Encouraged by the success in studying finite flat modules, see the preceding post, let’s think a bit about flat, finite type ring extensions.
Question: For which rings R is every finite type flat ring map R —> S of finite presentation?
A Noetherian ring satisfies this property. In the paper by Raynaud and Gruson they prove that this holds if R is a domain. I recently added this result to the stacks project (with a purely algebraic proof), see Algebra, Proposition Tag 053G. If R is a local ring whose maximal ideal is nilpotent then the result is true as well. But I don’t know what happens if the maximal ideal is only assumed to be locally nilpotent, i.e., every element of the maximal ideal is nilpotent, i.e., the maximal ideal is √(0). Do you?
By the way, I still want more ideas about the question I posted here! [Edit: this question has now been answered.]
[Edit on August 23, 2010: As David Rydh points out in a comment below any ring which has finitely many associated primes satisfies the condition. This follows trivially from Raynaud-Gruson Theorem 3.4.6. Don’t know why I did not see this! Anyway, so a local ring whose maximal ideal is locally nilpotent is an example too.]