In differential geometry, given two manifolds, only special types of morphisms between them are, submersions/immersions/and some one or two other types of maps.
In Algebriac geometry, given two schemes, there are more than 10 types of maps between schemes that are of interest.
- Separated
- Quasi compact
- Locally of finite presentation
- Proper
- Affine
- Finite
- Flat
- Smooth
- Unramified
- Etale
- Embedding
- Closed embedding
- fpqc morphism
and many more whose names itself far from my reach. It is difficult to even remember some names, let alone how they are defined.
Question : Why is it the case that maps between schemes are super different from that of manifolds? Or, are there analogues of above maps in differential geometry setup as well?