Question:
The i'th left derived functor $L_iF$ of an additive functor $F:\mathcal{A\to B}$ is defined to be $H_iFP$,where $P$ is taking the projective resolution of $A$(and drop $A$),$F$ acts degreewise and $H_i$ taking the i'th homology.
If we have a short exact sequence $0\to A\to B\to C\to 0$ in $\mathcal{A}$,then $0\to P(A)\to P(B)\to P(C)\to 0$ is split exact in $\mathcal{K_{\geqslant 0}(Proj(A))}$,which denotes the homotopy category of projectives in $\mathcal{A}$.
Since $F$ is additive,it preserves split exact(degreewise).So $0\to FP(A)\to FP(B)\to FP(C)\to 0$ is split exact.
The above works well,but I've got trouble with the following process:
1)Since $0\to FP(A)\to FP(B)\to FP(C)\to 0$ is exact,we apply the homology long exact sequence theorem and get a long exact sequence:$...\to H_i(A)\to H_i(B)\to H_i(C)\to H_{i-1}(A)\to ...$
2)But we know that $H_i$ is an additive functor from $\mathcal{Ch(A)}$ to $\mathcal{A}$,so it preserves direct sum of complexes,which is defined degreewise.Therefore,$H_i(B)\simeq H_i(A\oplus C)\simeq H_i(A)\oplus H_i(C)$,but this will force $H_{i-1}(A)=0$(or to say we will get a short exact sequence for each $H_i$),a contradiction to 1).
I wonder which step fails in 2).Thanks in advance!