I'm trying to solve the following exercise, namely Exercise 8.1.H, from Vakil's FOAG:
As hints, there are three approaches provided, of which I'm following roughly the first. The way I want to do this is to first define schemes $Y_i \cong \text{Spec }A_i/I(A_i)$, for each affine open $\text{Spec }A_i\subset X$. Next I wish to glue them in the obvious way.
For any distinguished open $\text{Spec }A_f\subset \text{Spec A}$, suppose we have, for some indices $A_i = A$ and $A_j = A_f$. Since we're given that $I(A_i)_f \cong I((A_i)_f)$, we have isomorphisms $(A_i/I(A_i))_f\cong (A_i)_f/I(A_i)_f\cong (A_i)_f/ I((A_i)_f)$. We use this to define isomorphisms $\phi_{ij}: \text{Spec } A_j/I(A_j)\to \text{Spec } (A_i/I(A_i))_f$, where $\text{Spec } (A_i/I(A_i))_f$ is an open subscheme of $Y_i$.
Next I want to define $\phi_{ij}:Y_i\to Y_j$ for arbitrary $Y_i$ and $Y_j$. To that end, in $X$, I cover the intersection of $\text{Spec }A_i$ and $\text{Spec }A_j$ with affine opens which are simultaneously distinguished opens for both $\text{Spec }A_i$ and $\text{Spec }A_j$. Now, if $\text{Spec }A_k$ is one of these distinguished opens, then I already have isomorphisms from an open subscheme of $Y_i$ to $Y_k$, and from $Y_k$ to an open subscheme of $Y_j$, from the previous paragraph. Finally, I want to show that it is possible to glue of all of these composites, to obtain $\phi_{ij}$. The $\phi_{ij}$ I just defined can be made free of the choice of the cover by distinguished open sets, by simply taking all such distinguished opens in the intersection, as the cover for the same.
Lastly, I need to check that the $\phi_{ij}$ agree on triple intersections (i.e. this is the cocycle condition). I'm not entirely sure how to do this, but I think it suffices to cover the triple intersection by affine opens which are distinguished in all three intersecting affine opens?
To conclude, my requests are as follows:
Am I heading in the right direction?
As is evident from the above, I haven't yet worked out the full details. Thus, I would appreciate feedback on the approach I've taken, and possible ways of making the solution cleaner.
I would be really grateful if someone can either post a complete solution to this or provide a reference where this has been worked out in detail, with machinery restricted to what Vakil has developed so far in the text.
Thank you.