I'm following Vakils book 'Foundations of Algebraic Geometry', and I'm currently in section 8.1 on closed subschemes. More precisely, I'm trying to do Exercise 8.1.H, which is the following:
Let $(Y,\mathcal{O}_Y)$ be a scheme, and suppose that for every affine open $U=\operatorname{Spec} A_U\subseteq Y$ we are given an ideal $I_U\subseteq A_U$, such that for every $f\in A_U$, the extension of $I_U$ along the composed morphism $A_U\to A_{U,f}\overset{\cong}{\to}A_{D_U(f)}$ is exactly $I_{D_U(f)}$. Show that this data uniquely determines a closed subscheme of $Y$, whose associated sheaf of ideals is exactly $I_U$ on the affine opens.
Vakil proposes three approaches to the problem. The first one is using glueing, and I'm not feeling too comfortable with glueing (yet), so I would like to use another approach. Thus I am following the second approach he proposes, which is to show that the above data determines a sheaf of ideals $\mathcal{J}$ of $\mathcal{O}_Y$ such that $\mathcal{J}(U)=I_U$ and $(\mathcal{O}_Y/\mathcal{J})(U)\cong A_U/I_U$ for all affine opens $U\subseteq Y$. I was able to show that by defining $$ \mathcal{J}(V):=\{s\in\mathcal{O}_Y(V)\ |\ s|_U\in I_U\text{ for every affine open }U\subseteq V\} $$ we obtain a sheaf of ideals with these required properties. However, I'm struggling to see how this can be used to explicitly give a closed subscheme of $Y$, but as this is all he proposed to do I suspect that it would have to be quite direct. Because all I see how to construct a closed subscheme from there is to define a closed subset $X\subseteq Y$ such that $X\cap U=V(I_U)$ (I succeeded in showing that this is well-defined), and put its structure sheaf to be $\mathcal{O}_X:=(\mathcal{O}_Y/\mathcal{J})|_X$, but I have no idea how to show that this gives $X$ the structure of a closed subscheme of $Y$.