Ringed Space - Tangent Spaces

Tangent Spaces

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_X; we want to define the tangent space T_x at the point x ∈ X. Take the local ring (stalk) R_x at the point x, with maximal ideal m_x. Then k_x := R_x/m_x is a field and m_x/m_x2 is a vector space over that field (the cotangent space). The tangent space T_x is defined as the dual of this vector space.

The idea is the following: a tangent vector at x should tell you how to "differentiate" "functions" at x, i.e. the elements of R_x. Now it is enough to know how to differentiate functions whose value at x is zero, since all other functions differ from these only by a constant, and we know how to differentiate constants. So we only need to worry about m_x. Furthermore, if two functions are given with value zero at x, then their product has derivative 0 at x, by the product rule. So we only need to know how to assign "numbers" to the elements of m_x/m_x2, and this is what the dual space does.