Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc

Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these spaces interesting?

In particular, for what reasons would one like to consider spaces over

1. Complex-analytic spaces,
2. Rigid-analytic spaces, or
3. Formal schemes?

Note: Maybe it is worth mentioning that schemes over formal schemes have been considered in references [2-5] below.

References

[1] Hakim, Monique. "Topos annelés et schémas relatifs. Volume 64 of." Ergebnisse der Mathematik und ihrer Grenzgebiete (1972). [Link]

[2] Lan, Kai-Wen. Arithmetic compactifications of PEL-type Shimura varieties. No. 36. Princeton University Press, 2013. [PDF]

[3] Faltings, Gerd, & Chai, Ching-Li (2013). Degeneration of abelian varieties (Vol. 22). Springer Science & Business Media. [Link]

[4] Chai, Ching-Li. Compactification of Siegel moduli schemes. Vol. 107. Cambridge University Press, 1985. [Link]

• Note: This question was originally Question III of another MO question. The reason for splitting it is to (hopefully) improve its focus. – Sofia Mar 5 '20 at 21:23

This a small example and a couple of reflections.

Consider an adic map of formal schemes $$f \colon \mathfrak{X} \to \mathfrak{Y}$$. Its fiber are schemes. So, I would consider this a "relative scheme" over $$\mathfrak{Y}$$ where $$\mathfrak{Y}$$ might be some interesting completion of a scheme, perhaps of a scheme of germs at a point of some base scheme completed at its maximal ideal, say. A lot of nice properties of schemes carry over to adic maps of formal schemes. This is not so surprising if you ponder that, after all, they are relative schemes.

On the other hand, non adic map of formal schemes do arise and are important. An instance of this is the completion of an ambient smooth scheme over a field along a singular closed subscheme. This provides a smooth formal scheme that supports the underlying non-smooth scheme. It allows, for instance to compute its correct De Rham cohomology in characteristic 0. But in this case you don't have a relative scheme. The classic reference for this is:

Hartshorne, Robin: On the De Rham cohomology of algebraic varieties. Inst. Hautes Études Sci. Publ. Math. No. 45 (1975), 5–99. http://www.numdam.org/item/?id=PMIHES_1975__45__5_0

A further point is to think about rigid-analytic spaces as generic fibers of formal schemes. I cannot give specific examples but I suspect this may be related to the previous considerations.

• Thanks! I really like this example and the "philosophy"/idea behind it ((as I undestand it) that a map of "geometric objects" whose fibres are all schemes "is"/[gives rise to] a relative scheme)! – Sofia Mar 23 '20 at 4:01
• Also, your third paragraph sounds very interesting! Are there references in which I could read more about this idea of computing de Rham cohomology of singular schemes by lifting them to smooth formal schemes? – Sofia Mar 23 '20 at 4:04
• For De Rham cohomology on non-adic schemes the reference is Hartshorne's classical paper, I'll edit the answer accordingly. The adic vs non-adic issue arose when we studied infinitesimal properties of maps o formal schemes. In a few words, the adic case is quite similar to the case of schemes but completions along a closed subset provide a different kind of map that does not arise on schemes. – Leo Alonso Mar 23 '20 at 10:36
• Great! Thanks for the elaboration! – Sofia Mar 25 '20 at 3:15