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In Thomason-Trobaugh in Remark 2.4.4 it is written: "On a general scheme, the perfect complexes are locally finitely presented objects in the "homotopy stack" of derived categories."

I was wondering if someone here could explain this a little bit to me (i.e. roughly has the knowledge of chapters II and III of Hartshorne + knows the definition of stack). In particular I would like to know what is meant by "homotopy stack". Thanks!

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In modern language, one would say that $D_{qcoh}(-)$ is a sheaf of $(\infty,1)$-categories on the scheme $X$ (so "homotopy stack" = "sheaf of $(\infty,1)$-categories").

If $X$ is affine, or more generally has an ample family of line bundles, the perfect complexes on $X$ are exactly the finitely presented objects (aka compact objects) in $D_{qcoh}(X)$: this is what Thomason proves. Nowadays, this is sometimes taken as the definition of perfect complex in the affine case, see for instance: Lurie, Higher Algebra, section 7.2.4.

Thomason's assertion can be rephrased by saying that the sub-presheaf $D_{perf}(-) \subset D_{qcoh}(-)$ is in fact a sheaf. Equivalently, if $F\in D_{qcoh}(X)$ is such that $F|U_i$ is perfect for every $U_i$ in some open covering of $X$, then $F$ is perfect. This is obvious with Thomason's definition of "perfect". In particular, $F$ is perfect iff $F|U\in D_{qcoh}(U)$ is finitely presented for every affine $U\subset X$.

Another point of view on perfect complexes is that they are precisely the dualizable objects in the symmetric monoidal $(\infty,1)$-category $D_{qcoh}(X)$. One way to prove this is to check it for $X$ affine and observe that dualizable objects also form a subsheaf (for categorical reasons: the functor that sends a symmetric monoidal $(\infty,1)$-category to its subcategory of dualizable objects is limit-preserving). A reference for this is Proposition 6.2.6.2 in Lurie's book Spectral Algebraic Geometry.

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  • $\begingroup$ Thanks Marc! I found this answer very enlightening. Also, thank you for the references! :) $\endgroup$ – Anette Sep 8 '16 at 2:24

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