A group scheme over a base scheme 
is a group object in the category of schemes over 
. It consists of an 
-scheme 
with multiplication, identity, and inverse morphisms
satisfying the group axioms. Over a base affine
scheme 
,
where 
is a commutative ring, a group scheme whose underlying
scheme is an affine scheme
is equivalently represented by a commutative 
-Hopf algebra.
For a group scheme 
over a field 
and a commutative ring 
equipped with a unital ring
homomorphism 
,
the 
-valued points of 
are the morphisms 
over 
. They form a group 
, and these groups vary naturally
with 
(Milne 2017, pp. 7-8).
The scheme-theoretic kernel of a homomorphism 
of group schemes over 
is the fiber product 
, where 
is the identity section (Milne 2017, p. 23).
This scheme can have more structure than the set
of points mapped to the identity element. For
example, the power map 
on the multiplicative
group 
,
for a positive integer 
, has kernel ![mu_n=Spec(k[t,t^(-1)]/(t^n-1))](/images/equations/GroupScheme/Inline25.svg)
. Its coordinate
ring has vector space dimension 
over 
, so the kernel is a finite group scheme. If 
has characteristic 
and 
, then 
. The kernel has one point over an algebraic
closure, but its coordinate ring contains
the nonzero nilpotent element 
and is not a reduced ring.
