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. Let ''X'' be a scheme over a field ''k''.

:To give a $k$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 $(\mathfrak_/\mathfrak_^2)^*$; i.e., a tangent vector at ''p''.
(To see this, use the fact that any local homomorphism $\mathcal_p \to k$epsilon(\epsilon)^2 must be of the form
:$\delta_p^v: u \mapsto u(p) + \epsilon v(u), \quad v \in \mathcal_p^*.$)
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 $\pi: F(k$epsilon(\epsilon)^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., $F = \operatorname_(\operatorname-, X)$), then each ''v'' as above may be identified with a derivation at ''p'' and this gives the identification of $\pi^(p)$ 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 $T_X = X(k$epsilon(\epsilon)^2). Then, for any morphism $f: X \to Y$ of schemes over ''k'', one sees $f^(\delta_p^v) = \delta_^$; this shows that the map $T_X \to T_Y$ that ''f'' induces is precisely the differential of ''f'' under the above identification.

References

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*{{Hartshorne AG

Algebraic geometry