Multiplicative group Totally Explained

Everything about Group Scheme Of Roots Of Unity totally explained

In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context

any group

scriptstylemathfrak

doesn't 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 $mu_p$ over a finite field with

p

elements for any prime number

p

.
This phenomenon isn't 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

p

(theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.

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