I have some questions related to formal schemes. Essentially I would like to understand how much formal schemes are different from usual schemes. First of all, let me specify the definition I prefere of formal schemes. I call a locally ringed space $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ a locally Noetherian formal scheme if it admits a covering given by affinoid formal schemes ${\text{Spf}(A_i)}$, where each $A_{i}$ is a Noetherian adic ring with ring of definition $I_i$.
1) Is there a functorial interpretation of formal schemes? I mean, is it possible to say, like for schemes, that a formal scheme is the representing object of a functor from the opposite category of Noetherian adic rings to set? I guess it is possible, since we know that $\text{Hom}_{\text{FSchemes}}(\mathfrak{X},\text{Spf}(A))\simeq\text{Hom}_{\text{ cont }}(A,\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}))$.
I know it is possible to introduce quasi-coherent sheaves in the context of formal schemes. For affinoid formal schemes, quasi-coherent sheaves are given by completions of usual modules, and it is in general possible to glue these sheaves over the whole formal scheme. I guess that also the definition of a quasi-coherent sheaf of adic algebras goes in the same way.
2) Is there a notion of relative affinoid spectrum? I mean, consider a formal scheme $(\mathfrak{X},\mathcal{O}_{\mathfrak{X}})$ and a quasi coherent sheaf of adic $\mathcal{O}_{\mathfrak{X}}$-algebras $\mathcal{E}$. Is it possible to define $\underline{\text{Spf}}_{\mathfrak{X}}(\mathcal{E})$ simply glueing along the intersections of affine pieces things like $\text{Spf}(\Gamma(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}))$? Again I cannnot see any problem in this construction. I am particularly interested in this construction expecially in the case when I construct line bundles. In fact, once we have a notion of coherent sheaf, we also have a notion of invertible sheaf, as coherent modules locally free of rank $1$. So, given an invertible sheaf $\mathfrak{L}$, we can take $\text{Sym}(\mathfrak{L})$. Now, since the construction of symmetric algebra is not left exact, maybe we have to complete before defining $\underline{\text{Spf}}_{\mathfrak{X}}(\widehat{\text{Sym}(\mathfrak{L})})$, where hat means completion. Does this construction, in this particular case, give the total space of a line bundle, i.e. satisfy all the formal properties of the equivalent construction in scheme theory?
3) I know that one example of formal scheme is given by the formal completion of a scheme along a closed subscheme. Is the reverse true? I.e. is it true that every formal scheme comes as the completion of a usual scheme along a closed subscheme?
4) I know that fiber product of formal schemes is locally constructed via completed tensor product. Which properties of usual tensor product does completed tensor product satisfy? In particular, I am interested in the following situation: once I have an invertible sheaf $\mathfrak{L}$ over a formal scheme, may I define the inverse as $\text{Hom}_{\mathcal{O}_{\mathfrak{X},\text{ cont}}}(\mathfrak{L},\mathcal{O}_{\mathfrak{X}})$, where I mean continuous homomorphisms of $\mathcal{O}_{\mathfrak{X}}$-modules?
Thanks for any suggestion, I know there are a lot of points here, but also if you can suggest me good references, I would appreciate it!
