can one define the pullback between stacks of coherent sheaves for non-flat morphisms? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:41:04Z http://mathoverflow.net/feeds/question/74247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/74247/can-one-define-the-pullback-between-stacks-of-coherent-sheaves-for-non-flat-morph can one define the pullback between stacks of coherent sheaves for non-flat morphisms? Yosemite Sam 2011-09-01T10:32:40Z 2011-09-02T11:52:15Z <p>Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}_X, \mathcal{M}_Y$.</p> <p>Does one have for free an induced pullback morphism $f^*: \mathcal{M}_X \to \mathcal{M}_Y$?</p> <p>I guess the question reduces to: if $S$ is a base scheme and $E$ is a family of sheaves flat over $S$ on $X$, and if $f_S: Y_S \to X_S$ is the induced morphism, then is $f_S^* E$ still a flat family of coherent sheaves? (as usual, for ugly $S$ coherent $E$ should be replaced by quasi-coherent (locally) of finite presentation).</p> <p>Thanks</p> http://mathoverflow.net/questions/74247/can-one-define-the-pullback-between-stacks-of-coherent-sheaves-for-non-flat-morph/74347#74347 Answer by euklid345 for can one define the pullback between stacks of coherent sheaves for non-flat morphisms? euklid345 2011-09-02T11:52:15Z 2011-09-02T11:52:15Z <p>(I had to delete my earlier answer; the following answer is based on a comment by ulrich which vanished when I deleted the answer.)</p> <p>The answer to your question is <em>no</em>. </p> <p><em>Counterexample:</em> Let $X$ be the plane and $Y$ the blow-up of $X$ at the origin. Consider the tautological family of length 1 skyscraper sheaves on $X$, parametrized by $X$. This means $S=X$, and $E=\Delta_\ast\mathscr{O}_X$, where $\Delta:X\to X\times X$ is the diagonal. Then $f_S^\ast E=(\Gamma_f)_\ast\mathcal{O}_Y$, where $\Gamma_f:Y\to Y\times X$ is the graph of $f$. This is not flat over $X$. </p> <p>This makes sense, geometrically: there is no way to pull back the skyscraper sheaf at the origin in $X$ to a skyscraper sheaf in $Y$, in a way which is compatible with families of skyscraper sheaves, for example along lines through the origin. </p>