Let $A \subseteq B \subseteq C$ be commutative rings such that: $(1)$ $B$ is a projective $A$-module $(2)$ $C$ is a finitely presented $B$-module and $(3)$ $C$ is a flat $A$-module.
Does it follow that $C$ is a projective $A$-module??
Intuitively this kind of scenario might occur when we mange to "normalize" a flat finitely presented algebra, in which case the question asks if the projectiveness of the normalization would imply the projectiveness of the original algebra.
Recall that a module is projective iff it is Mittag-Leffler, flat, and it decomposes as a direct sum of countably generated modules. So the meat of the question is whether the hypotheses are enough to transfer the property of being Mittag-Leffler as an $A$-module from $B$ to $C$.
Being Mittag-Leffler is, as far as I know, not a transitive property.
On the other hand, being finitely presented is a transitive property, so in case $B$ is finitely generated the conclusion is immediately yes.
