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 ${\displaystyle k[\epsilon ]/(\epsilon )^{2}}$-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 ${\displaystyle ({\mathfrak {m}}_{X,p}/{\mathfrak {m}}_{X,p}^{2})^{*}}$; i.e., a tangent vector at p.

(To see this, use the fact that any local homomorphism ${\displaystyle {\mathcal {O}}_{p}\to k[\epsilon ]/(\epsilon )^{2}}$ must be of the form

${\displaystyle \delta _{p}^{v}:u\mapsto u(p)+\epsilon v(u),\quad v\in {\mathcal {O}}_{p}^{*}.}$)

Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point ${\displaystyle p\in F(k)}$, the fiber of ${\displaystyle \pi :F(k[\epsilon ]/(\epsilon )^{2})\to F(k)}$ 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 ${\displaystyle \pi ^{-1}(p)}$ 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.^[3] Let ${\displaystyle T_{X}=X(k[\epsilon ]/(\epsilon )^{2})}$. Then, for any morphism ${\displaystyle f:X\to Y}$ of schemes over k, one sees ${\displaystyle f^{\#}(\delta _{p}^{v})=\delta _{f(p)}^{df_{p}(v)}}$; this shows that the map ${\displaystyle T_{X}\to T_{Y}}$ that f induces is precisely the differential of f under the above identification.

• Borel, Armand (1991) [1969], Linear algebraic groups, Graduate Texts in Mathematics, vol. 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8, MR 1102012
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. Zbl 0960.14002.
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157