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 $$0 \rightarrow \mathfrak{m}_x \rightarrow \mathcal{O}_{X,x} \rightarrow k \rightarrow 0.$$

  • $\begingroup$ 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) $\endgroup$ – Hurkyl Oct 2 '18 at 17:22

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