Representability of Hom Implies Flatness

[摘要] Let 𝑋 be a projective scheme over a noetherian base scheme 𝑆, and let $mathcal{F}$ be a coherent sheaf on 𝑋. For any coherent sheaf $mathcal{E}$ on 𝑋, consider the set-valued contravariant functor $hom_{(mathcal{E},mathcal{F})}$ on 𝑆-schemes, defined by $hom_{(mathcal{E},mathcal{F})}(T)=mathrm{Hom}(mathcal{E}_T,mathcal{F}_T)$ where $mathcal{E}_T$ and $mathcal{F}_T$ are the pull-backs of $mathcal{E}$ and $mathcal{F}$ to $X_T=X×_s T$. A basic result of Grothendieck ([EGA], III 7.7.9) says that if $mathcal{F}$ is flat over 𝑆 then $hom_{(mathcal{E},mathcal{F})}$ is representable for all $mathcal{E}$.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 $hom_{(L^{-n},mathcal{F})}$ is representable for infinitely many positive integers 𝑛, then $mathcal{F}$ is flat over 𝑆. As a corollary, taking $X=S$, it follows that if $mathcal{F}$ is a coherent sheaf on 𝑆 then the functor $Tmapsto H^0(T,mathcal{F}_T)$ on the category of 𝑆-schemes is representable if and only if $mathcal{F}$ 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.

[学科分类] 数学（综合）

[关键词] Flattening stratification;𝑄-sheaf;group-scheme;base change.