Tangent space to a functor

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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[edit]

  1. ^ Hartshorne 1977, Exercise II 2.8
  2. ^ Eisenbud–Harris 1998, VI.1.3
  3. ^ Borel 1991, AG 16.2