An algebraic torus over a field 
is an algebraic group which
becomes a direct product of copies of the multiplicative
group after extension to an algebraic closure.
A split torus over 
is already isomorphic over 
to 
. Over 
, its complex points are 
. Its characters are algebraic homomorphisms
to 
,
and its cocharacters are algebraic homomorphisms
from 
.
For a split torus, both lattices are isomorphic to
.
On an affine scheme with an algebraic torus group action, a cocharacter induces a graded algebra structure on its coordinate ring, indexed by integers. The fixed subscheme for this cocharacter has coordinate ring given by the quotient ring by the ideal generated by every nonzero-degree piece in this grading. The quotient ring may retain nilpotent functions (Milne 2017).
