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Let $X$ be an algebraic stack. In Perfect complexes on algebraic stacks (4.1) a perfect complex $P \in \mathsf{D}_{\mathrm qc}(X)$ is defined to be a complex such that, for any smooth morphism $\operatorname{Spec}(A) \to X$ (where $A$ is a commutative ring), the complex $\mathbf{R}\Gamma(X,P_{|\operatorname{Spec}(A)})$ is a perfect complex of $A$-modules.

On the other hand, if $X$ is just a scheme, then a complex $P \in \mathsf{D}_{\mathrm qc}(X)$ is called perfect if it is quasi-isomorphic to a bounded complex of locally free sheaves of finite rank.

Question: if $X$ is a scheme, how can I prove that the two definitions agree? Is there a reference with a proof of this fact?

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  • $\begingroup$ Thomason-Trobaugh? $\endgroup$ – Jason Starr Oct 10 '16 at 17:21
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    $\begingroup$ Perhaps I have misunderstood, but you seem to be asking why the property of being a perfect complex is preserved by smooth pullbacks. Given a bounded complex of vector bundles on a scheme $X$, their pullback under any smooth affine map $Spec(A) \to X$ is a bounded complex of locally free $A$-modules. On the other hand, the property of being a bounded complex of vector bundles can be checked locally (even in the Zariski topology). Thus the two definitions agree. $\endgroup$ – Sam Gunningham Oct 10 '16 at 19:14
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    $\begingroup$ I think at least part of the OP's question comes from the fact that the OP has the wrong definition of perfect complex in general. In general (as Leo Alonso says), a perfect complex is only \emph{locally} (in say the Zariski topology) quasi-iso. to a bounded complex of vector bundles. For "nice" schemes, it's a theorem (probably in Thomason-Trobaugh), that one gets a global quasi-iso. I learned about it from remark 1.7 of arxiv.org/pdf/math/0302304v2.pdf, which should give suitable references. $\endgroup$ – Daniel Pomerleano Oct 11 '16 at 9:52
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A perfect complex is one that it is locally quasi-isomorphic to a bounded complex of locally free sheaves of finite rank. On schemes you use Zariski topology, but it makes sense verbatim on any ringed topos. On stacks, you may use one of the big topologies with all of their drawbacks (and some advantages like obvious functoriality) or a small one, like the lisse-étale site. In this case, the translation of the definition given above is more or less what you say. That on schemes both notions agree is a consequence of flat descent for quasi-coherent sheaves.

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