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 given a plane curve ''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 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 $\mathfrak$ is defined to be
:$\mathfrak/\mathfrak^2$
where $\mathfrak$² is given by the product of ideals. It is a vector space over the residue field ''k:= R/$\mathfrak$''. 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 ''$\mathfrak$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 $T_P^*(X)$ to a scheme ''X'' at a point ''P'' is the (co)tangent space of $\mathcal_$. Due to the functoriality of Spec, the natural quotient map $f:R\rightarrow R/I$ induces a homomorphism $g:\mathcal_\rightarrow \mathcal_$ for ''X''=Spec(''R''), ''P'' a point in ''Y''=Spec(''R/I''). This is used to embed $T_P(Y)$ in $T_(X)$.''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 fields induced by ''g'' is an isomorphism. Then a morphism ''k'' of the cotangent spaces is induced by ''g'', given by

:$\mathfrak_P/\mathfrak_P^2$
:$\cong (\mathfrak_/I)/((\mathfrak_^2+I)/I)$
:$\cong \mathfrak_/(\mathfrak_^2+I)$
:$\cong (\mathfrak_/\mathfrak_^2)/\mathrm(k).$

Since this is a surjection, the transpose $k^*:T_P(Y) \rarr T_(X)$ 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 ''m/m²'' ≧ 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 $C^1(\mathbf)$ be the ring of continuously differentiable real-valued functions on $\mathbf$. Define $R = C_0^1(\mathbf)$ 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 $m/m^2$, so the dimension of $m/m^2$ is at least the $\mathfrak$, the cardinality of the continuum. The dimension of the Zariski tangent space $(m/m^2)^*$ is therefore at least $2^\mathfrak$. 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

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External links

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

Algebraic geometry
Differential algebra