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!



Perfect chain complexes
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](http://www.math.harvard.edu/~lurie/papers/HA.pdf), 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](http://www.math.harvard.edu/~lurie/papers/SAG-rootfile.pdf).


