# Rlim versus tensor product

Let $$R$$ be a coherent ring, and let $$(M_n)_{ngeq 1}$$ and $$(N_n)_{ngeq 1}$$ be two inverse systems of finitely generated flat $$R$$-modules. If $$R^1 lim M_n=R^1 lim N_n = 0$$, is it true also that $$R^1 lim (M_n otimes_R N_n)=0$$? This seems very reasonable but I’m having trouble seeing it.

A more conceptual question: is there a reasonably general "Kunneth formula" for $$Rlim$$ on inverse systems of $$R$$-modules?