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]

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]

M. Hakim developed in her doctoral thesis 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?

Notes:

1. I'm not sure how useful this information is, but, for the last case, it seems that schemes over formal schemes have been considered in the following references:

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

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

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

2. There are GAGA-type theorems for both schemes over rigid spaces and schemes over complex-analytic spaces.

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]

M. Hakim developed in her doctoral thesis 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 and rigid-analytic spaces?

Note: This question was Question III of this one, which was split into two to make it more focused.

M. Hakim developed in her doctoral thesis 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?

Notes:

1. I'm not sure how useful this information is, but, for the last case, it seems that schemes over formal schemes have been considered in the following references:

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

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

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

2. There are GAGA-type theorems for both schemes over rigid spaces and schemes over complex-analytic spaces.