# Sheaves on the product of algebraic varieties

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.

• For the sheaf of 1-forms you should take direct sum instead of tensor product. – danneks Apr 4 '18 at 5:30
• I edited the question: by $\Omega$ I meant the sheaf of top degree differential forms. Are the other ones correct? – Federico Apr 4 '18 at 8:40
• A structure sheaf is not a $\mathbb C$-module, it is a sheaf of them. For global sections your formula is correct, by the Kunneth theorem. For top-degree forms all is correct. For differential operators I don't know. – danneks Apr 4 '18 at 9:33