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?

• 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. – user76758 Dec 27 '13 at 6:07
• 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). – user76758 Dec 27 '13 at 20:06

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.

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.)

• I think this is fine as a stand-alone 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.) – Todd Trimble Dec 30 '13 at 2:11