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Let $X,Y$ be smooth algebraic varieties over $\mathbb{C}$, $M^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_X)$, $N^\bullet\in\mathsf{D}^b_{\text{coh}}(\mathcal{D}_Y)$, and denote by $\mathrm{D}_X$ the Verdier duality functor over $X$.

I wonder if it's true that $\mathrm{D}_{X\times Y}(M^\bullet\boxtimes N^\bullet)=\mathrm{D}_X(M^\bullet)\boxtimes \mathrm{D}_Y(N^\bullet)$.

This is true for constructible sheaves, so Riemann-Hilbert implies it at least for regular holonomic modules. (I can explain it if someone wants.) But there should be a direct proof in this generality.

(I've put complex geometry as a tag because the same result should hold for D-modules over complex varieties as well and, I hope, with the same proof.)

Gabriel
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  • By the Verdier dual of a $\mathcal{D}$-mod $M\in \mathrm{D}^b_{\mathrm{coh}}(\mathcal{D}X)$ do you mean $\mathrm{D}{X}M = \mathrm{RHom}{\mathcal{D}_X}(M, \mathcal{D}_X)\otimes{\mathcal{O}_X} \omega_X^{-1}[\mathrm{dim}(X)]$ or something else? – MPos Feb 17 '24 at 22:27

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