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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]].
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== 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<sup>a</sup>Y<sup>b''</sup> 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 number|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 [[Mathematical singularity|singular point]] ([[double point]], [[cusp (singularity)|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'' is defined to be
:<math>\mathfrak{m}/\mathfrak{m}^2</math>
where ''m<sup>2</sup>'' is given by the [[product of ideals]]. It is a [[vector space]] over the [[residue field]] ''k := R/m''. Its [[dual vector space|dual]] (as a ''k''-vector space) is called '''tangent space''' of ''R''.<ref>{{harvnb|Eisenbud|1998|loc=I.2.2, pg. 26}}</ref>
 
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<sup>2</sup>'' 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 <math>T_P(X)</math> and cotangent space <math>T_P^*(X)</math> to a scheme ''X'' at a point ''P'' is the (co)tangent space of <math>\mathcal{O}_{X,P}</math>. Due to the [[Spectrum of a ring#Functoriality|functoriality of Spec]], the natural quotient map <math>f:R\rightarrow R/I</math> induces a homomorphism <math>g:\mathcal{O}_{X,f^{-1}(P)}\rightarrow \mathcal{O}_{Y,P}</math> for ''X''=Spec(''R''), ''P'' a point in ''Y''=Spec(''R/I''). This is used to embed <math>T_P(Y)</math> in <math>T_{f^{-1}P}(X)</math>.<ref>''Smoothness and the Zariski Tangent Space'', James McKernan, [http://math.mit.edu/~mckernan/Teaching/10-11/Spring/18.726/lectures.html 18.726 Spring 2011] Lecture 5</ref> Since morphisms of fields are injective, the surjection of the [[residue field]]s induced by ''g'' is an isomorphism. Then a morphism ''k'' of the cotangent spaces is induced by ''g'', given by
 
:<math>\mathfrak{m}_P/\mathfrak{m}_P^2</math>
:<math>\cong (\mathfrak{m}_{f^{-1}P}/I)/((\mathfrak{m}_{f^{-1}P}^2+I)/I)</math>
:<math>\cong \mathfrak{m}_{f^{-1}P}/(\mathfrak{m}_{f^{-1}P}^2+I)</math>
:<math>\cong (\mathfrak{m}_{f^{-1}P}/\mathfrak{m}_{f^{-1}P}^2)/\mathrm{Ker}(k).</math>
 
Since this is a surjection, the [[Transpose#Transpose of linear maps|transpose]] <math>k^*:T_P(Y) \rarr T_{f^{-1}P}(X)</math> is an injection.
 
(One often defines the [[tangent space|tangent]] and [[cotangent space]]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<sub>n</sub>/I'', where ''F<sub>n</sub>'' is the ring of smooth/analytic/holomorphic functions on this vector space.  The Zariski tangent space at ''x'' is
:''m<sub>n</sub> / ( I+m<sub>n</sub><sup>2</sup> ),
where ''m<sub>n</sub>'' is the maximal ideal consisting of those functions in ''F<sub>n</sub>'' vanishing at ''x''.
 
In the planar example above, ''I'' = <''F''>, and ''I+m<sup>2</sup> = <L>+m<sup>2</sup>.
 
== Properties ==
If ''R'' is a [[noetherian ring|Noetherian]] local ring, the dimension of the tangent space is at least the [[Krull dimension|dimension]] of ''R'':
:dim ''m/m<sup>2</sup>'' ≧ dim ''R''
''R'' is called [[regular local ring|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'',
 
:''K[t]/t<sup>2</sup>'':
 
in the parlance of [[scheme (mathematics)|schemes]], morphisms ''Spec K[t]/t<sup>2</sup>'' 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''.<ref>{{harvnb|Hartshorne|1977|loc=Exercise II 2.8}}</ref> Therefore, one also talks about '''tangent vectors'''. See also: [[tangent space to a functor]].
 
== See also ==
 
* [[Tangent cone]]
* [[Jet (mathematics)]]
 
==References==
{{reflist}}
 
*{{Hartshorne AG}}
*{{cite book
| author = [[David Eisenbud]]
| coauthors = [[Joe Harris (mathematician)|Joe Harris]]
| year = 1998
| title = The Geometry of Schemes
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| isbn = 0-387-98637-5
}}
 
== External links ==
* [http://www.encyclopediaofmath.org/index.php/Zariski_tangent_space Zariski tangent space]. V.I. Danilov (originator), Encyclopedia of Mathematics.  
 
[[Category:Algebraic geometry]]
[[Category:Differential algebra]]

Latest revision as of 06:52, 13 November 2014

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