# Restricted closed subscheme and reduced induced closed subscheme

$$(X,\mathcal O_X)$$ is a scheme, and $$Y \subset X$$ is a closed subset. So on $$Y$$ we can define a canonical structure sheaf, i.e. the restriction structure sheaf as $$(Y,\mathcal O_X|_Y)$$. I guess it is not hard to see this is a closed subscheme of $$(X,\mathcal O_X)$$. However, we know that,(Hartshorne Ex 3.2.6) there are many different closed subschemes with underlying space $$Y$$ but different structure sheaves. In that Ex 3.2.6, it defines reduced induced closed subscheme which is "smaller" (universal) than any other closed subschemes with space $$Y$$.

So my question is for those two specific constructions of closed subschemes, are they the same or if not, related in some way? For example, if $$X=Spec \, A$$, then we know that the reduced induced closed subscheme $$Y=Spec \, A/I$$, where $$I$$ is the largest ideal for which $$V(I)=Y$$ (Also in Ex 3.2.6), then what will be the ideal $$I$$ corresponding to the restricted subscheme $$Y, \mathcal O_X|_Y$$??

Thank you so much for any help!

• How are you defining $\mathcal{O}_X|_Y$? Nov 16, 2020 at 2:58
• Sorry, I think that's defined in any textbook right? it is the inverse image sheaf of the inclusion map from $Y$ to $X$. Nov 16, 2020 at 3:04
• I ask because that does not even give a scheme - taking $X=\Bbb A^1_k$ and $Y$ to be the origin, you get a point with the constant sheaf $k[t]_{(t)}$, and $\operatorname{Spec} k[t]_{(t)}$ has two points. Nov 16, 2020 at 3:06
• @KReiser Oh sorry! I don't understand your example, but I see your point that the restricted sheaf doesn't always give you a scheme. Can you explain that example a little bit more? Nov 16, 2020 at 3:15
• Sure - a scheme is a locally ringed space where every point has an affine neighborhood isomorphic to an affine scheme (this is a definition). In particular, any scheme which has underlying space a point must have structure sheaf which is a dimension-zero local ring. I showed that your proposed definition applied to the case where $X=\operatorname{Spec} k[t]$ and $Y$ is the closed point $(t)\in X$ gives something not of this form - that is, your claim that $(Y,\mathcal{O}_X|_Y)$ is a scheme is false. Nov 16, 2020 at 3:24

Here's a recap of the discussion from the comments and chat: the locally-ringed space $$(Y,\mathcal{O}_X|_Y)$$ is not in general a scheme, because it is not the case that every point has an open neighborhood isomorphic as a locally ringed space to $$\operatorname{Spec} A$$ for some ring $$A$$ (this is the defining property for schemes among locally ringed spaces). In particular, taking $$X=\operatorname{Spec} k[t]$$ for $$k$$ a field and $$Y$$ to be the set consisting of the single closed point $$(t)\in X$$, we have that $$(Y,\mathcal{O}_X|_Y)$$ is the locally ringed space which is a point with the structure sheaf the constant sheaf with value $$k[t]_{(t)}$$. If this were to be a scheme, then it must be an affine scheme, and it would need to be isomorphic to the affine scheme $$\operatorname{Spec} k[t]_{(t)}$$. But this is impossible, because this has two points while $$Y$$ is a singleton.