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In these notes one can read on page $5$: "it may seem surprising, but one cannot define left $D$-modules for non-smooth schemes". Could someone elaborate on it?

If I'm not mistaken, to define $D$-modules on a singular scheme $Z$, we embed $Z$ into a smooth scheme $Z$ and then define the $D$-modules on $Z$ to be the full subcategory of $D$-modules on $X$ with support in $Z$. Why this definition makes sense for right $D$-modules but not left $D$-modules?

Gabriel
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curious
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  • crossposted on MO : https://mathoverflow.net/questions/382461/left-d-modules-for-singular-varieties – curious Jan 28 '21 at 17:28
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    I would like to remark that the crosspost doesn't seem to exist anymore. – Gabriel Feb 21 '23 at 17:52
  • I am confused by the statement since neither talking about left- nor right-modules is meaningful after restriction to a singular scheme. Perhaps the author is just trying to say that the correspondence used to motivate the definition naturally applies to right-modules, locally, when the thing we restrict to is smooth? – Ben Feb 28 '23 at 09:08
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    Dear @Ben, I agree that the notion of left / right-module is meaningless in the singular case, since all that we have is the category of D-modules (up to equivalence). Now, I don't really understand the second part of your comment. By "correspondence" do you mean Kashiwara's equivalence? If so, it surely applies naturally to left D-modules. – Gabriel Feb 28 '23 at 13:03
  • Dear @Gabriel, yeah, I guess you're right, Kashiwara is no less natural for left-modules. I guess I tried too hard to make sense of the statement. – Ben Feb 28 '23 at 17:18

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