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In this MSE answer adapted from this answer the infinitesimal extension $\Bbbk[\varepsilon]\to \Bbbk$ is given as an example of an open morphism which is not flat.

  1. Are all open non-flat morphisms "of this kind" i.e fail because of square-zero ideals somewhere? Why?
  2. If not, what are some geometric examples of open maps non-flat morphisms of a different nature?
  3. What are some criteria making open morphisms flat apart from 'being an immersion'?
Pedro
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Arrow
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1 Answers1

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Here is one result, which answers your Question 3:

Theorem [EGA IV$_{3}$, Cor. 15.2.3]. Let $Y$ be a locally noetherian scheme, and let $f \colon X \to Y$ be a morphism locally of finite type. Let $x \in X$ be a point with image $y = f(x)$. Suppose the following conditions hold:

  1. the morphism $f$ is universally open at the generic point of every irreducible component of $f^{-1}(y)$ containing $x$;
  2. the fiber $f^{-1}(y)$ is geometrically reduced (over $\kappa(y)$) at $x$; and
  3. the local ring $\mathcal{O}_{Y,y}$ is reduced.

Then, $f$ is flat at the point $x$.

This says that universally open morphisms that are not flat must have some non-reducedness somewhere, either in the fibers or on the base, thereby answering your Question 1, at least if you strengthen "open" to "universally open." There are surely examples of open but not universally open morphisms, too, though.

Arrow asked two more questions in the comments.

  1. My intuition for a non-reduced base is that for the morphism $f$ to be flat, the morphism should look like a restriction of a nice family on a larger base space. So you can't expect flatness to be captured purely topologically. Depending on your taste, non-reduced fibers are less geometric, since this won't happen if $X$ is smooth and everything is over a field of characteristic $0$ by generic smoothness.
  2. Again depending on your taste, open but not universally open morphisms are less geometric, since they cannot happen if the base is reduced and normal [EGA IV$_{3}$, Cor. 14.4.9]. I find this MathOverflow question insightful. In particular, it mentions an example of an open morphism which is not universally open, hence not flat; there is another example [EGA IV$_{3}$, Rem. 14.3.9(i)] in the comments.
Takumi Murayama
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  • Your very interesting answer has inspired two follow up questions: 1. what is your geometric intuition behind the "necessity" of non-reducedness somewhere? 2. What is your geometric intuition for the (seemingly non-topological) notion of universally open morphisms? – Arrow Jan 01 '18 at 11:32
  • @Arrow I addressed your extra questions. Hopefully it helps somewhat! – Takumi Murayama Jan 01 '18 at 19:15