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Joyal and Tierney proved that morphisms of rings which are of effective descent are exactly those morphisms $\phi:R\to S$ such that $\phi$ presents $S$ as a pure $R$-module. … Grothendieck had originally shown that being faithfully flat implied being of effective descent, but had not entirely characterized such morphisms. …

3) Instead of starting out with a sheaf on $Y$, can I start off with an effective descent datum for $\phi$ and determine all twisted forms of that descent datum? … How is their canonical descent datum (coming from pulling back) related to the descent datum on $F''$? 4) What can I do/say if $\phi$ is not faithfully flat? …

$B\otimes_AB$ also fits into the Amitsur complex for $\phi$, which lets us compute the set of descent data cohomologically. … They then go on to show that flat connections are equivalent to descent data for the map $\phi$. …

The question is essentially the title. In other words, is there some universal property that the Amitsur complex for a morphism of rings $\phi:A\to B$ satisfies as a cosimplicial ring, or cosimplicial …

It's known that a pure morphism of commutative rings $\phi:A\to B$ is of effective descent for the stack of modules. … Presumably there are more maps which are of effective descent for $Mod^{fin}(-)$ than there are for $Mod(-)$. Is there anywhere that this question is addressed or solved? …

One then can ask which morphisms of commutative ring spectra are of effective descent for modules. … Is there a classification of effective descent morphisms in this setting? Thanks! …

Descent cohomology for a comonad is defined at degrees 0 and 1 by Mesablishvili in his paper "On Descent Cohomology" (as well as by many other authors in many other contexts). … The 1st descent cohomology is the pointed set of isomorphism classes of $\bot$-comodule structures on $b$, if I'm reading him correctly. …

For a homomorphism of commutative rings $f:R\to S$, there are at least two notions of a descent datum for this map. … I have written down some vague things about how these two are the same, but is there a functorial equivalence between the two categories of descent data? …