In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the Zariski tangent space is a construction that defines a
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
at a point ''P'' on an
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...
''V'' (and more generally). It does not use
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, being based directly on
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, and in the most concrete cases just the theory of a
system of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
.
Motivation
For example, suppose given a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...
''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 mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
. 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
.)
It is easy to see that over the
real field we can obtain ''L'' in terms of the first
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s of ''F''. When those both are 0 at ''P'', we have a
singular point (
double point
In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied.
Algebraic curves in the plane
Algebraic curv ...
,
cusp
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurc ...
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 In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic num ...
''R'', with
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals cont ...
is defined to be
:
where
2 is given by the
product of ideals. It is a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
over the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
''k:= R/
''. 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 ''
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
.
[''Smoothness and the Zariski Tangent Space'', James McKernan]
18.726 Spring 2011
Lecture 5 Since morphisms of fields are injective, the surjection of the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
s 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
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
is an injection.
(One often defines the
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
and
cotangent space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
s 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+m
2 ='' (''L''(''X,Y''))''+m
2.''
Properties
If ''R'' is a
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
local ring, the dimension of the tangent space is at least the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
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'' 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
In algebra, the dual numbers are a hypercomplex number system first introduced in the 19th century. They are expressions of the form , where and are real numbers, and is a symbol taken to satisfy \varepsilon^2 = 0 with \varepsilon\neq 0.
Du ...
for ''K''; in the parlance of
schemes,
morphisms
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
from ''Spec'' ''K''
't'''/''(''t
2'') to a scheme ''X'' over ''K'' correspond to a choice of a
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
''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
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.
See also
*
Tangent cone
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
Definitions in nonlinear analysis
In nonlinear analysis, there are many definitions for a tangen ...
*
Jet (mathematics) In mathematics, the jet is an operation that takes a differentiable function ''f'' and produces a polynomial, the truncated Taylor polynomial of ''f'', at each point of its domain. Although this is the definition of a jet, the theory of jets regards ...
Notes
Citations
Sources
*
*
*
{{refend
External links
Zariski tangent space V.I. Danilov (originator), Encyclopedia of Mathematics.
Algebraic geometry
Differential algebra