Zariski tangent space

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In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally). It does not use differential calculus, being based directly on abstract algebra, and in the most concrete cases just the theory of a system of linear equations.

1 Example: plane curve
2 Definition
3 Properties
4 See also
5 References

[edit] Example: plane curve

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

F(X,Y) = 0

and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get a 'linearised' equation reading

L(X,Y) = 0

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.

[edit] 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.

[edit] Properties

If R is a noetherian local ring, the dimension of the tangent space is at least the dimension of R:

dim m/m² ≧ dim R

By definition, R is 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,

K[t]/[t²]:

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. Therefore, one also talks about tangent vectors.