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Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies all the properties

Let $Y\subset X$ a closed subscheme of $X$, do we know if $Y$ is locally of finite type?

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  • $\begingroup$ maybe we also need to assume that $X$ is a indscheme over a countable set. $\endgroup$
    – prochet
    Commented Jun 8, 2013 at 8:11
  • $\begingroup$ By $T(C-x)$ do you mean $\text{Hom}(C-x,T)$ ? $\endgroup$
    – Matthieu Romagny
    Commented Jun 8, 2013 at 9:48
  • $\begingroup$ yes, that's right. $\endgroup$
    – prochet
    Commented Jun 8, 2013 at 10:11
  • $\begingroup$ SGA3 uses the notation $\underline{\operatorname{Hom}}_k(C-x,T)$ to indicate the Hom sheaf. The underline is to keep people from mistaking it for the Hom-set. $\endgroup$
    – S. Carnahan ♦
    Commented Jun 9, 2013 at 2:27

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