2003/08/05 by Nitin Nitsure, Nitsure, Nitin
Engineering · Mathematics · #14A15 (Primary) 14F05 #14L15 (Secondary) #Algebraic Geometry (math.AG) #Dynamics and Control of Mechanical Systems #FOS: Mathematics #Mathematics and Applications #Metal Forming Simulation Techniques #math.AG #msc:14A15 #msc:14F05 #msc:14L15
paper · pdf · doi:10.48550/arxiv.math/0308036
9 pages, LaTeX
arxiv created 2003/08/05 · openalex publication_date 2003/08/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 HomE,F on S-schemes, defined by HomE,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 HomE,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 HomL-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↦ H0(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 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.