Definition
The cotangent space of a local ring R, with maximal ideal m is defined to be
- m/m2
It is a vector space over the residue field k := R/m. Its dual (as a k-vector space) is called tangent space of R.
This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic variety V and a point v of V. Morally, modding out m2 corresponds to dropping the non-linear terms from the equations defining V inside some affine space, therefore giving a system of linear equations that define the tangent space.
(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)
Read more about this topic: Zariski Tangent Space
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