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Remark (Associativity Isomorphisms). Let $(C_{\ast }, \partial )$, $(D_{\ast }, \partial )$, and $(E_{\ast }, \partial )$ be chain complexes of abelian groups. Then there is a unique isomorphism of graded abelian groups

\[ \alpha : C_{\ast } \boxtimes (D_{\ast } \boxtimes E_{\ast } ) \rightarrow (C_{\ast } \boxtimes D_{\ast }) \boxtimes E_{\ast } \]

satisfying the identity $\alpha ( x \boxtimes (y \boxtimes z) ) = (x \boxtimes y) \boxtimes z$. Moreover, $\alpha $ is an isomorphism of chain complexes: this follows from the observation that $\alpha ( \partial (x \boxtimes (y \boxtimes z) ) )$ and $\partial \alpha ( x \boxtimes (y \boxtimes z) )$ are both given by the sum

\[ (\partial x \boxtimes y) \boxtimes z + (-1)^{m} (x \boxtimes \partial y) \boxtimes z + (-1)^{m+n} ((x \boxtimes y) \boxtimes \partial z) \]

for $x \in C_{m}$, $y \in D_{n}$, $z \in E_{p}$.