Higher Local Class Field Theory
For a higher dimensional local field there is a higher local reciprocity map which describes abelian extensions of the field in terms of open subgroups of finite index in the Milnor K-group of the field. Namely, if is an -dimensional local field then one uses or its separated quotient endowed with a suitable topology. When the theory becomes the usual local class field theory. Unlike the classical case, Milnor K-groups do not satisfy Galois module descent if . Higher dimensional class field theory was pioneered by A.N. Parshin in positive characteristic and K. Kato, I. Fesenko, Sh. Saito in the general case.
Read more about this topic: Local Class Field Theory
Famous quotes containing the words higher, local, class, field and/or theory:
“The imagination, give it the least license, dives deeper and soars higher than Nature goes.”
—Henry David Thoreau (1817–1862)
“This is the only “wet” community in a wide area, and is the rendezvous of cow hands seeking to break the monotony of chuck wagon food and range life. Friday night is the “big time” for local cowboys, and consequently the calaboose is called the “Friday night jail.””
—Administration in the State of Texa, U.S. public relief program (1935-1943)
“One of the strengths I derive from my class background is that I am accustomed to contempt.”
—Dorothy Allison (b. 1949)
“Is not the tremendous strength in men of the impulse to creative work in every field precisely due to their feeling of playing a relatively small part in the creation of living beings, which constantly impels them to an overcompensation in achievement?”
—Karen Horney (1885–1952)
“The theory [before the twentieth century] ... was that all the jobs in the world belonged by right to men, and that only men were by nature entitled to wages. If a woman earned money, outside domestic service, it was because some misfortune had deprived her of masculine protection.”
—Rheta Childe Dorr (1866–1948)