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In SAG Remark 6.3.3.8, Lurie asserts that if we have a representable (by Spectral Deligne-Mumford stacks) natural transformation $X\to Y$ where $Y$ is a functor satisfying fpqc descent, then so too does $X$.

In particular, if we let $Y$ be the representable functor $\operatorname{Spec}(\mathbb{S})$ the sphere spectrum, this functor of points satisfies fpqc descent, so this statement would imply that the functor of points of any spectral Deligne-Mumford stack also satisfies fpqc descent.

However, later on, in the paragraph immediately following Remark 9.1.4.2, he says that the functor of points $h_X$ of a spectral Deligne-Mumford stack need not satisfy fpqc descent.

Is this a mistake? Is it true, at any rate, that such a functor at least always satisfies fppf descent (where fppf means 'faithfully flat and locally almost of finite presentation')? Do we need to require that the functor is representable by a relative Deligne-Mumford n-stack for n finite?

(It seems like at least classically, this also isn't true.)

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  • $\begingroup$ I proved something weaker today. I don't know if it's the strongest thing you can say, but I showed that if $\mathcal{X}$ is a spectral Deligne-Mumford stack whose underlying $\infty$-topos is a bounded $\infty$-topos in the sense of SAG Appendix A, then $\mathcal{X}$ satisfies fppf descent. The proof goes by showing that the functor sending a spectral DM stack to its category of truncated étale sheaves is a stack for the fppf topology (and this goes through Toën's theorem for derived algebraic stacks, exploiting the nil-invariance of this functor to reduce to the case of ordinary rings). $\endgroup$
    – Harry Gindi
    Commented Sep 11, 2020 at 17:08

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