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. Let X be a scheme over a field k.
To give a k [ ϵ ] / ( ϵ ) 2 {\displaystyle k[\epsilon ]/(\epsilon )^{2}} -point of X is the same thing as to give a k-rational point p of X (i.e., the residue field of p is k) together with an element of ( m X , p / m X , p 2 ) ∗ {\displaystyle ({\mathfrak {m}}_{X,p}/{\mathfrak {m}}_{X,p}^{2})^{*}} ; i.e., a tangent vector at p. (To see this, use the fact that any local homomorphism O p → k [ ϵ ] / ( ϵ ) 2 {\displaystyle {\mathcal {O}}_{p}\to k[\epsilon ]/(\epsilon )^{2}} must be of the form
δ p v : u ↦ u ( p ) + ϵ v ( u ) , v ∈ O p ∗ . {\displaystyle \delta _{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in {\mathcal {O}}_{p}^{*}.} ) Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point p ∈ F ( k ) {\displaystyle p\in F(k)} , the fiber of π : F ( k [ ϵ ] / ( ϵ ) 2 ) → F ( k ) {\displaystyle \pi :F(k[\epsilon ]/(\epsilon )^{2})\to F(k)} over p is called the tangent space to F at p. 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., F = Hom Spec k ( Spec − , X ) {\displaystyle F=\operatorname {Hom} _{\operatorname {Spec} k}(\operatorname {Spec} -,X)} ), then each v as above may be identified with a derivation at p and this gives the identification of π − 1 ( p ) {\displaystyle \pi ^{-1}(p)} 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 in the following way. Let T X = X ( k [ ϵ ] / ( ϵ ) 2 ) {\displaystyle T_{X}=X(k[\epsilon ]/(\epsilon )^{2})} . Then, for any morphism f : X → Y {\displaystyle f:X\to Y} of schemes over k, one sees f # ( δ p v ) = δ f ( p ) d f p ( v ) {\displaystyle f^{\#}(\delta _{p}^{v})=\delta _{f(p)}^{df_{p}(v)}} ; this shows that the map T X → T Y {\displaystyle T_{X}\to T_{Y}} that f induces is precisely the differential of f under the above identification.
References
Borel, Armand (1991) [1969], Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012 Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. Zbl 0960.14002. Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
