TITLE: Pure morphisms which are not faithfully flat
QUESTION [7 upvotes]: 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. Is there a specific example that drove this characterization? Is there a good family of examples of morphisms which are pure but not faithfully flat?

REPLY [6 votes]: Kiran Kedlaya wrote up a detailed exposition of the result you mention. His write-up also describes a bit of the history, mentioning Olivier as one of the first to state it. It is available as Section Tag 08WE in the Stacks project.  I think this result as part of the research in the late 60's and early 70's done by a group of mathematicians, including Raynaud, Ferrand, Lazard, Olivier, and others, around flatness, descent, etc. (Sorry, don't have the points to make this a comment.)