Zariski tangent space

Short description: Tangent spaces in algebraic geometry

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 given a plane curve C 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 [math]\displaystyle{ \mathfrak{m} }[/math] is defined to be

[math]\displaystyle{ \mathfrak{m}/\mathfrak{m}^2 }[/math]

where [math]\displaystyle{ \mathfrak{m} }[/math]² is given by the product of ideals. It is a vector space over the residue field k:= R/[math]\displaystyle{ \mathfrak{m} }[/math]. 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 [math]\displaystyle{ \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]\displaystyle{ T_P(X) }[/math] and cotangent space [math]\displaystyle{ T_P^*(X) }[/math] to a scheme X at a point P is the (co)tangent space of [math]\displaystyle{ \mathcal{O}_{X,P} }[/math]. Due to the functoriality of Spec, the natural quotient map [math]\displaystyle{ f:R\rightarrow R/I }[/math] induces a homomorphism [math]\displaystyle{ 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]\displaystyle{ T_P(Y) }[/math] in [math]\displaystyle{ T_{f^{-1}P}(X) }[/math]. 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]\displaystyle{ \mathfrak{m}_P/\mathfrak{m}_P^2 }[/math]

[math]\displaystyle{ \cong (\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^2+I)/I) }[/math]

[math]\displaystyle{ \cong \mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^2+I) }[/math]

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

Since this is a surjection, the transpose [math]\displaystyle{ 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 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 m/m² ≧ 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 [math]\displaystyle{ C^1(\mathbf{R}) }[/math] be the ring of continuously differentiable real-valued functions on [math]\displaystyle{ \mathbf{R} }[/math]. Define [math]\displaystyle{ 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]\displaystyle{ x^\alpha }[/math] for [math]\displaystyle{ \alpha \in (1, 2) }[/math] define linearly independent vectors in the Zariski cotangent space [math]\displaystyle{ m/m^2 }[/math], so the dimension of [math]\displaystyle{ m/m^2 }[/math] is at least the [math]\displaystyle{ \mathfrak{c} }[/math], the cardinality of the continuum. The dimension of the Zariski tangent space [math]\displaystyle{ (m/m^2)^* }[/math] is therefore at least [math]\displaystyle{ 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.

See also

• Tangent cone
• Jet (mathematics)

Sources

• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. https://archive.org/details/springer_10.1007-978-0-387-22639-2. 
• Hartshorne, Robin (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. New York: Springer-Verlag. ISBN 978-0-387-90244-9. 
• Zariski, Oscar (1947). "The concept of a simple point of an abstract algebraic variety". Transactions of the American Mathematical Society 62: 1–52. doi:10.1090/S0002-9947-1947-0021694-1. https://www.ams.org/journals/tran/1947-062-01/S0002-9947-1947-0021694-1/. 

External links

• Zariski tangent space. V.I. Danilov (originator), Encyclopedia of Mathematics.