A ug 2 00 3 Representability of Hom implies flatness

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(ET ,FT ) where ET and FT are the pull-backs of E and F to XT = X ×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 7→ H(T,FT ) 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 [N1]) that the automorphism group functor of a coherent sheaf on S is representable if and only if the sheaf is locally free. Let S be a noetherian scheme, and let X be a projective scheme over S. If E and F are coherent sheaves on X, consider the contravariant functor Hom(E,F) from the category of schemes over S to the category of sets which is defined by putting Hom(E,F)(T ) = HomXT (ET ,FT ) for any S-scheme T → S, where XT = X×ST , and ET and FT denote the pull-backs of E and F under the projection XT → X. This functor is clearly a sheaf in the fpqc topology on Sch/S. It was proved by Grothendieck that if F is flat over S then the above functor is representable (see [EGA] III 7.7.8, 7.7.9). Our main theorem is as follows, which is a converse to the above. Theorem 1 Let S be a noetherian scheme, X a projective scheme over S, and L a relatively very ample line bundle on X over S. Let F be a coherent sheaf on X. Then the following three statements are equivalent: (1) The sheaf F is flat over S.