Hartshorne defines the Zariski tangent space at a point $$x$$ of a scheme $$X$$ to be $$(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$$, where the star $$*$$ means taking the dual, i.e. homomorphisms $$\mathfrak{m}_x / \mathfrak{m}_x^2 \rightarrow k(x)$$. Exercise II 2.8 claims that if $$X$$ is a scheme over a field $$k$$, and $$x$$ is a point rational over $$k$$ (i.e. $$k(x) = k$$), then there is a one to one correspondence of elements in the tangent space and morphisms of schemes $$\text{Spec }k[\epsilon] \rightarrow X$$ which hit $$x$$. Here $$k[\epsilon]$$ means the dual numbers, i.e. $$\epsilon^2 = 0$$.
Can this characterisation be extended to points which are not rational over $$k$$, if we replace $$k[\epsilon]$$ by $$k(x)[\epsilon]$$? This is also the definition used in the stacks-project, but I was not successful in showing that both definitions are the same.
I did something similar to this solution, but there the inclusion $$k \hookrightarrow \mathcal{O}_{X,x}$$ is used to split the sequence $$0 \rightarrow \mathfrak{m}_x \rightarrow \mathcal{O}_{X,x} \rightarrow k \rightarrow 0.$$
• There are three nontrivial group homomorphisms $\mathbb{Z}/4 \to \mathbb{Z}/2 \oplus \mathbb{Z}/2$, one for each nonzero choice of image for the class of $1$. (and a unique ring homomorphism, assuming by $\mathbb{Z}/2 \oplus \mathbb{Z}/2$ you mean the product ring) – Hurkyl Oct 2 '18 at 17:22