Inverse systems of projective schemes appear in several contexts, for example:
- in constructing the Zariski-Riemann space of a projective variety,
- in studying subvarieties of a projective variety passing through a given point by the crutch of infinitesimal thickenings (e.g. the formal group of an abelian variety),
Almost everywhere in the literature, inverse systems are studied under the assumption that the transition morphisms in the system are affine (mostly because in this case the limit is representable).
Do you know some references (articles, books...) where inverse systems are studied without the affineness assumption?
The natural object one wants to study is the limit of the system as an fppf sheaf, and one wants to do some geometry with it, that is, attach geometric quantities to it like a topological space, a dimension, define regularity properties etc.