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?