I want to show that for any exact functor $F\colon {}_R\mathrm{Mod}\rightarrow {}_S\mathrm{Mod}$ there is a natural isomorphism $$F\circ H_n \cong H_n \circ \mathbf{Ch}(F)$$ where $\mathbf{Ch}(F)$ denotes the induced map on chain complexes. I have already done this the boring way, writing out an isomorphism and showing commutativity of the appropriate diagrams.
Question: Is there some better (read: computation-free) way of doing this? Specifically, is it possible to realize the isomorphism as a connecting homomorphism in an LES of homology modules (and then just refer to naturality of connecting homomorphisms)?
My efforts: Consider a chain complex $C_\bullet$. We have associated chain complexes
- $Z_\bullet$ = chain complex of cycles with all differentials $0$.
- $B_\bullet$ = chain complex of boundaries with all differentials $0$.
- $H_\bullet$ = chain complex of homology modules (!) with all differentials $0$.
These assemble into an SES of chain complexes: $$ 0 \rightarrow B_\bullet \rightarrow Z_\bullet \rightarrow H_\bullet \rightarrow 0. $$ An easy argument shows that $F$ exact $\Rightarrow \mathbf{Ch}(F)$ exact. Hence we have the following SES of chain complexes in ${}_S\mathrm{Mod}$: $$ 0\rightarrow FB_\bullet \rightarrow FZ_\bullet \rightarrow FH_\bullet \rightarrow 0. $$ The corresponding LES in homology is then (since all differentials are zero): $$ \dots \rightarrow FB_n\rightarrow FZ_n \rightarrow FH_n \xrightarrow \partial FB_{n-1}\rightarrow FZ_{n-1}\rightarrow \dots $$ Here I realize that what I really want is for $\partial$ to go from e.g. $FH_n$ to $H_n(FC_\bullet )$ (or the other way around), not to mention the fact that I want flanking zero maps. In that case, I really want to look at a different SES of chain complexes, probably involving $FC_\bullet$ directly. It also seems that the "right" SES of chain complexes will involve one of the chain complexes already mentioned, but shifted so that degrees match up.
Can someone give me a hint as to the chain complexes I should be looking at?
Update (solved). As suggested by Max in the comments, one can consider the SES of chain complexes $$ 0 \rightarrow Z_\bullet\rightarrow C_\bullet \xrightarrow d B_{\bullet - 1} \rightarrow 0, $$ where $d$ is the differential. Proceeding as above, we now find the following LES in homology (after first applying $F$): $$ \dots \rightarrow H_{n+1} (FC_\bullet )\xrightarrow 0 FB_n \xrightarrow \partial FZ_n \rightarrow H_n(FC_\bullet ) \xrightarrow 0 \dots $$ from which we extract $$ 0 \rightarrow FB_n \xrightarrow \partial FZ_n \rightarrow H_n(FC_\bullet )\rightarrow 0. $$ The conclusion should now follow from naturality of $\partial$.