6
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Let $\mathsf{D}^b_h(X)$ be the full subcategory of $\mathsf{D}^b(D_X\textsf{-Mod})$ (the bounded derived category of left $D_X$-modules) consisting of the complexes with holonomic cohomology. The natural tensor product of $D_X$-modules seems to be $M\otimes_{\mathscr{O}_X}N$ with the $D_X$-action given by the Leibniz rule. But this tensor product is such that $f^!$ becomes monoidal, instead of $f^*$ as in most six-functor formalisms.

I wonder if there are natural tensor products and inner homs in $\mathsf{D}^b_h(X)$ satisfying

  1. The tensor-hom adjunction
  2. $\mathsf{D}^b_h(X)$ is symmetric monoidal
  3. $f^*$ is strong symmetric monoidal (that is, $f^*(M\otimes N)\cong f^*M\otimes f^*N$ and $f^* 1\cong 1$)
  4. The projection formula $M\otimes f_!N\cong f_!(f^*M\otimes N)$.
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  • $\begingroup$ What about the *-tensor product? i.e. the *-restriction of the external product of sheaves to the diagonal (or equivalently conjugating by Verdier duality)? see Sam Gunningham's answer here mathoverflow.net/a/115026/582 $\endgroup$ Sep 30, 2021 at 2:30
  • $\begingroup$ Dear @DavidBen-Zvi, I indeed thought about that. That seems a plausible answer but I couldn't verify most of the properties I've said. For example, what is the internal hom in this case? $\endgroup$
    – Gabriel
    Sep 30, 2021 at 4:53

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