TITLE: Flatness of $\Omega_{B/K}$ over $B$.
QUESTION [2 upvotes]: Let $K$ be a field of characteristic zero. Assume that $K \subset A \subseteq B$ are noetherian integral domains, with $A$ regular (= all its localizations at maximal ideals are regular local rings). Assume also that $B$ is a flat $A$-module.
I do not mind to further assume integrality of $B$ over $A$ if that helps-- in that case $B$ is faithfully flat over $A$ (since integral+flat implies faithfully flat).
My question: Is it possible to show that $B$ is regular? I suspect there exists a counterexample.
Corollary on page 201 of this paper 
says that $B$ is regular iff $\Omega_{B/K}$ is flat over $B$.

I wonder if there is a connection between $A$-flatness of $\Omega_{A/K}$ (= which is known, since $A$ is regular) and $B$-flatness of $\Omega_{B/K}$.

Recall: $\Omega_{B/K}$ is the $B$-module of $K$-differentials = $\ker g/(\ker g)^2$, where $g: B\otimes_K B \to B$ is defined by $g(b_1 \otimes_K b_2)=b_1b_2$.
If I am not wrong, the converse of what I ask is true, namely: regularity of $B$ implies regularity of $A$, by Proposition 8, page 59 of Bourbaki's book "Commutative Algebra", chapter 10.

REPLY [3 votes]: I don't think this is necessarily true.
Let $A=K[t]$ and $B=K[x,y,t]/(xy-t^2)$. Then $B$ is flat over $A$ (this is because the Hilbert polynomial is constant in this family). Of course they are flat as $K$-modules as well, since $K$-modules are vector spaces, and vector spaces are free.
Then all the conditions are satisfied, but $B$ is singular at the origin. 
-- 
Here's another example, which satisfies all conditions, including integral and faithfully flat, thanks to Jake Levinson's comment below:
$$k[x] \to k[x,t]/(x^2-t^3)$$
--
Read the below musings with skepticism:
I don't know if B is regular if we further impose integrality or faithful flatness. I suspect it is true, however, for the following reason: a ring being integral (and finite type) over another ring says that the corresponding maps of varities are covering maps, and since covering maps are local diffeomorphisms, smoothness of A should imply smoothness of B.
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