Page "Ringed space" Paragraph 13

Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces.

Let X be locally ringed space with structure sheaf O < sub > X </ sub >; we want to define the tangent space T < sub > x </ sub > at the point x ∈ X.

Take the local ring ( stalk ) R < sub > x </ sub > at the point x, with maximal ideal m < sub > x </ sub >.

Then k < sub > x </ sub > := R < sub > x </ sub >/ m < sub > x </ sub > is a field and m < sub > x </ sub >/ m < sub > x </ sub >< sup > 2 </ sup > is a vector space over that field ( the cotangent space ).

The tangent space T < sub > x </ sub > is defined as the dual of this vector space.