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:
“For me, as a beginning novelist, all other living writers form a control group for whom the world is a placebo.”
—Nicholson Baker (b. 1957)
“Your scheme must be the framework of the universe; all other schemes will soon be ruins.”
—Henry David Thoreau (1817–1862)
“If the national security is involved, anything goes. There are no rules. There are people so lacking in roots about what is proper and what is improper that they don’t know there’s anything wrong in breaking into the headquarters of the opposition party.”
—Helen Gahagan Douglas (1900–1980)
“Jesus abolished the very concept of “guilt”Mhe denied any cleavage between God and man. He lived this unity of God and man as his “glad tidings” ... and not as a prerogative!”
—Friedrich Nietzsche (1844–1900)