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 X^{a}Y^{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 is defined to be
where ^{2} is given by the product of ideals. It is a vector space over the residue field k := R/. 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 ^{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 and cotangent space to a scheme X at a point P is the (co)tangent space of . Due to the functoriality of Spec, the natural quotient map induces a homomorphism for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed in .[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
Since this is a surjection, the transpose 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_{n}^{2} ),
where m_{n} is the maximal ideal consisting of those functions in F_{n} vanishing at x.
In the planar example above, I = <F>, and I+m^{2} = <L>+m^{2}.
Properties
If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R:
- dim m/m^{2} ≧ dim R
R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V in 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 homomorphisms to the dual numbers for K,
- K[t]/t^{2}:
in the parlance of schemes, morphisms Spec K[t]/t^{2} 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.
See also
References
- Eisenbud 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
Books
- Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
- David Eisenbud; Joe Harris (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5.
External links
- Zariski tangent space. V.I. Danilov (originator), Encyclopedia of Mathematics.