Let $X, Y$ be smooth algebraic varieties. I am trying to figure out which are the relations between sheaves on $X$,$Y$ and on $Z = X \times Y$. I am particularly interested in the structure sheaves, the sheaves of differential operators and the sheaves of differential forms. I think I have proved the relations below:
$\mathcal{D}_{Z} \cong p_1^{*}\mathcal{D}_X \otimes_{\mathcal{O}_Z} p_2^{*}\mathcal{D}_Y$
$\Omega_{Z} \cong p_1^{*}\Omega_{X} \otimes_{\mathcal{O}_{Z}} p_2^{*}\Omega_{Y}$
$\mathcal{O}_{Z} \cong \mathcal{O}_{X} \otimes_{\mathbb{C}} \mathcal{O}_Y$ as $\mathbb{C}$-modules
Can someone tell me whether I am right or wrong, and point me to some reference?
EDIT: $\Omega_{\bullet}$ is the sheaf of top degree differential forms.