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.
, 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.
