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?



Pure morphisms which are not faithfully flat
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.


1. Normal Toric singularities (or seminormal toric singularities are ok too).
2. $k[x,y,z]/\langle x^3 + y^3 + z^3 \rangle$ if $\text{char } k = 1 \text{ mod } 3$. More generally, normal affine cones over ordinary Abelian varieties.
3. Nodes (ie, $k[x,y]/\langle y^2 - x^3-x^2\rangle$ in characteristic $\neq 2$)
4. Singularities that show up on Schubert varities (and various generalizations).
5. Direct summands of regular rings (including many quotient singularities).
6. And many more.


