For example, suppose given a plane curveC defined by a polynomial equation and take P to be the origin. Erasing terms of higher order than 1 would produce a 'linearised' equation reading in which all terms XaYb have been discarded if a + b > 1. We have two cases: L may be 0, or it may be the equation of a line. In the first case the tangent space to C at is the whole plane, considered as a two-dimensional affine space. In the second case, the tangent space is that line, considered as affine space. It is easy to see that over the real field we can obtain L in terms of the first partial derivatives of F. When those both are 0 at P, we have a singular point. The general definition is that singular points of C are the cases when the tangent space has dimension 2.
Definition
The cotangent space of a local ringR, with maximal ideal is defined to be where 2 is given by the product of ideals. It is a vector space over the residue fieldk := R/. Its dual is called tangent space of R. This definition is a generalization of the above example to higher dimensions: suppose given an affine algebraic varietyV and a point v of V. Morally, modding out2 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. The tangent space and cotangent space to a scheme X at a point P is the tangent space of. Due to the functoriality of Spec, the natural quotient map induces a homomorphism for X=Spec, P a point in Y=Spec. This is used to embed in. Since morphisms of fields are injective, the surjection of the residue fields induced by g is an isomorphism. Then a morphism k of the cotangent spaces is induced by g, given by Since this is a surjection, the transpose is an injection.
Analytic functions
If V is a subvariety of an n-dimensional vector space, defined by an ideal I, then R = Fn/I, where Fn is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is where mn is the maximal ideal consisting of those functions in Fn vanishing at x. In the planar example above, I = , and ''I+m2 = +m2.
Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R: R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V in v, one also says that v is a regular point. Otherwise it is called a singular point. The tangent space has an interpretation in terms of homomorphisms to the dual numbers for K, in the parlance of schemes, morphisms Spec K/t2 to a scheme X over K correspond to a choice of a rational pointx ∈ X and an element of the tangent space at x. Therefore, one also talks about tangent vectors. See also: tangent space to a functor.
