Six functor formalism for quasi-coherent $D$-modules

Question

Let $X$ be a smooth scheme over a field $k$ and let $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$ be the full subcategory of $\mathsf{D}(\mathcal{D}_X\mathsf{-Mod})$ composed of the complexes of $\mathcal{D}_X$-modules with quasi-coherent cohomology.

If $f:X\to Y$ is a morphism between such schemes, we have natural functors $$f_*:\mathsf{D}_{\text{qc}}(\mathcal{D}_X)\leftrightarrows \mathsf{D}_{\text{qc}}(\mathcal{D}_Y):f^!.$$

If we restrict to the subcategory of complexes with holonomic cohomology, then the Verdier duality functor allows us to find left adjoints $f^*$ and $f_!$ of $f_*$ and $f^!$, respectively.

I wonder if those adjoints already exist in $\mathsf{D}_{\text{qc}}(\mathcal{D}_X)$.

Perhaps we can use Brown representability for left adjoints (as in Neeman's book about triangulated categories) or some other adjoint functor theorem...

In Neeman's paper The Grothendieck Duality Theorem via Bousfield's Techniques, it is proven that $f_*$ has a right adjoint (even though in the text it is said that it's a left adjoint). Perhaps we can use this functor to construct our left adjoints?

Answer 1

They do not exist in general. The simplest example is maybe to take $X\rightarrow Y$ to be the closed embedding of the origin inside $\mathbb{A}^1.$ Then $f_*$ sends a vector space $V$ to the $\mathcal{D}_{\mathbb{A}^1}$-module $V\otimes\delta_0$, where $\delta_0$ is the irreducible $\mathcal{D}_{\mathbb{A}^1}$-module set-theoretically supported at $0$. The point is that because $\delta_0$ is infinite-dimensional as a vector space, this functor does not commute with products and hence cannot have a left adjoint.