Representability of Hom implies flatness

Abstract

Let $X$ be a projective scheme over a noetherian base scheme $S$, and let $F$ be a coherent sheaf on $X$. For any coherent sheaf $E$ on $X$, consider the set-valued contravariant functor $Hom_{E,F}$ on $S$-schemes, defined by $Hom_{E,F}(T) = Hom(E_T,F_T)$ where $E_T$ and $F_T$ are the pull-backs of $E$ and $F$ to $X_T = X\times_S T$. A basic result of Grothendieck ([EGA] III 7.7.8, 7.7.9) says that if $F$ is flat over $S$ then $Hom_{E,F}$ is representable for all $E$. We prove the converse of the above, in fact, we show that if $L$ is a relatively ample line bundle on $X$ over $S$ such that the functor $Hom_{L^{-n},F}$ is representable for infinitely many positive integers $n$, then $F$ is flat over $S$. As a corollary, taking $X=S$, it follows that if $F$ is a coherent sheaf on $S$ then the functor $T\mapsto H^0(T, F_T)$ on the category of $S$-schemes is representable if and only if $F$ is locally free on $S$. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author's earlier result (see arXiv.org/abs/math.AG/0204047) that the automorphism group functor of a coherent sheaf on $S$ is representable if and only if the sheaf is locally free.