Construction 2.5.5.1 (00QA): The Moore Complex

Construction 2.5.5.1 (The Moore Complex). Let $A_{\bullet }$ be a semisimplicial abelian group (Definition 1.1.1.2). For each $n \geq 1$, we define a group homomorphism $\partial : A_{n} \rightarrow A_{n-1}$ by the formula

\[ \partial (\sigma ) = \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ), \]

where $d^{n}_{i}: A_{n} \rightarrow A_{n-1}$ is the $i$th face operator (Construction 1.1.1.4). For $n \geq 2$ and $\sigma \in A_{n}$, we compute

\begin{eqnarray*} \partial ^2( \sigma ) & = & \partial ( \sum _{i = 0}^{n} (-1)^{i} d^{n}_ i(\sigma ) ) \\ & = & \sum _{i = 0}^{n} \sum _{j = 0}^{n-1} (-1)^{i+j} (d^{n-1}_{j} d^{n}_ i)(\sigma ) \\ & = & 0 \end{eqnarray*}

where the final equality follows from the identity $d^{n-1}_{i} \circ d^{n}_{j} = d^{n-1}_{j-1} \circ d^{n}_{i}$ for $0 \leq i < j \leq n$ (see Remark 1.1.1.7). We let $\mathrm{C}_{\ast }(A)$ denote the chain complex of abelian groups given by

\[ \mathrm{C}_{n}(A) = \begin{cases} A_{n} & \text{ if } n \geq 0 \\ 0 & \text{otherwise,} \end{cases} \]

where the differential is given by $\partial $. We will refer to $\mathrm{C}_{\ast }(A)$ as the Moore complex of the semisimplicial abelian group $A_{\bullet }$.

If $A_{\bullet }$ is a simplicial abelian group, we let $\mathrm{C}_{\ast }(A)$ denote the Moore complex of the semisimplicial abelian group underlying $A_{\bullet }$ (Remark 1.1.1.3).