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:
“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another group is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”
—Gore Vidal (b. 1925)
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“Though of erect nature, man is far above the plants. For man’s superior part, his head, is turned toward the superior part of the world, and his inferior part is turned toward the inferior world; and therefore he is perfectly disposed as to the general situation of his body. Plants have the superior part turned towards the lower world, since their roots correspond to the mouth, and their inferior parts towards the upper world.”
—Thomas Aquinas (c. 1225–1274)
“The unity of effect or impression is a point of the greatest importance. It is clear, moreover, that this unity cannot be thoroughly preserved in productions whose perusal cannot be completed at one sitting.”
—Edgar Allan Poe (1809–1849)