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According to the usual definition, an induced D-module on a complex manifold $X$ is a right D-module of the form $L \otimes_{\mathscr{O}_X} \mathscr{D}_X$, for $L$ a coherent sheaf of $\mathscr{O}_X$-modules. Moreover, Saito writes that "induced D-modules are naturally defined as right D-modules" (in his paper Induced D-modules and differential complexes), and one finds similar comments elsewhere.

Why are induced D-modules naturally right D-modules? In other words, what is wrong with considering left D-modules of the form $\mathscr{D}_X \otimes_{\mathscr{O}_X} L$ and calling these "induced"?

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I don't know whether thats the reason, but $f_*$ is more naturally defined for right modules and I think $f_*$ commutes with induction for right modules? –  Jan Weidner Apr 5 '12 at 9:33

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