# Direct image of structure sheaf under base change

Dear all,

Here is a problem I came across recently. Let $f:X\to Y$ be a proper morphism of Noetherian schemes and assume that $f_*\mathcal{O}_X\cong \mathcal{O}_Y$. Now let $S$ be a subscheme of $Y$, denote $T:=S\times_YX$ and let $f':T\to S$ be the projection.

Is it then true that $f'_*\mathcal{O}_T\cong \mathcal{O}_S$? If not, can you give an example where this does not work? In that case, are there may be general extra conditions on $X,Y$ and $f$ such that it does work?

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It's not true in general. Basically, you ask whether $f$ is cohomologically flat (in degree zero), see mathoverflow.net/questions/61289/… For a specific counterexample and some discussion, see mathoverflow.net/questions/56019/… Namely (quoting from Allen Knutson's answer to last question): let $f$ be a flat family of curves degenerating into a curve with an embedded point. –  t3suji Apr 12 '11 at 12:42
An example is an elliptic fibration with wild fibres. See paper of Bombieri and Mumford on classification of surfaces in positive characteristic. –  Torsten Ekedahl Apr 12 '11 at 13:05
@Ekedahl. Thank you, that is a nice example! Do you also happen to know an example in characteristic 0? –  Bart van den Dries Apr 19 '11 at 12:13
@Bart: Not for elliptic fibrations, wild fibres only exist in positive characteristic. –  Torsten Ekedahl May 3 '11 at 17:54

Let $Y$ be normal, $S\subset Y$ be non-normal and $f:X\to Y$ an embedded resolution of $S$. (To make it concrete, you may choose $Y=\mathbb P^2$, $S=Z(x^2-y^2)$ (or a nodal cubic) and $f:X\to Y$ the blow up of the singular point of $S$).
Then $T$ is the resolution of $S$ and since it is not-normal, $f'_*\mathscr O_T\not\simeq \mathscr O_S$, but $Y$ is normal, so $f_*\mathscr O_X\simeq \mathscr O_Y$.