$(X,\mathcal O_X)$ 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!
Best Answer
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.