Let $X$ denote a set. Let $C_{n}(X)$ denote the free abelian group generated by $(n+1)$-tuples of elements of $X$. Define $\partial_n (x_0, x_1, \ldots, x_n) = \sum_{k=0}^n (-1)^k (x_0,x_1, \ldots, \widehat{x_k}, \ldots, x_n).$ It is not difficult to show that $\partial_{n-1}\circ \partial_n =0$. I am reading an author who claims that it is clear that this complex is acyclic. While I believe that it is true that it is acyclic, I don't see that it is clear.
It seems to me that one needs to explicitly find a basis for the kernel and determine elements in the image of the previous differential. I can do so for a small set $X$ and for small values of $n$, but I have yet to see a general pattern in either the kernel or the image.
If it is true in general, then there must be a nice reason for it, but I don't get it yet.