ScalingStacks

Proof. By Lemma 3.20, we know that compact and dualizable objects in $\qc(X)$ coincide. Thus we need only to check that $\qc(X)$ is compactly generated. The pullbacks $p^{*}M$ of compact (hence dualizable) objects on $Y$ are dualizable, hence compact, as are the line bundles $\mathcal{L}$ in the given relatively ample family. We claim the $\infty$-category of compact objects $p^{*}M\otimes\mathcal{L}$ generates $\qc(X)$. The argument is as above in the case of an external product: let $N$ be right orthogonal to the compact objects, so that $\Hom(p^{*}M\otimes\mathcal{L},N)=0$. We first find by adjunction and the fact that $Y$ is compactly generated that $p_{*}{\mathcal{H}om}(\mathcal{L},N)=0$. Since the objects $\mathcal{L}$ form a relatively ample family of line bundles this forces $N=0$. ∎