TITLE: Definition of exact split complex
QUESTION [6 upvotes]: I am reading "an introduction to homological algebra by Charles A.Weibel" and the author deifnes split exact complex to be exact complex $\{C_n,d_n:C_n\to C_{n-1}\}$ such that there exists a sequence of maps $s_n:C_n\to C_{n+1}$ such that $d_{n}s_{n-1}d_{n}=d_{n}.$ He then claims that this is equivalent to the condition $ds+sd=id$. I can see why $ds+sd=id$ implies that the complex is split exact. I tried to prove the other direction but I dont see why it is true. So I Tried to write $C_n=Ker(sd)+Ker(sd-Id)$ which holds because $(sd)^2=sd$ then if $x\in Ker(sd)$ then I can show that $sdx+dsx=x$ but if $x\in Ker(sd-id)$ I cant prove that $dsx=0$. I would appreciate any help.

REPLY [5 votes]: The thing is, you can't keep the same $s$ for the null-homotopy.
The maps $s_n$ are built thanks to the splitting but are not the splitting $s$ itself.