Over a field, taking homology commutes with the tensor product of two chain complexes by the Künneth formula: $H(C_*\otimes D_*)\cong H(C_*)\otimes H(D_*)$.
Does this extend to arbitrary tensor products of chain complexes, i.e. do we have $H(\otimes_iC^i_*) \cong \otimes_iH(C_*^i)$?