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:
“Once it was a boat, quite wooden
and with no business, no salt water under it
and in need of some paint. It was no more
than a group of boards. But you hoisted her, rigged her.
She’s been elected.”
—Anne Sexton (1928–1974)
“Television programming for children need not be saccharine or insipid in order to give to violence its proper balance in the scheme of things.... But as an endless diet for the sake of excitement and sensation in stories whose plots are vehicles for killing and torture and little more, it is not healthy for young children. Unfamiliar as yet with the full story of human response, they are being misled when they are offered perversion before they have fully learned what is sound.”
—Dorothy H. Cohen (20th century)
“There is nothing but is related to us, nothing that does not interest us,—kingdom, college, tree, horse, or iron show,—the roots of all things are in man.”
—Ralph Waldo Emerson (1803–1882)
“Civilization is a process in the service of Eros, whose purpose is to combine single human individuals, and after that families, then races, peoples and nations, into one great unity, the unity of mankind. Why this has to happen, we do not know; the work of Eros is precisely this.”
—Sigmund Freud (1856–1939)