The vector space over the residue field
of the point
that is dual to the space
,
where
is the maximal ideal of the
local ring
of
on
.
If
is defined by a system of equations
where
,
then the Zariski tangent space at a rational point
is defined by the system of linear equations
A variety
is non-singular at a rational point
if and only if the dimension of the Zariski tangent space to
at
is equal to the dimension of
.
For a rational point
,
the Zariski tangent space is dual to the space
— the
stalk at
of the cotangent sheaf
.
An irreducible variety
over a perfect field
is smooth if and only if the sheaf
is locally free. The vector bundle
associated with
is called the
tangent bundle
of
over
;
it is functorially related to
.
Its sheaf of sections is called the
tangent sheaf
to
.
The Zariski tangent space was introduced by
O. Zariski
[1].