Zariski tangent space

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.

Motivation
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 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 (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 $$\mathfrak{m}$$ is defined to be
 * $$\mathfrak{m}/\mathfrak{m}^2$$

where $$\mathfrak{m}$$2 is given by the product of ideals. It is a vector space over the residue field k:= R/$$\mathfrak{m}$$. 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 $$\mathfrak{m}$$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 $$\mathcal{O}_{X,P}$$. Due to the functoriality of Spec, the natural quotient map $$f:R\rightarrow R/I$$ induces a homomorphism $$g:\mathcal{O}_{X,f^{-1}(P)}\rightarrow \mathcal{O}_{Y,P}$$ 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


 * $$\mathfrak{m}_P/\mathfrak{m}_P^2$$
 * $$\cong (\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^2+I)/I)$$
 * $$\cong \mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^2+I)$$
 * $$\cong (\mathfrak{m}_{f^{-1}P}/\mathfrak{m}_{f^{-1}P}^2)/\mathrm{Ker}(k).$$

Since this is a surjection, the transpose $$k^*:T_P(Y) \rarr T_{f^{-1}P}(X)$$ 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 = Fn / I, where Fn is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at x is
 * mn / (I+mn2),

where mn is the maximal ideal consisting of those functions in Fn vanishing at x.

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

Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R:
 * $$\dim{\mathfrak{m}/\mathfrak{m}^2 \geq \dim{R}}$$

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. 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 $$C^1(\mathbf{R})$$ be the ring of continuously differentiable real-valued functions on $$\mathbf{R}$$. Define $$R = C_0^1(\mathbf{R})$$ 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 $$x^\alpha$$ for $$\alpha \in (1, 2)$$ define linearly independent vectors in the Zariski cotangent space $$\mathfrak{m}/\mathfrak{m}^2$$, so the dimension of $$\mathfrak{m}/\mathfrak{m}^2$$ is at least the $$\mathfrak{c}$$, the cardinality of the continuum. The dimension of the Zariski tangent space $$(\mathfrak{m}/\mathfrak{m}^2)^*$$ is therefore at least $$2^\mathfrak{c}$$. 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.