Consider a morphism $f: Y \to X$ between two varieties and consider the stacks parametrizing coherent sheaves on them $\mathcal{M}\_X, \mathcal{M}\_Y$.


Does one have for free an induced pullback morphism $f^\*: \mathcal{M}\_X \to \mathcal{M}\_Y$?


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).


Thanks



can one define the pullback between stacks of coherent sheaves for non-flat morphisms?
(I had to delete my earlier answer; the following answer is based on a comment by ulrich which vanished when I deleted the answer.)


The answer to your question is *no*. 


*Counterexample:* 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 $X$. 


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. 


