Representability of Hom implies flatness |
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Authors: | Email author" target="_blank">Nitin?NitsureEmail author |
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Institution: | (1) School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005 Mumbai, India |
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Abstract: | LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, 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
andF
T are the pull-backs of ε andF toX
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 Komε,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, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F
t) on the category ofS-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. |
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Keywords: | Flattening stratification Q-sheaf group-scheme base change |
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