How to exchange left and right $\mathcal{D}_X$-modules when $X$ is not smooth?

Question

When $X$ is a smooth scheme (over something of characteristic $0$), one can exchange left and right $\mathcal{D}_X$-modules ($\mathcal{D}_X$ means the sheaf of differential operators) by tensor with (resp. putting Hom from) $\omega_X$, the canonical sheaf of $X$. However, as I know, to do this, one need

the right $\mathcal{D}_X$-module structure on the tensor product $R\otimes_{\mathcal{O}} L$ whenever $R$ is a right $\mathcal{D}_X$-module and $L$ is a left $\mathcal{D}_X$-module, and the left $\mathcal{D}_X$-module strucuture of $\mathcal{H}om_{\mathcal{O}}(R,S)$ whenever both $R$ and $S$ are right $\mathcal{D}_X$-modules.

However, I don't know how to do this in the non-smooth situation, because in general $\mathcal{D}_X$ is not generated by $\mathcal{O}_X$ and derivations on $X$.

Question: Do those structures in the above box still exist even when $\mathcal{D}_X$ is not generated by $\mathcal{O}_X$ and derivations? If so, how to define them? If not, can we still obtain an equivalence between the category of left $\mathcal{D}_X$-modules and the category of right $\mathcal{D}_X$-modules?

Answer 1

Even in the non-smooth situation, left and right $D$-modules are (derived) equivalent by tensoring by the dualizing complex. The real issue is in defining "$D$-modules" on a smooth variety. It is not the category of modules over the ring of differential operators on $X$ unless $X$ is smooth.

In the classical approach to the subject, $D$-modules on a singular variety $X$ are defined by choosing an embedding $X \to Y$ into a smooth variety $Y$, and then defining the category of $D$-modules on $X$ to be those $D$-modules on $Y$ supported on $X$. This is well-defined thanks to Kashiwara's lemma. Then the (derived) equivalence between left and right $D$-modules is given by tensoring with the dualizing complex of $X$.

A more high-tech approach, suitable not just for singular varieties but also for (higher) stacks, is contained in "Crystals and D-modules" by Gaitsgory and Rozenblyum. They state in §0.4 that left and right $D$-modules are equivalent via tensoring by the dualizing complex.