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
k [ ϵ ] / ( ϵ ) 2 -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
( m X , p / m X , p 2 ) ∗ ; i.e., a tangent vector at
p.
(To see this, use the fact that any local homomorphism O p → k [ ϵ ] / ( ϵ ) 2 must be of the form
δ p v : u ↦ u ( p ) + ϵ v ( u ) , v ∈ O p ∗ . )
Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point p ∈ F ( k ) , the fiber of π : F ( k [ ϵ ] / ( ϵ ) 2 ) → F ( k ) 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., F = Hom Spec k ( Spec − , X ) ), then each v as above may be identified with a derivation at p and this gives the identification of π − 1 ( p ) 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 T X = X ( k [ ϵ ] / ( ϵ ) 2 ) . Then, for any morphism f : X → Y of schemes over k, one sees f # ( δ p v ) = δ f ( p ) d f p ( v ) ; this shows that the map T X → T Y that f induces is precisely the differential of f under the above identification.
