# ag.algebraic geometry – Is there a functor of points approach to defining the tangent sheaf?

Let me motivate my question by recalling two equivalent definitions of tangent spaces in algebraic geometry.

Let $$X$$ be scheme over a field $$k$$. Let $$p in X(k)$$, equivalently a closed point $$p in X$$ with residue field $$k$$. There are two definitions of the tangent space $$T_p X$$.

1. The traditional definition (due to Zariski?) is $$(m_p / m_p^2)^*$$ where $$m_p$$ is the maximal ideal in the local ring at $$p$$. This definition is based on the idea of a tangent vector as a “directional derivative” of a function.
2. The functor of points definition is to consider $$X(k(varepsilon)) rightarrow X(k)$$ (here and later $$varepsilon$$ satisfies $$varepsilon^2 = 0$$) and then we define the tangent space as the fibre over $$p$$. This definition is based on the idea of a tangent vector as a first order approximation to a curve passing through $$p$$.

Now, let us consider the tangent sheaf. We could either try to define the sheaf of sections of this bundle or its total space. Traditionally (Hartshorne, Vakil, etc) define the cotangent sheaf $$Omega_{X/k}$$ and then by duality the tangent sheaf. We can also define the total space of the “tangent bundle” $$TX$$ by $$TX = Spec_X( Sym Omega_{X/k})$$. (It is not always a vector bundle, since $$Omega_{X/k}$$ is not always locally free.)

I would like to know if there is a way to define the total space of the tangent bundle $$TX$$ as a functor. It seems that
$$TX(k) = X(k(varepsilon))$$,
since the tangent bundle is the union of the tangent spaces. So we could try the following:
$$TX(R) = X(R(varepsilon))$$
for any $$k$$-algebra $$R$$.

In other words, $$TX$$ is obtained from $$X$$ by Weil restriction.

Is there some class of schemes (at least smooth varieties over an algebraically closed field), for which this construction works?

My motivation for this question is as follows. In Hartshorne and Vakil, there is a rather involved proof in local coordinates for the Euler sequence describing the cotangent bundle of projective space. If we had a functor of points construction, the proof would be much more natural!

After writing this question, I found the following related question
Relationship between Tangent bundle and Tangent sheaf
and Laurent Moret-Bailly’s answer there. According to what he wrote, it seems that it always works! So why isn’t this more well-known and is there a reference for this?

Posted on Categories Articles