Representability of Hom implies flatness

Abstract

LetX be a projective scheme over a noetherian base scheme S, and let F be a coherent sheaf on X. For any coherent sheaf ε on X, consider the set-valued contravariant functor Hom_(ε,F)S-schemes, defined by Hom_(ε,F) (T)= Hom(ε_T ,F_T) where ε_T and F_T are the pull-backs of ε and F to X_T =X x_S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then hom_(ε,F) is representable for all ε. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom_{(L ^-n ,F)} is representable for infinitely many positive integersn, 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 → H°(T, F_t) on the category of S-schemes is representable if and only ifF is locally free onS. 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 [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.