Multiplicative group explained
In mathematics and group theory, the term multiplicative group refers to one of the following concepts:
Examples
. When
n is not prime, there are elements other than zero that are not invertible.
is an
abelian group with 1 its
identity element. The
logarithm is a
group isomorphism of this group to the
additive group of real numbers,
.
- The multiplicative group of a field
is the set of all nonzero elements:
, under the multiplication operation. If
is
finite of order
q (for example
q =
p a prime, and
), then the multiplicative group is cyclic:
.
Group scheme of roots of unity
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.
The resulting group scheme is written μn (or
[2]). It gives rise to a
reduced scheme, when we take it over a field
K,
if and only if the
characteristic of
K does not divide
n. This makes it a source of some key examples of non-reduced schemes (schemes with
nilpotent elements in their
structure sheaves); for example μ
p over a
finite field with
p elements for any
prime number p.
This phenomenon is not easily expressed in the classical language of algebraic geometry. For example, it turns out to be of major importance in expressing the duality theory of abelian varieties in characteristic p (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.
See also
Notes
- See Hazewinkel et al. (2004), p. 2.
- Book: Milne, James S. . Étale cohomology . Princeton University Press . 1980 . xiii, 66 .
References
- Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004.