# 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[edit]

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 ^{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[edit]

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[edit]

If *V* is a subvariety of an *n*-dimensional vector space, defined by an ideal *I*, then *R = F _{n}* /

*I*, where

*F*is the ring of smooth/analytic/holomorphic functions on this vector space. The Zariski tangent space at

_{n}*x*is

*m*(_{n}/*I+m*)_{n}^{2}*,*

where *m _{n}* is the maximal ideal consisting of those functions in

*F*vanishing at

_{n}*x*.

In the planar example above, *I* = (*F*(*X,Y*)), and *I+m ^{2} =* (

*L*(

*X,Y*))

*+m*

^{2}.## Properties[edit]

If *R* is a Noetherian local ring, the dimension of the tangent space is at least the dimension of *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*]*/*(*t ^{2}*), the dual numbers for

*K*; in the parlance of schemes, morphisms from

*Spec*

*K*[

*t*]

*/*(

*t*) to a scheme

^{2}*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 be the ring of continuously differentiable real-valued functions on . Define 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 for define linearly independent vectors in the Zariski cotangent space , so the dimension of is at least the , the cardinality of the continuum. The dimension of the Zariski tangent space is therefore at least . 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]}

## See also[edit]

## Notes[edit]

### Citations[edit]

**^**Eisenbud & Harris 1998, I.2.2, pg. 26.**^**James McKernan,*Smoothness and the Zariski Tangent Space*, 18.726 Spring 2011 Lecture 5**^**Hartshorne 1977, Exercise II 2.8.

## Sources[edit]

- Eisenbud, David; Harris, Joe (1998).
*The Geometry of Schemes*. Springer-Verlag. ISBN 0-387-98637-5 – via Internet Archive. - Hartshorne, Robin (1977).
*Algebraic Geometry*. Graduate Texts in Mathematics. Vol. 52. New York: Springer-Verlag. ISBN 978-0-387-90244-9. MR 0463157. - 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. MR 0021694. Zbl 0031.26101.

## External links[edit]

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