The exterior tensor product of sheaves of modules is defined as:

$M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N \cong \mathcal{O}_{X \times Y} \otimes_{p_1^{-1}\mathcal{O}_X \otimes_{\mathbb{C}_{X \times Y}}p_2^{-1}\mathcal{O}_Y} \left( p_1^{-1}M \otimes_{\mathbb{C}_{X \times Y}} p_2^{-1}N \right) $

The exterior product of $D$-modules is defined as

$M \boxtimes N = \mathcal{D}_{X \times Y} \otimes_{p_1^{-1}\mathcal{D}_X \otimes_{\mathbb{C}_{X \times Y}} p_2^{-1}\mathcal{D}_Y} \left( p_1^{-1}M \otimes_{\mathbb{C}_{X \times Y}} p_2^{-1}N \right) $

To me it seems that this definition doesn’t agree with

$M \boxtimes N = p_1^{*}M \otimes_{\mathcal{O}_{X \times Y}} p_2^{*}N $

where the inverse image and the tensor product are the ones defined for $\mathcal{D}$-modules, and which seems the most natural to me. Are they isomorphic or is there a different reason to give that definition in spite of the second one? Moreover I would be grateful if you can provide any reference for the notion of exterior tensor product, both for sheaves and $D$-modules.


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