- Source: 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 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 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 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
m
{\displaystyle {\mathfrak {m}}}
is defined to be
m
/
m
2
{\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
where
m
{\displaystyle {\mathfrak {m}}}
2 is given by the product of ideals. It is a vector space over the residue field k:= R/
m
{\displaystyle {\mathfrak {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
m
{\displaystyle {\mathfrak {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
)
{\displaystyle T_{P}(X)}
and cotangent space
T
P
∗
(
X
)
{\displaystyle T_{P}^{*}(X)}
to a scheme X at a point P is the (co)tangent space of
O
X
,
P
{\displaystyle {\mathcal {O}}_{X,P}}
. Due to the functoriality of Spec, the natural quotient map
f
:
R
→
R
/
I
{\displaystyle f:R\rightarrow R/I}
induces a homomorphism
g
:
O
X
,
f
−
1
(
P
)
→
O
Y
,
P
{\displaystyle g:{\mathcal {O}}_{X,f^{-1}(P)}\rightarrow {\mathcal {O}}_{Y,P}}
for X=Spec(R), P a point in Y=Spec(R/I). This is used to embed
T
P
(
Y
)
{\displaystyle T_{P}(Y)}
in
T
f
−
1
P
(
X
)
{\displaystyle T_{f^{-1}P}(X)}
. 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
m
P
/
m
P
2
{\displaystyle {\mathfrak {m}}_{P}/{\mathfrak {m}}_{P}^{2}}
≅
(
m
f
−
1
P
/
I
)
/
(
(
m
f
−
1
P
2
+
I
)
/
I
)
{\displaystyle \cong ({\mathfrak {m}}_{f^{-1}P}/I)/(({\mathfrak {m}}_{f^{-1}P}^{2}+I)/I)}
≅
m
f
−
1
P
/
(
m
f
−
1
P
2
+
I
)
{\displaystyle \cong {\mathfrak {m}}_{f^{-1}P}/({\mathfrak {m}}_{f^{-1}P}^{2}+I)}
≅
(
m
f
−
1
P
/
m
f
−
1
P
2
)
/
K
e
r
(
k
)
.
{\displaystyle \cong ({\mathfrak {m}}_{f^{-1}P}/{\mathfrak {m}}_{f^{-1}P}^{2})/\mathrm {Ker} (k).}
Since this is a surjection, the transpose
k
∗
:
T
P
(
Y
)
→
T
f
−
1
P
(
X
)
{\displaystyle k^{*}:T_{P}(Y)\rightarrow T_{f^{-1}P}(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 = 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(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
m
/
m
2
≥
dim
R
{\displaystyle \dim {{\mathfrak {m}}/{\mathfrak {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
)
{\displaystyle C^{1}(\mathbf {R} )}
be the ring of continuously differentiable real-valued functions on
R
{\displaystyle \mathbf {R} }
. Define
R
=
C
0
1
(
R
)
{\displaystyle R=C_{0}^{1}(\mathbf {R} )}
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
α
{\displaystyle x^{\alpha }}
for
α
∈
(
1
,
2
)
{\displaystyle \alpha \in (1,2)}
define linearly independent vectors in the Zariski cotangent space
m
/
m
2
{\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
, so the dimension of
m
/
m
2
{\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}
is at least the
c
{\displaystyle {\mathfrak {c}}}
, the cardinality of the continuum. The dimension of the Zariski tangent space
(
m
/
m
2
)
∗
{\displaystyle ({\mathfrak {m}}/{\mathfrak {m}}^{2})^{*}}
is therefore at least
2
c
{\displaystyle 2^{\mathfrak {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.
See also
Tangent cone
Jet (mathematics)
Notes
= Citations
=Sources
External links
Zariski tangent space. V.I. Danilov (originator), Encyclopedia of Mathematics.