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The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group GL(1), considered as a group scheme.That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product of schemes, with the morphism e that serves as the identity.
foobar2000 [a] (often abbreviated as fb2k or f2k) is a freeware audio player for Microsoft Windows, iOS, Android, macOS, and formerly Windows Phone, developed by Peter Pawłowski. It has a modular design , which provides user flexibility in configuration and customization. [ 5 ]
This functor is representable by a finite flat S-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself. If G is a constant commutative group scheme, then its Cartier dual is the diagonalizable group D ( G ), and vice versa.
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field.
A smooth formal group scheme is a special case of a formal group scheme. Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections. The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the ...
Foo Camp is an annual hacker convention.; BarCamp, an international network of user-generated conferences.; During the United States v.Microsoft Corp. trial, evidence was presented that Microsoft had tried to use the Web Services Interoperability organization (WS-I) as a means to stifle competition, including e-mails in which top executives including Bill Gates and Steve Ballmer referred to ...
Every group G can be viewed as a category with a single object; morphisms in this category are just the elements of G. Given an arbitrary category C, a representation of G in C is a functor from G to C. Such a functor selects an object X in C and a group homomorphism from G to Aut(X), the automorphism group of X.
They are determined by an abelian coefficient group , and denoted by (,) (where is sometimes omitted, especially if it is ). Usually G {\displaystyle G} is the integers, the rationals, the reals, the complex numbers, or the integers mod a prime p {\displaystyle p} .