In the notes websites.math.leidenuniv.nl/geom/tate.ps (and probably in other places), there is a construction of the Tate curve, where the steps are summarized below.
1) Take $\mathbb{P}^{1}_{\mathbb{C}[[q]]}$. We could do it over $\mathbb{Z}[[q]]$, but that's not what I want to focus on.
2) Take the 0 and $\infty$ sections over the generic fiber, and then take the intersection of the closures with the special fiber. If we are working over $\mathbb{C}$, then the intersection is two points. Blow up at those two points.
3) Repeat step 2 over and over.
4) In the limit, we will get a scheme $T$ over $\mathbb{Z}[[q]]$ where over the special fiber we get a countable chain of $\mathbb{P}^{1}$.
5) Take $n^{th}$-order neighborhoods of the fiber over the generic point, quotient by $\mathbb{Z}$ to get schemes $X_n$, then take the limit to get a formal scheme. Apply Grothendieck's algebraization theorem to get the Tate curve.
I think I'm supposed to understand that we're taking the quotient "$K^{\times}/q^{\mathbb{Z}}$", where $T$ above is supposed to be a substitute for $K^{\times}$ in the formal scheme case.
Do you know of a reference where I can learn about formal schemes in order to understand this? To a beginner like me, $T$ looks ugly and unmotivated.