Construction 2.5.6.15 (The Normalized Moore Complex: Second Construction). Let $A_{\bullet }$ be a simplicial abelian group. For each $n \geq 0$, we let $\widetilde{ \mathrm{N} }_{n}( A)$ denote the subgroup of $\mathrm{C}_{n}(A) = A_ n$ consisting of those elements $x$ which satisfy $d^{n}_{i}(x) = 0$ for $1 \leq i \leq n$. Note that if $x$ satisfies this condition, then we have
\[ \partial (x) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(x) = d^{n}_0(x). \]
Moreover, the identity $d^{n-1}_{i} d^{n}_0(x) = d^{n-1}_0 d^{n}_{i+1}(x) = 0$ shows that $\partial (x) = d^{n}_0(x)$ belongs to the subgroup $\widetilde{ \mathrm{N} }_{n-1}(A) \subseteq \mathrm{C}_{n-1} = A_{n-1}$. We can therefore regard $\widetilde{N}_{\ast }( A)$ as a subcomplex of the Moore complex $\mathrm{C}_{\ast }(A)$.