Pure morphisms which are not faithfully flat 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?
 A: 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.)
A: While this is already answered in the comments, let me give you a large class of examples.
A ring $R$ of characteristic $p > 0$ is called $F$-pure if the Frobenius map $F : R \to R$ is a pure morphism.  
On the other hand, by a theorem of Kunz, a ring of characteristic $p > 0$ is regular if and only if the Frobenius morphism is flat.  
$F$-pure rings have something like log canonical singularities, in particular, they can have singularities.
Examples: What follows are examples of $F$-pure by not regular rings, so for each the Frobenius map is pure but not flat.


*

*Normal Toric singularities (or seminormal toric singularities are ok too).

*$k[x,y,z]/\langle x^3 + y^3 + z^3 \rangle$ if $\text{char } k = 1 \text{ mod } 3$.  More generally, normal affine cones over ordinary Abelian varieties.

*Nodes (ie, $k[x,y]/\langle y^2 - x^3-x^2\rangle$ in characteristic $\neq 2$)

*Singularities that show up on Schubert varities (and various generalizations).

*Direct summands of regular rings (including many quotient singularities).

*And many more.

