Let $k$ be a field, $C, D$ two complexes of $k$ modules and $C \otimes D$ the tensor product of $C$ and $D$ over $k$. ($(C \otimes D)^n = \oplus (C^p \otimes D^q)$ and its coboundary is given by $c \otimes d \mapsto \delta c \otimes d + (-1)^{-p} c \otimes \delta d $. ) Then $h^*(C\otimes D) \cong h^*(C) \otimes h^*(D)$?
I have shown that the Kunneth for two quasi-compact separated schemes over a field and quasi-coherent sheaves follows from this.