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I have a relatively naive question. Suppose that $f: X \rightarrow Y$ is a map of schemes. Then, we get a map of local rings $\mathcal{O}_{Y,f(x)} \rightarrow \mathcal{O}_{X,x}$ and thus for any sheaf $F$ on $X$ we can say that $F$ is flat over $Y$ if the stalk $F_x$ is flat as a $\mathcal{O}_{Y,f(x)}$-module. This notion is important, for example, because it is a prerequisite for applying many theorems of the form "If $F$ is flat over $Y$ and (hypothesis) then $f_*F$ is locally free on $Y$".

I would like an example of a sheaf $F$ which is flat on $X$ and still flat over $Y$ when $f$ is not flat, if such an example exists. If no such example exists, why not?

Bonus points if addressing the specific situation where $f$ is proper and birational (e.g. $X$ is a blow-up of $Y$).

Notice that if $G$ is locally free on $Y$ then $F = f^*G$ will be locally free on $X$ but will be flat over $Y$ if only if $f$ is flat so an example will not arise this way.

  • 0
    You mean if $F$ is flat over $X$, then $f_*F$ is...2012-07-30
  • 1
    The zero sheaf is a trivial example...2012-07-31

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