Zariski tangent space

Motivation

For example, suppose given a plane curve C defined by a polynomial equation

and take P to be the origin (0,0). Erasing terms of higher order than 1 would produce a 'linearised' equation reading

in which all terms X^aY^b 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 (Zariski) tangent space to C at (0,0) 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. (The question of the origin comes up, when we take P as a general point on C; it is better to say 'affine space' and then note that P is a natural origin, rather than insist directly that it is a vector 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 (double point, cusp or something more complicated). 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 ring R, with maximal ideal m is defined to be

where m ² is given by the product of ideals. 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 m ² 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 T P ( X ) and cotangent space T P ∗ ( X ) to a scheme X at a point P is the (co)tangent space of O X , P . Due to the functoriality of Spec, the natural quotient map f : R → R / I induces a homomorphism g : O X , f − 1 ( P ) → O Y , P for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed T P ( Y ) in T f − 1 P ( X ) . 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 k ∗ : T P ( Y ) → T f − 1 P ( X ) is an injection.

(One often defines the tangent and cotangent spaces for a manifold in the analogous manner.)

Analytic functions

If V is a subvariety of an n-dimensional vector space, defined by an ideal I, then R = F_n/I, where F_n is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is

where m_n is the maximal ideal consisting of those functions in F_n vanishing at x.

In the planar example above, I = <F>, and I+m² = <L>+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[t]/t² to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x. Therefore, one also talks about tangent vectors. See also: tangent space to a functor.

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