Tangent space to a functor

Short description: Concept in category theory

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

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

${\delta }_p^v:u\mapsto u(p)+ϵv(u),v\in {{𝒪}}_p^{*}.$)

Let F be a functor from the category of k-algebras to the category of sets. Then, for any k-point $p\in F(k)$, the fiber of $\pi :F(k[ϵ]/(ϵ{)}^2)\to F(k)$ over p is called the tangent space to F at p. 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., $F={\mathrm{Hom}}_{\mathrm{Spec}k}(\mathrm{Spec}-,X)$), then each v as above may be identified with a derivation at p and this gives the identification of ${\pi }^{-1}(p)$ 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. Let $T_X=X(k[ϵ]/(ϵ{)}^2)$. Then, for any morphism $f:X\to Y$ of schemes over k, one sees $f^{\mathrm{\#}}({\delta }_p^v)={\delta }_{f(p)}^{d{f}_p(v)}$; this shows that the map $T_X\to T_Y$ that f induces is precisely the differential of f under the above identification.

References

• Borel, Armand (1991), Linear algebraic groups, Graduate Texts in Mathematics, 126 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97370-8 
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 0-387-98637-5. https://archive.org/details/springer_10.1007-978-0-387-22639-2. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9