Tangent space to a functor

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 $$k[\epsilon]/(\epsilon)^2$$-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 $$(\mathfrak{m}_{X, p}/\mathfrak{m}_{X, p}^2)^*$$; i.e., a tangent vector at p.

(To see this, use the fact that any local homomorphism $$\mathcal{O}_p \to k[\epsilon]/(\epsilon)^2$$ must be of the form
 * $$\delta_p^v: u \mapsto u(p) + \epsilon v(u), \quad v \in \mathcal{O}_p^*.$$)

Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point $$p \in F(k)$$, the fiber of $$\pi: F(k[\epsilon]/(\epsilon)^2) \to F(k)$$ 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., $$F = \operatorname{Hom}_{\operatorname{Spec}k}(\operatorname{Spec}-, X)$$), then each v as above may be identified with a derivation at p and this gives the identification of $$\pi^{-1}(p)$$ 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 $$T_X = X(k[\epsilon]/(\epsilon)^2)$$. Then, for any morphism $$f: X \to Y$$ of schemes over k, one sees $$f^{\#}(\delta_p^v) = \delta_{f(p)}^{df_p(v)}$$; this shows that the map $$T_X \to T_Y$$ that f induces is precisely the differential of f under the above identification.