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I saw the question When is an almost geometric quotient flat? which said

"The quotient $\pi$ is flat if and only if $\pi$ is equidimensional and $X$ is smooth".

I am curious is there an example that $\pi$ is not equidimensional. Also where can I see the original reference for the statement?

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    $\begingroup$ As explained in the question you quote, the case where $\pi$ is equidimensional is actually quite rare. The paper Representations with a free module of covariants by V. Popov (Functional Anal. Appi. 10 (1976), 242-243) gives a complete list of these cases for irreducible representations of an almost simple, simply connected group; just take any representation which is not in that list. $\endgroup$
    – abx
    Commented May 5, 2022 at 4:15
  • $\begingroup$ @abx Thank you for the detail, because I am a beginner on this subject, I am not familiar with notations and concepts of the paper. Could you show me an easy examples? Thanks in advance! $\endgroup$
    – Mary Susy
    Commented May 5, 2022 at 9:07
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    $\begingroup$ Consider the action of the multiplicative group $\mathbb{G}_m =\text{Spec}\ k[t,t^{-1}]$ on $\mathbb{A}^3=\text{Spec}\ k[x,y,z]$ by $t\cdot(x,y,z) = (tx,ty,t^{-1}z)$. The categorical quotient is given by the morphism of affine $k$-schemes associated to the inclusion of rings $k[u,v]\hookrightarrow k[x,y,z]$ by $u\mapsto xz$ and $v\mapsto yz$. This morphism is flat of relative dimension $1$ over the complement of the origin in $\text{Spec}\ k[u,v]$, but the fiber over the origin has a component of dimension $2$. $\endgroup$
    – Jason Starr
    Commented May 5, 2022 at 13:14

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