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 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 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 [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.[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 [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].[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
- [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 = 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 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.[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 [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.[lower-alpha 1]
See also
Notes
Citations
- ↑ Eisenbud & Harris 1998, I.2.2, pg. 26.
- ↑ Smoothness and the Zariski Tangent Space, James McKernan, 18.726 Spring 2011 Lecture 5
- ↑ Hartshorne 1977, Exercise II 2.8.
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.
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