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:
“The proper study of mankind is man in his relation to his deity.”
—D.H. (David Herbert)
“True variety is in that plenitude of real and unexpected elements, in the branch charged with blue flowers thrusting itself, against all expectations, from the springtime hedge which seems already too full, while the purely formal imitation of variety ... is but void and uniformity, that is, that which is most opposed to variety....”
—Marcel Proust (1871–1922)
“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)