Proper Morphism of Formal Schemes
Let be a morphism between locally noetherian formal schemes. We say f is proper or is proper over if (i) f is an adic morphism (i.e., maps the ideal of definition to the ideal of definition) and (ii) the induced map is proper, where and K is the ideal of definition of .(EGA III, 3.4.1) The definition is independent of the choice of K. If one lets, then is proper.
For example, if is a proper morphism, then its extension between formal completions is proper in the above sense.
As before, we have the coherence theorem: let be a proper morphism between locally noetherian formal schemes. If F is a coherent -module, then the higher direct images are coherent.
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