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?

4$\begingroup$ Let's first recall what "pure" means: injective after tensoring against any module over the source ring. So a ring map with a section, or more generally one that acquires a section after a faithfully flat base change, is pure. Such maps are often not flat, since there is no condition on the augmentation ideal. $\endgroup$– user76758Dec 27, 2013 at 6:07

$\begingroup$ It is perhaps also worth noting that there is a distinct concept called "$A$pure" for modules over $A$algebras (under suitable finiteness hypotheses) for a general ring $A$ in the 1971 Inventiones paper of Raynaud and Gruson on flatness criteria (see 3.3.3 of part I of that paper). $\endgroup$– user76758Dec 27, 2013 at 20:06
2 Answers
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, $k[x,y]/\langle y^2  x^3x^2\rangle$ 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.
Kiran Kedlaya wrote up a detailed exposition of the result you mention. His writeup 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.)

2$\begingroup$ I think this is fine as a standalone answer (as opposed to a comment), answer_bot. Incidentally, your reputation of 1 probably reflects the fact that you have created separate accounts which should be merged into one as a registered user. (Edit: I have just filled out a form asking for merger of your 4 separate accounts. This should take place soon.) $\endgroup$– Todd Trimble ♦Dec 30, 2013 at 2:11