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.
[1] Let X be a scheme over a field k. (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
π :
over p is called the tangent space to F at p.[2] 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.,
{\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
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.
of schemes over k, one sees
; this shows that the map
that f induces is precisely the differential of f under the above identification.