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Construction (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 $d_{i}(x) = 0$ for $1 \leq i \leq n$. Note that if $x$ satisfies this condition, then we have

\[ \partial (x) = \sum _{i = 0}^{n} (-1)^{i} d_ i(x) = d_0(x). \]

Moreover, the identity $d_{i} d_0(x) = d_0 d_{i+1}(x) = 0$ shows that $\partial (x) = d_0$ 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 $\mathrm{C}_{\ast }(A)$.