Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Proposition 2.5.6.19. Let $A_{\bullet }$ be a simplicial abelian group. Then the composite map $\widetilde{ \mathrm{N} }_{\ast }(A) \hookrightarrow \mathrm{C}_{\ast }(A) \twoheadrightarrow \mathrm{N}_{\ast }(A)$ is an isomorphism of chain complexes. In other words, the Moore complex $\mathrm{C}_{\ast }(A)$ splits as a direct sum of the subcomplex $\widetilde{ \mathrm{N} }_{\ast }(A)$ of Construction 2.5.6.16 and the subcomplex $\mathrm{D}_{\ast }(A)$ of Proposition 2.5.5.6.

Proof. The surjectivity of the composite map $\widetilde{ \mathrm{N} }_{\ast }(A) \hookrightarrow \mathrm{C}_{\ast }(A) \twoheadrightarrow \mathrm{N}_{\ast }(A)$ follows from Lemma 2.5.6.17. Moreover, it follows by induction that the subgroup $\mathrm{D}_{n}(A) \subseteq A_ n$ is generated by the images of the maps

\[ \widetilde{\mathrm{N}}_{m}(A) \hookrightarrow A_{m} \xrightarrow { \alpha ^{\ast } } A_{n} \]

where $\alpha : [n] \twoheadrightarrow [m]$ is a nondecreasing surjection and $m < n$, so that the injectivity of $\rho $ also follows from Lemma 2.5.6.17. $\square$