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 -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 ; i.e., a tangent vector at p.
(To see this, use the fact that any local homomorphism must be of the form
Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point , the fiber of over p is called the tangent space to F at p. The tangent space may be given the structure of a vector space over k. If F is a scheme X over k (i.e., ), then each v as above may be identified with a derivation at p and this gives the identification of 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 . Then, for any morphism of schemes over k, one sees ; this shows that the map that f induces is precisely the differential of f under the above identification.