For a smooth Fano variety $Z$, let $X$ be the total space of its canonical bundle. Let $\operatorname{Coh}_Z(X)$ be the category of coherent sheaves that support on $Z$ set-theorically. How to show that the bounded derived category $D^b(\operatorname{Coh}_Z(X))$ is a Calabi-Yau category. Many papers mention this result, but I couldn't find any proof. The two problems I have are that there is no Serre duality and no finitely locally free resolution on a non-compact Calabi-Yau, so I cannot analogously apply the proof for Serre duality on smooth projective schemes.
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It is better to consider $D_Z^b(\mathrm{coh}(X))$ --- the full subcategory of the bounded derived category of $X$ formed by objects with cohomology sheaves supported on $Z$. This category does have Serre duality. This is mentioned (without proof), for instance in [Bridgeland, Tom. T-structures on some local Calabi-Yau varieties. J. Algebra 289 (2005), no. 2, 453–483], above Proposition 4.5.
To prove this you can embed $X$ into some smooth and proper variety $\bar{X}$ (for instance, you can use the projectivization of $\mathcal{O}_Z \oplus \omega_Z$), note that $D_Z^b(\mathrm{coh}(X))$ is a full subcategory in $D^b(\mathrm{coh}(\bar{X}))$, and use Serre duality on $\bar{X}$. This works well because the Serre functor of $\bar{X}$ preserves $D_Z^b(\mathrm{coh}(X))$.
