composing with a non-flat morphism

Question

Let $X,Y,Z$ be integral schemes and $f:X\to Y$, $g:Y\to Z$ be morphisms such that $g$ is flat and $f$ is not flat. Does this imply that the composition $g\circ f:X\to Z$ is necessarily non-flat?

[EDIT: ykm's example works but I'm also interested to know the answer in the case where $X,Y,Z$ have the same dimension].

Whoops, I didn't see the word integral. – Hoot Jul 17 '15 at 03:43 — Hoot, Jul 17 '15 at 03:43

score 3 · Accepted Answer · answered Jul 17 '15 at 04:15

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No. Take $Z=Spec k$ where $k$ is a field. Everything is flat over a field.

answered Jul 17 '15 at 04:15

ykm

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Thanks! what if I take $X,Y,Z$ to have the same dimension? – adrido Jul 17 '15 at 04:29
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@adrido I don't think that will change the answer. In general, if $g \circ f$ is flat and $g$ is unramified, then $f$ is flat. Requiring $g$ to be flat does not warrant this conclusion. – c_c_chaos Jul 17 '15 at 14:15
@c_c_chaos: thanks! – adrido Jul 17 '15 at 15:25

composing with a non-flat morphism

1 Answers1