Confusion on summand of Hochschild homology of D-modules

Question

I've encountered the following confusion.

Start with the category of perfect D-modules on $\mathbb{A} ^{1}$, which I'll denote $D$. We have the object $\mathcal{O}$, which is exceptional in the sense of having derived endomorphism ring $k$.

I believe that this implies that the subcategory generated by $\mathcal{O}$ is an admissible subcategory and so its Hochschild homology is a summand of that of $D$. On the other hand the Hochschild homology of the category $D$ ought to be the Hochschild homology of the algebra of differential operators on $\mathbb{A} ^{1}$, which has vanishing zero degree component, and thus the confusion.

Evidently I'm missing an assumption somewhere, but I've been unable to locate exactly where and help would be appreciated.

Edit: I should point out that Morita theory implies that the subcategory generated by $\mathcal{O}$ is isomorphic to perfect complexes over $\mathbb{R}End(\mathcal{O})=k$.

Answer 1

I think that if we consider the category perfect modules(which we have to in order to get a non-trivial answer for Hochschild homology) then the inclusion of the subcategory generated by $E$ doesn't have any adjoints. For example, the usual construction of the left adjoint is $M\mapsto RHom(M, E)^{*}\otimes_k E$ but for $M=\mathcal{D}$(free module of rank $1$) we have $$RHom(\mathcal{D},\mathcal{O})=Hom(\mathcal{D},{\mathcal{O}})=\mathcal{O}$$ but $\mathcal{O}$ is not a finite-dimensional vector space. One could probably turn this observation into a rigorous argument by showing that the adjoint, if exists, should be compatible with that on the category of all modules.

Edit: As noticed by EBz and Sam Gunningham in the comments, what follows is incorrect and either of the orthogonal complements to $\mathcal{O}$ in the category of perfect $\mathcal{D}$-modules is non-zero.

[Another way to see that there is no orthogonal decomposition of the form $D=\langle \langle \mathcal{O}\rangle, A\rangle$ is that the right(and also left) orthogonal $\mathcal{O}^{\perp}$ is zero in the category of perfect D-modules. Indeed, if $M$ is an object of the derived category of D-modules with finitely generated cohomology in a bounded range of degrees which is right orthogonal to $\mathcal{O}$, the spectral sequence $$E^{pq}_2=Ext^q(\mathcal{O},H^p(M))\Rightarrow H^{p+q}(RHom(\mathcal{O},M))$$ converges to $0$. Let $p$ be the minimal degree where $H^p(M)\neq 0$. There are no differentials coming in or out of the object $E^{p,0}_r$ for any $r$ so $E^{p,0}_2$ has to vanish for the spectral sequence to be converging to $0$. Since $\mathcal{O}$ is the cokernel of the right multiplication by $\partial_x$ on $\mathcal{D}$, it implies that $\partial_x$ is an automorphism of $H^p(M)$. But since $H^p(M)$ is finitely generated, it has to be $0$ which contradicts with the definition of $p$.]