Zariski tangent space

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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}}} {\displaystyle {\mathfrak {m}}} is defined to be

m / m 2 {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}} {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}

where m {\displaystyle {\mathfrak {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}}} {\displaystyle {\mathfrak {m}}}. 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 m {\displaystyle {\mathfrak {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)} {\displaystyle T_{P}(X)} and cotangent space T P ∗ ( X ) {\displaystyle 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}} {\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} {\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}} {\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)} {\displaystyle T_{P}(Y)} in T f − 1 P ( X ) {\displaystyle T_{f^{-1}P}(X)} {\displaystyle T_{f^{-1}P}(X)}.[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

m P / m P 2 {\displaystyle {\mathfrak {m}}_{P}/{\mathfrak {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)} {\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)} {\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).} {\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)} {\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}}} {\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.[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 C 1 ( R ) {\displaystyle C^{1}(\mathbf {R} )} {\displaystyle C^{1}(\mathbf {R} )} be the ring of continuously differentiable real-valued functions on R {\displaystyle \mathbf {R} } {\displaystyle \mathbf {R} }. Define R = C 0 1 ( R ) {\displaystyle R=C_{0}^{1}(\mathbf {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 }} {\displaystyle x^{\alpha }} for α ∈ ( 1 , 2 ) {\displaystyle \alpha \in (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}} {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}}, so the dimension of m / m 2 {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}} {\displaystyle {\mathfrak {m}}/{\mathfrak {m}}^{2}} is at least the c {\displaystyle {\mathfrak {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})^{*}} {\displaystyle ({\mathfrak {m}}/{\mathfrak {m}}^{2})^{*}} is therefore at least 2 c {\displaystyle 2^{\mathfrak {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.[a]

See also

Notes

Citations

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

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