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!
Best Answer
For 1)
The functorial interpretation is developed by Strickland in
Formal schemes and formal groups. Homotopy invariant algebraic structures (Baltimore, MD, 1998), 263–352, Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.
an expanded version is on his webpage
https://neil-strickland.staff.shef.ac.uk/research/subgp.pdf
On quasi-coherent sheaves, the issue is certainly delicate, I would recommend you the initial sections in
"Duality and flat base change on formal schemes", which is the first paper in:
Alonso Tarrío, Leovigildo; Jeremías López, Ana; Lipman, Joseph: Studies in duality on Noetherian formal schemes and non-Noetherian ordinary schemes. Contemporary Mathematics, 244. American Mathematical Society, Providence, RI, 1999.
For a readily available pdf (with some corrections incorporated)
http://www.math.purdue.edu/~lipman/papers/formal-duality.pdf
For 2)
The short answer is no, as far as I know.
For 3) there are famous counterexamples, see
Hironaka, Heisuke; Matsumura, Hideyuki: Formal functions and formal embeddings. J. Math. Soc. Japan 20 1968, 52–82.
and
Hartshorne, Robin: Ample subvarieties of algebraic varieties. Lecture Notes in Mathematics, Vol. 156 Springer-Verlag, Berlin-New York 1970
but also
Non-algebraizable Formal Scheme?
For 4)
As far as you stick to coherent sheaves, morphisms are automatically continuous, so invertible sheaves behave much as on ordinary schemes. Another completely different issue is ampleness, where counterexamples in hartshornes "algebraic Geometry" book show that the issue is subtle, and somehow connected to your second question.
Generalizing, you would like to seek for a nice system of generators of a substitute of the ill-behaved category of quasi-coherent sheaves. There are several possible candidates, but this line has not been pursued in these terms.