Zariski tangent space

Template:Short description Template:Use American EnglishIn 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.

MotivationEdit

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

F(X,Y) = 0

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

DefinitionEdit

The cotangent space of a local ring R, with maximal ideal <math>\mathfrak{m}</math> is defined to be

<math>\mathfrak{m}/\mathfrak{m}^2</math>

where <math>\mathfrak{m}</math>² is given by the product of ideals. It is a vector space over the residue field k:= R/<math>\mathfrak{m}</math>. Its dual (as a k-vector space) is called tangent space of R.Template:Sfn

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 <math>\mathfrak{m}</math>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 <math>T_P(X)</math> and cotangent space <math>T_P^*(X)</math> to a scheme X at a point P is the (co)tangent space of <math>\mathcal{O}_{X,P}</math>. Due to the functoriality of Spec, the natural quotient map <math>f:R\rightarrow R/I</math> induces a homomorphism <math>g:\mathcal{O}_{X,f^{-1}(P)}\rightarrow \mathcal{O}_{Y,P}</math> for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed <math>T_P(Y)</math> in <math>T_{f^{-1}P}(X)</math>.<ref>James McKernan, Smoothness and the Zariski Tangent Space, 18.726 Spring 2011 Lecture 5</ref> 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

<math>\mathfrak{m}_P/\mathfrak{m}_P^2</math>

<math>\cong (\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^2+I)/I)</math>

<math>\cong \mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^2+I)</math>

<math>\cong (\mathfrak{m}_{f^{-1}P}/\mathfrak{m}_{f^{-1}P}^2)/\mathrm{Ker}(k).</math>

Since this is a surjection, the transpose <math>k^*:T_P(Y) \rarr T_{f^{-1}P}(X)</math> is an injection.

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

Analytic functionsEdit

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

m_n / (I+m_n²),

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

In the planar example above, I = (F(X,Y)), and I+m² = (L(X,Y))+m².

PropertiesEdit

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

<math>\dim{\mathfrak{m}/\mathfrak{m}^2 \geq \dim{R}}</math>

R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V at a point 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 K[t]/(t²), the dual numbers for K; in the parlance of schemes, morphisms from 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.Template:Sfn Therefore, one also talks about tangent vectors. See also: tangent space to a functor.

In general, the dimension of the Zariski tangent space can be extremely large. For example, let <math>C^1(\mathbf{R})</math> be the ring of continuously differentiable real-valued functions on <math>\mathbf{R}</math>. Define <math>R = C_0^1(\mathbf{R})</math> to be the ring of germs of such functions at the origin. Then R is a local ring, and its maximal ideal m consists of all germs which vanish at the origin. The functions <math>x^\alpha</math> for <math>\alpha \in (1, 2)</math> define linearly independent vectors in the Zariski cotangent space <math>\mathfrak{m}/\mathfrak{m}^2</math>, so the dimension of <math>\mathfrak{m}/\mathfrak{m}^2</math> is at least the <math>\mathfrak{c}</math>, the cardinality of the continuum. The dimension of the Zariski tangent space <math>(\mathfrak{m}/\mathfrak{m}^2)^*</math> is therefore at least <math>2^\mathfrak{c}</math>. On the other hand, the ring of germs of smooth functions at a point in an n-manifold has an n-dimensional Zariski cotangent space.Template:Efn