Group Scheme of Roots of Unity
The group scheme of -th roots of unity is by definition the kernel of the -power map on the multiplicative group, considered as a group scheme. That is, for any integer we can consider the morphism on the multiplicative group that takes -th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism that serves as the identity.
The resulting group scheme is written . It gives rise to a reduced scheme, when we take it over a field, if and only if the characteristic of does not divide . This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example over a finite field with elements for any prime number .
This phenomenon is not easily expressed in the classical language of algebraic geometry. It turns out to be of major importance, for example, in expressing the duality theory of abelian varieties in characteristic (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.
Read more about this topic: Multiplicative Group
Famous quotes containing the words group, scheme, roots and/or unity:
“No group and no government can properly prescribe precisely what should constitute the body of knowledge with which true education is concerned.”
—Franklin D. Roosevelt (1882–1945)
“We doubt not the destiny of our country—that she is to accomplish great things for human nature, and be the mother of a nobler race than the world has yet known. But she has been so false to the scheme made out at her nativity, that it is now hard to say which way that destiny points.”
—Margaret Fuller (1810–1850)
“He who sins easily, sins less. The very power
Renders less vigorous the roots of evil.”
—Ovid (Publius Ovidius Naso)
“Hearing, seeing and understanding each other, humanity from one end of the earth to the other now lives simultaneously, omnipresent like a god thanks to its own creative ability. And, thanks to its victory over space and time, it would now be splendidly united for all time, if it were not confused again and again by that fatal delusion which causes humankind to keep on destroying this grandiose unity and to destroy itself with the same resources which gave it power over the elements.”
—Stefan Zweig (18811942)