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. Let X be a scheme over a field k.

To give a [math]\displaystyle{ k[\epsilon]/(\epsilon)^2 }[/math]-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 [math]\displaystyle{ (\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]\displaystyle{ \mathcal{O}_p \to k[\epsilon]/(\epsilon)^2 }[/math] must be of the form

[math]\displaystyle{ \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]\displaystyle{ p \in F(k) }[/math], the fiber of [math]\displaystyle{ \pi: F(k[\epsilon]/(\epsilon)^2) \to F(k) }[/math] 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., [math]\displaystyle{ 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]\displaystyle{ \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 in the following way. Let [math]\displaystyle{ T_X = X(k[\epsilon]/(\epsilon)^2) }[/math]. Then, for any morphism [math]\displaystyle{ f: X \to Y }[/math] of schemes over k, one sees [math]\displaystyle{ f^{\#}(\delta_p^v) = \delta_{f(p)}^{df_p(v)} }[/math]; this shows that the map [math]\displaystyle{ T_X \to T_Y }[/math] that f induces is precisely the differential of f under the above identification.

References

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8 
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. https://archive.org/details/springer_10.1007-978-0-387-22639-2. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9