Take the tour ×  All rings are assumed to be commutative and
unital, with all homomorphisms unital as well.  On last week's
homework, there was a mistake in one of the questions:  (2.5)
Let $R\to S$ be a morphism of commutative rings giving $S$ an
$R$-algebra structure. Suppose that the induced maps $R\to
S_{\mathfrak{p}}$ are formally étale for all prime ideals
$\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.
According to our professor, the exercise should have stated
additionally that $S$ was finitely presented over $R$ (which
allows us to prove that $S$ is in fact étale over $R$ rather
than just formally étale). This is not too hard to do and is
left as an exercise. (You can also find it in EGA).  However,
I'm interested in seeing either a counterexample or a proof for
the stronger claim in the grey box.  share|improve this question
Why not post this as an answer to Giant Laser Cannons question?
Just to get reputation? –  Steven Gubkin Mar 10 '10 at 22:02 I'm
sure Bjorn Poonen would have been ame