Let $\texttt{R}$ be an $\texttt{E}$-infinity ring and let $\texttt{M,N}$ be $\texttt{E}$-infinity modules. Under what conditions do we have


$$ \texttt{[M, N] ≅ [M,R] ⊗ N}$$


Under ordinary circumstances (with modules), we can show that a canonical map is an isomorphism using rank or dimension. What is the easiest approach here?


One application is to the (higher) trace $\mathbb{Tr}$:


$$ \texttt{R → [M, M] → [M,R] ⊗ M → R}.$$



[M,N]≅ [M,R] ⊗ N for E-infinity modules
See page 70 of EKMM. The map $$F\_R(M,N) \wedge\_R F\_R(M',N') \rightarrow F\_R(M \wedge\_R M', N \wedge\_R N')
$$ is an isomorphism if $M,M'$ are strongly dualizable or if $M$ is strongly dualizable and $N=R$. Being strongly dualizable is equivalent to being a wedge summand of a finite cell $R$ module.


