Let $F:(Sch/k)^{o}\to Sets$ be a functor, where $Sch/k$ is the category of schemes over a field $k$. Suppose that $F$ admits a coarse moduli space, let it be $M$. Consider a $k$-point $x\in M$ (which corresponds to an element $\alpha\in F(Spec(k))$ ). Define the "restriction" functor $F_x$ from the category $Art(k)$ of local artin $k$-algebras with residue field $k$ to $Sets$ : $$A\mapsto \{ \beta\in F(Spec(A)): \beta|_{Spec(k)}=\alpha \} $$ Does $F$ admit a hull? If it does admit a hull $R$, is it true that the dimension of $R$ is equal to the dimension of the completed local ring $\hat{\mathcal O}_{M,x}$? I am asking this because I would like to bound the dimension of the coarse moduli space as in the case of a fine moduli space, where this construction works and gives the desired answers.


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