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I'm reading a paper on algebraic stacks and in some part is stated the following:

Let $X$ be an algebraic stack and let $P\in D_{qc}(X)$ be a perfect complex. Then, for every $x\in |X|$, there exists a flat morphism of locally of finite presentation $f: \mbox{Spec}(A)\rightarrow X$ with image containing $x$ such that $R\Gamma(\mbox{Spec}(A), Lf^ ∗_{qc}P)$ is a strictly perfect complex of $A$-modules. In particular, every perfect complex on an affine scheme is strictly perfect.

My question is: How is deduced from this that every perfect complex on an affine scheme is strictly perfect?

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  • $\begingroup$ No, it is used. A perfect complex on an affine scheme is strictly perfect: just take a projective resolution. It is easy to see that it may be taken bounded so "is" strictly perfect in the derived category. $\endgroup$
    – Leo Alonso
    Commented Nov 11, 2019 at 20:34

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