Direct image of structure sheaf under base change - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T07:36:54Z http://mathoverflow.net/feeds/question/61390 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61390/direct-image-of-structure-sheaf-under-base-change Direct image of structure sheaf under base change Bart van den Dries 2011-04-12T12:33:08Z 2011-04-12T14:35:01Z <p>Dear all,</p> <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/61390/direct-image-of-structure-sheaf-under-base-change/61394#61394 Answer by Keerthi Madapusi Pera for Direct image of structure sheaf under base change Keerthi Madapusi Pera 2011-04-12T13:02:45Z 2011-04-12T13:02:45Z <p>The property you want is called cohomological flatness. See <a href="http://mathoverflow.net/questions/61289/cohomological-flatness-in-degree-0" rel="nofollow">this question</a>. You can check out Hartshorne's chapter on semi-continuity and base change, but I would recommend Mumford's quick treatment in his book 'Lectures on curves on an algebraic surface' (this is just a <a href="http://press.princeton.edu/titles/2404.html" rel="nofollow">great book</a> in general). There are also notes by Nitin Nitsure in <a href="http://www.amazon.co.uk/Fundamental-Algebraic-Geometry-Grothendiecks-Mathematical/dp/0821842455" rel="nofollow">FGA Explained</a> in his chapter on the Hilbert scheme. Of course, you can also always look at EGA III.</p>