View source for Tangent space to a functor

{{Short description|Concept in category theory}}
In algebraic geometry, the '''tangent space to a functor''' generalizes the classical construction of a tangent space such as the [[Zariski tangent space]]. The construction is based on the following observation.<ref>{{harvnb|Hartshorne|1977|loc=Exercise II 2.8}}</ref> Let ''X'' be a scheme over a field ''k''.

:To give a <math>k[\epsilon]/(\epsilon)^2</math>-point of ''X'' is the same thing as to give a ''k''-[[Rational point|rational point]] ''p'' of ''X'' (i.e., the residue field of ''p'' is ''k'') together with an element of <math>(\mathfrak{m}_{X, p}/\mathfrak{m}_{X, p}^2)^*</math>; i.e., a tangent vector at ''p''.
(To see this, use the fact that any local homomorphism <math>\mathcal{O}_p \to k[\epsilon]/(\epsilon)^2</math> must be of the form
:<math>\delta_p^v: u \mapsto u(p) + \epsilon v(u), \quad v \in \mathcal{O}_p^*.</math>)
Let ''F'' be a functor from the category of ''k''-algebras to the category of sets. Then, for any ''k''-point <math>p \in F(k)</math>, the fiber of <math>\pi: F(k[\epsilon]/(\epsilon)^2) \to F(k)</math> over ''p'' is called the '''tangent space''' to ''F'' at ''p''.<ref>{{harvnb|Eisenbud|Harris|1998|loc=VI.1.3}}</ref>
If the functor ''F''  preserves fibered products (e.g. if it is a scheme), the tangent space may be given the structure of a vector space over ''k''. If ''F'' is a scheme ''X'' over ''k'' (i.e., <math>F = \operatorname{Hom}_{\operatorname{Spec}k}(\operatorname{Spec}-, X)</math>), then each ''v'' as above may be identified with a derivation at ''p'' and this gives the identification of <math>\pi^{-1}(p)</math> with the space of derivations at ''p'' and we recover the usual construction.

The construction may be thought of as defining an analog of the [[Tangent bundle|tangent bundle]] in the following way.<ref>{{harvnb|Borel|1991|loc=AG 16.2}}</ref> Let <math>T_X = X(k[\epsilon]/(\epsilon)^2)</math>. Then, for any morphism <math>f: X \to Y</math> of schemes over ''k'', one sees <math>f^{\#}(\delta_p^v) = \delta_{f(p)}^{df_p(v)}</math>; this shows that the map <math>T_X \to T_Y</math> that ''f'' induces is precisely the differential of ''f'' under the above identification.

== References ==
{{Reflist}}
*{{Citation | last1=Borel | first1=Armand |  title=Linear algebraic groups | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-97370-8 | mr=1102012 | year=1991 | orig-year=1969 |volume=126}}
*{{cite book
 | first1 = David 
 | last1= Eisenbud
 | first2=Joe 
 |last2=Harris 
 | year = 1998
 | title = The Geometry of Schemes
 | url = https://archive.org/details/springer_10.1007-978-0-387-22639-2
 | publisher = [[Company:Springer Science+Business Media|Springer-Verlag]]
 | isbn = 0-387-98637-5
 | zbl = 0960.14002
}}
*{{Hartshorne AG}}

[[Category:Algebraic geometry]]

{{Sourceattribution|Tangent space to a functor}}