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Let $S$ be a Noetherian scheme and $X,Y$ be $S$-schemes of finite type. Consider the functor $X^Y$ given by $T \mapsto Mor_S(Y \times_S T,X)$. When is this functor representable by an $S$-scheme of finite type?

Is it true when $Y \rightarrow S$ is finite and flat?

In http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Sept17(Bun(G)).pdf (Thm 1.6) it is shown that $X^Y$ is representable if $X \rightarrow S$ is quasi-projective and $Y \rightarrow S$ is projective. Dennis Gaitsgory explained to us that this implies that $X^Y$ is representable by a locally finite type scheme.

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    $\begingroup$ Is your functor not the same as the Hom-functor $T\mapsto \mathrm{Hom}_T(Y_T,X_T)$, by the universal property of fiber products? (I let $Y_T = Y\times_S T$ and $X_T = X\times_S T$.) If so, then this is representable by a locally finitely presented scheme over $S$ if $Y$ is proper and flat over $S$; see math.berkeley.edu/~molsson/homstackfinal.pdf (this is just one reference). You could also have a look at Nitsure's chapter in FGA explained. If you want the Hom-scheme to be of finite type, you will need some "boundedness" result. (Representability fails when $X=Y$ is the affine) $\endgroup$
    – Ariyan Javanpeykar
    Feb 20, 2018 at 21:40
  • $\begingroup$ Small correction: locally finitely presented scheme should have been locally finitely presented algebraic space. (You get a scheme if you impose projectivity assumptions.) $\endgroup$
    – Ariyan Javanpeykar
    Feb 21, 2018 at 0:59

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