Let $X,Y$ be (smooth, say) spaces. Is it true that $$\Omega^r_{X\times Y}=\bigoplus_{i+j=r}\Omega^i_X\otimes_{\mathcal{O}_{X\times Y}}\Omega^j_Y\,\,?$$
This seems rather strong in general, but I am certain that this is true when $Y=|\Delta_p|$ is the smooth $p$-simplex. However, I cannot find either a proof of the general statement or a proof of this specific case. (It's annoyingly tricky to find references about forms on product spaces, since the moment you have the keywords 'product' and 'forms' you invariably find only references for products of forms.)