A few days ago, I came across the notion of the tangent sheaf of an affine $k$-scheme $X = \operatorname{Spec} A $, with $k$ an arbitrary field. The natural question that arose was: what is the relationship between this and the tangent space $$T_p(X) = \operatorname{Hom}_{k(p)}(\mathfrak{m}_p / \mathfrak{m}_p^2, k(p)) \cong \operatorname{Der}_k(A,k),$$ where $ p \in X $ is a closed point? My hypothesis was that they should coincide with the fiber of the tangent sheaf at that point:
$$T_X(p) = T_{X,p} \otimes_{A_{p}} k(p) = \operatorname{Der}_k(A)_p \otimes_{A_{p}} k(p).$$
Unfortunately, i couldn't find a proof by myself, even with simple hipothesis.
For example, if A is a finitely generated $k$-álgebra, then $$ T_{X,p} = \operatorname{Der}_k(A)_p \cong \operatorname{Der}_k(A_p). $$ We can also assume $k=k(p)$. In that case, we can define $\Phi: \operatorname{Der}_k(A_p) \otimes_{A_{p}} k \longrightarrow \operatorname{Der}_k(A,k)$ as follows:
$$ \Phi(d \otimes 1) = d: A_p \longrightarrow A_p \longrightarrow k. $$
However, I have big problems to show that $\ker \Phi = 0.$
Thanks in advance
