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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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maybe we also need to assume that $X$ is a indscheme over a countable set. –  prochet Jun 8 '13 at 8:11
By $T(C-x)$ do you mean $\text{Hom}(C-x,T)$ ? –  Matthieu Romagny Jun 8 '13 at 9:48
yes, that's right. –  prochet Jun 8 '13 at 10:11
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. –  S. Carnahan Jun 9 '13 at 2:27
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