Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$?
If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after replacing $Y$ by $f(X)$, we can assume $f$ is faithfully flat. I'd be happy to understand even the case where $X$ and $Y$ are local:
Suppose $R$ and $S$ are local rings and $R\to S$ is a local homomorphism with $S$ (faithfully) flat over $R$. If $S$ is regular, must $R$ be regular?
Note that I'm not asking if smoothness is "flat local"; there are certainly flat morphisms from singular things to smooth things (e.g. $k[x,y]/(x^2-y^2)$ is flat over $k[x]$). The question is whether there are flat morphisms from smooth schemes which hit singular points.
Best Answer
EGA 0-IV, 17.3.3 has the second claim: if $A \to B$ is a local homomorphism of local noetherian rings, and $B$ is regular and $A$-flat, then $A$ is regular. The strategy is to use the fact that if $B$ is faithfully flat over $A$, then the (global) projective dimension of $A$ is at most the projective dimension of $B$, and Serre's characterization of regular local rings as those with finite global dimension. For instance, suppose that $\mathrm{proj} \dim B = n$, and consider a resolution
$$0 \to M_0 \to M_1 \to \dots \to M_n \to M \to 0$$ of $A$-modules, where all the $M_i$ except possibly $M_0$ are projective. We can assume without loss of generality that everything is finitely generated. Tensoring with $B$ gives a resolution: $$0 \to M_0 \otimes_A B \to M_1 \otimes_A B \to \dots \to M_n \otimes_A B\to M\otimes_A B \to 0$$ where, by the condition on $B$, we find that $M_0 \otimes_A B$ is projective. Thus $M_0$ is projective over $A$.
For the last step, I used the fact that projectivity descends under faithfully flat extensions; in general, this is a theorem of Raynaud-Gruson, but it follows directly if everything is noetherian and finitely generated, since then projectivity is equivalent to flatness.