When is the push-forward of the structure sheaf locally free - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:22:49Z http://mathoverflow.net/feeds/question/23891 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free When is the push-forward of the structure sheaf locally free Ariyan Javanpeykar 2010-05-07T20:13:50Z 2011-06-08T14:46:02Z <p>Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?</p> <p><strong>Example 1</strong>. Suppose that $f$ is affine. Then $f_\ast\mathcal{O}_X$ is a quasi-coherent $\mathcal{O}_Y$-module.</p> <p><strong>Example 2</strong>. Suppose that $f$ is finite. Then $f_\ast \mathcal{O}_X$ is even coherent.</p> <p><strong>Example 3</strong>. Suppose that $f:X\longrightarrow Y$ is a finite morphism of regular integral 1-dimensional schemes. Then <code>$f_\ast \mathcal{O}_X$</code> is coherent and locally free. (The local rings $\mathcal{O}_{Y,y}$ are discrete valuation rings.)</p> <p>In view of the above examples, I'm basically looking for a higher-dimensional analogue of Example 3. But, I don't require quasi-coherence in my question. (Although this is quite unnatural.)</p> <p><strong>Idea</strong>. For any finite morphism $f:X\longrightarrow Y$, we have that $f_\ast \mathcal{O}_X$ is locally free. Only what are the precise conditions on $X$ and $Y$?</p> http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free/23895#23895 Answer by Matthieu Romagny for When is the push-forward of the structure sheaf locally free Matthieu Romagny 2010-05-07T20:32:46Z 2011-06-08T14:46:02Z <p>It is of course not true that for any finite morphism $f:X\to Y$ we have $f_*\mathcal{O}_X$ locally free : think about a closed immersion.</p> <p>In fact, your question is about the important topic of "base change and cohomology of sheaves" for proper morphisms, which is treated by Grothendieck in EGA3. The simplest answer one can give, I think, is that if $f$ is proper [EDIT : and flat, as t3suji points out] and for all $y\in Y$ we have $H^1(X_y,\mathcal{O}_{X_y})=0$ then $f_*\mathcal{O}_X$ is locally free.</p> <p>You may want to avoid going to find the exact reference in EGA3, since as Mumford says, that result is "unfortunately buried there in a mass of generalizations". In that case, go to chapter 0, section 5 of <em>Geometric Invariant Theory</em> (3rd ed.) by Mumford, Fogarty and Kirwan. This is where Mumford's comment is taken from.</p> http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free/23896#23896 Answer by Peter Bruin for When is the push-forward of the structure sheaf locally free Peter Bruin 2010-05-07T20:40:42Z 2010-05-07T20:40:42Z <p>If $f\colon X\to Y$ is a finite morphism, then $f_*O_X$ is usually not locally free; for example, consider the inclusion of a point into the affine line. The question is: given a ring homomorphism $A\to B$ such that $B$ is a finitely generated $A$-module, is $B$ locally free? There is not really a useful property of $f$ that would imply this non-tautologically.</p> <p>As Matthieu Romagny said (his answer arrived while I was writing mine), the real question is about proper morphisms. You may also be interested in Hartshorne, <i>Algebraic Geometry</i>, III.12.</p> http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free/23903#23903 Answer by Matthew Morrow for When is the push-forward of the structure sheaf locally free Matthew Morrow 2010-05-07T22:08:51Z 2010-05-07T22:51:47Z <p>Assuming that $f$ is finite (I'm not quite sure if you are), then $f_*\mathcal{O}_X$ is locally free if and only if $f$ is a flat morphism.</p> <p>Like Peter Bruin says, you are asking the following: given a finite ring homomorphism $A\to B$, when is $B$ a locally free $A$-module? But there is a natural characterization of this! A finitely generated $A$-module is locally free if and only if it is flat, because a finitely generated module over a local ring is free if and only if it is flat.</p> <p>Example 3 is a special case of this, because <em>any</em> surjective morphism to a regular one dimensional scheme is automatically flat (because any injection of a dvr into another ring is flat).</p> <p>Edit: Worth noting that if $X,Y$ are regular and $f$ is finite and surjective, then $f$ is flat.</p> http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free/23904#23904 Answer by Karl Schwede for When is the push-forward of the structure sheaf locally free Karl Schwede 2010-05-07T22:13:03Z 2010-05-08T02:41:37Z <p>Well, if you have a finite flat morphism as Matthew Morrow says above.</p> <p>Also, this may or may not be relevant eventually, but with regards to analogues of (3) with the <em>higher</em> direct images (and in higher relative dimension, ie non-finite morphisms), you might also want to check out Steenbrink's paper (and Du Bois's earlier paper).</p> <p><a href="http://www.numdam.org/item?id=CM_1980__42_3_315_0" rel="nofollow">http://www.numdam.org/item?id=CM_1980__42_3_315_0</a></p> <p><a href="http://www.numdam.org/item?id=BSMF_1981__109__41_0" rel="nofollow">http://www.numdam.org/item?id=BSMF_1981_<em>109</em>_41_0</a></p> <p>See in particular Theorem 1 (and Theorem 4.6). </p> <p>It says that if $f : X \to Y$ is flat (EDIT: and <em>proper</em>) and the fibers have nice enough singularities, then $R^i f_* O_X$ is locally free for all $i$. There's also a recent preprint on the arXiv of Kollar and Kovacs on Du Bois singularities which deals with some things related to this at the end, see:</p> <p><a href="http://front.math.ucdavis.edu/0902.0648" rel="nofollow">http://front.math.ucdavis.edu/0902.0648</a></p> http://mathoverflow.net/questions/23891/when-is-the-push-forward-of-the-structure-sheaf-locally-free/23931#23931 Answer by David Speyer for When is the push-forward of the structure sheaf locally free David Speyer 2010-05-08T13:09:24Z 2010-05-08T13:09:24Z <p>One should probably also mention the "miracle flatness" theorem: If $f: X \to Y$ is finite, $X$ and $Y$ have the same dimension, $X$ is Cohen-Macaulay and $Y$ is regular, then $f$ is flat. As everyone has mentioned above, finite and flat implies locally free, so this theorem can be one useful way to get the flatness hypothesis.</p>