# Does miracle flatness always fail for a non-regular base?

In [1], Huybrechts and Mauri argue that a holomorphic Lagrangian fibration $$f: X \to B$$ with smooth base $$B$$ is flat. This is an application of so called miracle flatness [2, Thm 23.1], because Lagrangian fibrations have equidimensional fibers. Then they write

Remark 1.18. Note that the conclusion that $$f$$ is flat really needs the base to be smooth. In fact, by miracle flatness, $$f$$ is flat if and only if $$B$$ is smooth.

I see that smoothness of $$B$$ is needed to apply miracle flatness, but I don't see the converse. Why does flatness necessarily fail if $$B$$ is not smooth?

[1] Huybrechts, Mauri, Lagrangian fibrations, arXiv, 2022

[2] Matsumura, Commutative Ring Theory, 1986

Theorem 23.7. Let $$(A, \mathfrak m, k)$$ and $$(B, \mathfrak n, k')$$ be local Noetherian local rings, and $$A \to B$$ a local homomorphism [...]. We assume that $$B$$ is flat over $$A$$. (i) If $$B$$ is regular then so is $$A$$.
As $$X$$ is always assumed to be smooth, the flatness of $$f$$ will imply the smoothness of the base $$B$$.
• Maybe I am confused, but does this imply that the regularity satisfies fpqc descent for Noetherian ring (I have not heard results of this kind before)? Namely, given a faithfully flat map $R\to S$ of Noetherian rings. If $S$ is regular, then so is $R$.