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.
Read more about this topic: Proper Morphism
Famous quotes containing the words proper, formal and/or schemes:
“I should like to know what is the proper function of women, if it is not to make reasons for husbands to stay at home, and still stronger reasons for bachelors to go out.”
—George Eliot [Mary Ann (or Marian)
“I will not let him stir
Till I have used the approvèd means I have,
With wholesome syrups, drugs, and holy prayers,
To make of him a formal man again.”
—William Shakespeare (1564–1616)
“Science is a dynamic undertaking directed to lowering the degree of the empiricism involved in solving problems; or, if you prefer, science is a process of fabricating a web of interconnected concepts and conceptual schemes arising from experiments and observations and fruitful of further experiments and observations.”
—James Conant (1893–1978)