Chain duality IV (Ex)

Show that for $M$$<wikitex>; Show that for $M$, $N$ in $\Zz_\ast (K)$ we have $$ (M \otimes_{\Zz_\ast (K)} N)_{r} = \sum_{\sigma \in K} \sum_{\sigma \leq \lambda,\mu} (M(\lambda) \otimes_{\Zz} N(\mu))_{r-|\sigma|} $$ </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]M$, $N$$N$ in $\Zz_\ast (K)$$\Zz_\ast (K)$ we have

$\displaystyle (M \otimes_{\Zz_\ast (K)} N)_{r} = \sum_{\sigma \in K} \sum_{\sigma \leq \lambda,\mu} (M(\lambda) \otimes_{\Zz} N(\mu))_{r-|\sigma|}$ $$\displaystyle (M \otimes_{\Zz_\ast (K)} N)_{r} = \sum_{\sigma \in K} \sum_{\sigma \leq \lambda,\mu} (M(\lambda) \otimes_{\Zz} N(\mu))_{r-|\sigma|}$$