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

- Complex-analytic spaces,
- Rigid-analytic spaces, or
- 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]

Question IIIof another MO question. The reason for splitting it is to (hopefully) improve its focus. $\endgroup$ – Sofia Mar 5 '20 at 21:23