Zariski tangent space explained

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

ak{m}

is defined to be

ak{m}/ak{m}²

where

ak{m}

² is given by the product of ideals. It is a vector space over the residue field k:= R/
ak{m}

. 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

ak{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

	*(X)
T
	P

to a scheme X at a point P is the (co)tangent space of

l{O}_X,P

. Due to the functoriality of Spec, the natural quotient map

f:R → R/I

induces a homomorphism

g:l{O}
	X,f^-1(P)

→ l{O}_Y,P

for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed

T_P(Y)

T
	f^-1P

(X)

.^[1] 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

ak{m}_P/ak{m}

	2

	P

\cong

(ak{m}
	f^-1P

	2+I)/I)
/I)/((ak{m}
	f^-1P

\cong

ak{m}
	f^-1P

	2+I)
/(ak{m}
	f^-1P

\cong

(ak{m}
	f^-1P

	2)/Ker(k).
/ak{m}
	f^-1P

	*:T
k
	P(Y)

\rarr

T
	f^-1P

(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{ak{m}/ak{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. 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(R)

be the ring of continuously differentiable real-valued functions on

. Define

	1(R)
C
	0

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

ak{m}/ak{m}²

, so the dimension of

ak{m}/ak{m}²

is at least the

ak{c}

, the cardinality of the continuum. The dimension of the Zariski tangent space

(ak{m}/ak{m}²⁾^*

is therefore at least

2^ak{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.

Notes

Citations

Sources

Book: The Geometry of Schemes . Eisenbud . David . Harris . Joe . David Eisenbud . Joe Harris (mathematician) . 1998 . . . 0-387-98637-5.
Book: Hartshorne, Robin . . 1977 . Robin Hartshorne . Springer-Verlag . New York . 52 . . 978-0-387-90244-9 . 0463157.
The concept of a simple point of an abstract algebraic variety . Zariski . Oscar . Oscar Zariski . . 1947 . 62 . 1–52 . 10.1090/S0002-9947-1947-0021694-1 . 0021694 . 0031.26101. free.

External links

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

Notes and References

[James McKernan]