# 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.^{[1]} 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*.^{[2]}
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.^{[3]} 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.

## References

- A. Borel,
*Linear algebraic groups* - {{#invoke:citation/CS1|citation

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