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 $\mathbb{R}End(\mathcal{O})=k$.

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 $\mathbb{R}End(\mathcal{O})=k$.

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.

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 $\mathbb{R}End(\mathcal{O})=k$.