ScalingStacks

Proof. The result for ordinary (non-derived) schemes $X$ is a theorem of Neeman [N2], extending ideas of Thomason [TT]. (In fact, Neeman proves that for quasi-compact, quasi-separated schemes, $\qc(X)$ is compactly generated, and dualizable and compact objects coincide. We assume $X$ has affine diagonal only because the definition of perfect stack requires it.)

A modified exposition of Neeman’s argument appears in the work of Bondal-Van den Bergh [BV], who in fact prove that $\qc(X)$ is generated by a single perfect object. One can translate the latter proof, which occupies [BV, Section 3.3], directly into the derived setting, substituting Lemma 3.17 above for its underived version [BV, Corollary 3.3.4], and using the natural identification $\qc(\Spec A)\simeq\Mod_{k}$ instead of [BV, Corollary 3.3.5]. (In the derived setting, there is no general notion of the abelian category of quasi-coherent sheaves, so we do not need to worry about the potential distinction between its derived category and the quasi-coherent derived category). In what follows, we sketch the argument for the reader’s convenience, keeping the notation from [BV].

The proof that $\qc(X)$ is generated by a single perfect (dualizable) object is an induction on the number of opens in an affine cover of $X$. The base case of an affine derived scheme is Lemma 3.5. For the inductive step, we write $X=Y\cup U$ with $Y$ open and $U$ affine (putting us in the context of Lemma 3.18), and assume that $\qc(Y)$ has a perfect generator $E$. By [BoN, Proposition 6.1], there is an explicit compact generator $Q$ for the kernel of the restriction from $U$ to the intersection $S=Y\cap U$. (One can think of $Q$ as a form of the structure sheaf of the closed complement $V=U\setminus S$). The key to the inductive step is Neeman’s abstract categorical form [N1, Theorem 2.1] of Thomason’s extension theorem for compact objects. This allows us to extend $E\oplus E[1]|_{S}$ to a compact (hence perfect) object on $U$, and then to glue the latter to $E\oplus E[1]$ to obtain a perfect object $P$ on all of $X$. (Note that we extend $E\oplus E[1]$ rather than $E$ itself since K-theoretic obstructions vanish for the former.) One then checks by a Mayer-Vietoris argument that the sum of $P$ and the pushforward of $Q$ (which is itself compact and perfect by support considerations) to $X$ generates all of $\qc(X)$.

By Lemma 3.17, we know that $\mathcal{O}_{X}$ is compact, and hence that dualizable complexes are compact. The assertion that compact objects are dualizable follows from [N1]: if a set $\mathcal{C}^{\circ}$ of compact objects generates $\mathcal{C}$, then all compact objects of $\mathcal{C}$ are summands of finite colimits of objects of $\mathcal{C}^{\circ}$ and their shifts. Since $\qc(X)$ is generated by a perfect object, we conclude that all compact objects are summands of perfect objects, which allows one to check locally that they are indeed perfect. ∎