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 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 ${\mathfrak {m}}$ 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.^[1]

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 $T_{f^{-1}P}(X)$ .^[2] 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 $k^{*}:T_{P}(Y)\rightarrow T_{f^{-1}P}(X)$ 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

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.

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.^[3] 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.^[a]