the resolutions

Let \phi:A \to B be a morphism between two objects with injective resolutions A_{\bullet},B_{\bullet}. In the figure below, the map \phi_{0} is constructed from the fact that d_{0}:A \to I_{0} is a monomorphism and there is a map A \to B \to I'_{0} from A to an injective object.

df1

Now, there is a monomorphism I_{0}/ker(d_{1})=I_{0}/Im(d_{0})=Coker(A \to I_{0}) \hookrightarrow I_{1}. Next, note that by the commutativity of the square already defined, \phi_{0} takes Im(d_{0})=Ker(d_{1}) to Ker(d'_{1})=Im(d'_{0}) by the fact that d'{1}d'_{0}=0 by exactness of the lower sequence. This means that the map \phi_{0} induces a morphism h_{0}: Coker(A \to I_{0}) \to Coker(B \to I'_{0}) and by exactness, we can compose this with B/Ker(d'_{1}) to get a map Coker(A \to I_{0}) \to I'_{1}. Since I'_{1}, we get the required map \phi_{1}. Inductively continue this process to get the entire chain map. Note that all the maps defined from the injective object property are not unique.

Step 2:Proving that any two such extensions are chain-homotopic

Let f_{n},g_{n} be two chain maps from A \to I_{\bullet} to B \to I'_{\bullet}.

df2

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