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There is a well understood bifibration of $\infty$-categories over the $\infty$-category of commutative ring spectra whose fiber over a ring $R$ is the category of $R$-module spectra. This is in analogy to the usual Grothendieck fibration over commutative rings in discrete algebra. One then can ask which morphisms of commutative ring spectra are of effective descent for modules. In other words, one can ask whether or not the category of $R$-modules is equivalent to the category of descent data for some morphism $R\to S$ (which is constructed as as the homotopical totalization of the Amitsur complex).

Rognes has introduced a notion of faithful morphism of ring spectra which is the following: if $\phi:R\to S$ is a morphism of ring spectra, then it is faithful if for every $R$-module $M$, $M\wedge_RS\simeq \ast$ implies that $M\simeq\ast$. Lurie has also introduced a notion of faithfully flat: the morphism $\phi$ is faithfully flat if $\pi_0(\phi):\pi_0R\to\pi_0S$ is faithfully flat and $\pi_iS\cong \pi_iR\otimes_{\pi_0R}\pi_0 S$. Are either of these sufficient to imply effective decent for modules?

What if we say that morphism is faithful in the sense of Rognes but also a Galois extension for some stably dualizable group $G$ or even a Hopf-Galois extension?

Is there a classification of effective descent morphisms in this setting?

Thanks!

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  • $\begingroup$ You're going to be at MSRI next week, right? I seem to remember seeing that Rognes is giving a talk. Maybe you could ask him, if you don't get an answer here. $\endgroup$ – David White Jan 21 '14 at 23:34
  • $\begingroup$ I won't be there, unfortunately. $\endgroup$ – Jonathan Beardsley Jan 22 '14 at 1:30
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    $\begingroup$ For posterity, I should add that someone just told me that faithful Galois extensions of ring spectra are of effective descent for modules, though I've not yet found a reference. $\endgroup$ – Jonathan Beardsley Jan 22 '14 at 2:03
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    $\begingroup$ Furthermore, that person is Tyler Lawson, and he pointed me towards Brooke Shipley's paper on Morita theory: homepages.math.uic.edu/~bshipley/tilting7.pdf where you want to use the fact that a descent datum is the same thing as an $S\langle G\rangle$-module. $\endgroup$ – Jonathan Beardsley Jan 22 '14 at 18:10
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In DAG7, Theorem 6.1, Lurie proves faithfully flat descent for modules: the functor $A \mapsto \mathrm{Mod}_A$ is a hypercomplete sheaf for the fpqc topology on $E_\infty$-rings. This, in particular, gives effective descent for module spectra along faithfully flat maps.

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  • $\begingroup$ Thanks anonymous. I thought that might be the case, otherwise I don't know why he would call it that. Presumably a weaker condition holds as well. $\endgroup$ – Jonathan Beardsley Jan 22 '14 at 1:31

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