Zariski tangent space
Encyclopedia
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.
C defined by a polynomial equation
and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get a 'linearised' equation reading
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 derivative
s 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.
R, with maximal ideal
m is defined to be
It is a vector space
over the residue field
k := R/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 m2 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.
(One often defines the tangent
and cotangent space
s for a manifold in the analogous manner.)
where mn is the maximal ideal consisting of those functions in Fn vanishing at x.
In the planar example above, I = <F>, and I+m2 =+m2.
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 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 homomorphism
s to the dual numbers for K,
in the parlance of schemes
, morphisms 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. Therefore, one also talks about tangent vectors.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, the Zariski tangent space is a construction that defines a tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
at a point P on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
V (and more generally). It does not use differential calculus
Differential calculus
In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus....
, being based directly on abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, and in the most concrete cases just the theory of a system of linear equations.
Example: plane curve
For example, suppose given a plane curvePlane curve
In mathematics, a plane curve is a curve in a Euclidean plane . The most frequently studied cases are smooth plane curves , and algebraic plane curves....
C defined by a polynomial equation
- F(X,Y) = 0
and take P to be the origin (0,0). When F is considered only in terms of its first-degree terms, we get 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
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
. 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
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
.)
It is easy to see that over the real field
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
we can obtain L in terms of the first partial derivative
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...
s of F. When those both are 0 at P, we have a singular point
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
(double point, cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
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 ringLocal ring
In abstract algebra, more particularly in 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 number fields examined at a particular place, or...
R, with maximal ideal
Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
m is defined to be
- m/m2
It is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the residue field
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...
k := R/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 m2 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.
(One often defines the tangent
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
and cotangent space
Cotangent space
In differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
s for a manifold in the analogous manner.)
Example extended
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>, and I+m2 =
Properties
If R is a NoetherianNoetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
local ring, the dimension of the tangent space is at least the dimension
Krull dimension
In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
of R:
- dim m/m2 ≧ dim R
R is called regular
Regular local ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
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 homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s to the dual numbers for K,
- K[t]/[t2]:
in the parlance of schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
, morphisms 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. Therefore, one also talks about tangent vectors.