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 it's Hochschild homology is a summand of that of $D$. On the other hand the hochschild homology of the category $D$ aught 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 $\mathbb{R}End(\mathcal{O})=k$.