I am trying to understand how much it is possible to extend the notion of spectrum of a qcoh sheaf of algebras to stacks.

More precisely, given a scheme $S$ and a stack $F$ of cohomological cdga's that are quasi-coherent $\mathcal{O}_S$-modules (with cohomology in non-negative degrees) I would like to define a stack over over S by $(f:Spec(A) \to S) \mapsto Map_{cdga(A)}(f^*F,A)$, generalizing the usual notion of relative spectrum https://stacks.math.columbia.edu/tag/01LQ.

What bothers me is how to prove that the defined presheaf satisfies the hyperdescent condition.

Now, given a quasi-coherent complex $L$ on $S$, Toen defines the linear stack associated to L by $(f:Spec(A) \to S) \mapsto Map_{dg A-mod}(f^*L,A)$, and this is a stack as the quasi-coherent complexes form a stack of infinity categories (this I also don't understand why, and this might be the fist question). Hence, if $F$ is a stack of free cdg $\mathcal{O}_S$-algebras (say $F=Symm_{O_S}(L)$), then my relative spectrum satisfies the hyperdescent just by applying the free functor to the mapping spaces. On the other hand, if the cohomology of F is concentrated in the non-positive degrees, than this relative spectrum should be well defined and give a derived scheme and this is also something I more or less understand (I could use the ringed spaces approach or I think it is also possible to embed nicely $Map_{cdga(A)}(f^*F,O_U)$ into $Map_{A-mod}(f^*F,O_U)$ so that the embedding respects homotopy limits). However, when F has cohomology in non-negative degrees I get lost.

Does anybody know if the functor: $Aff/S\to\infty-cat: Spec(A) \mapsto L(cdga(A))$ is a stack of infty-categories (where affine schemes are non-derived) as then my prestack would be automatically a stack, or even simpler question, how do you prove that quasi-coherent derived categories form a stack (this is mentioned at the beginning of section 3.1 in Toen's derived algebraic geometry), so I can try to do a similar thing for algebras (if possible)


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