Zariski Tangent Space and $k[\varepsilon]/\varepsilon^2$

Question

Let $X$ be a scheme over a field $k$, and let $x\in X$ be a rational point, that is, we have $k(x):=\mathcal{O}_x/\mathfrak{m}_x\cong k$. Let $\alpha:\mathfrak{m}_x/\mathfrak{m}_x^2\rightarrow k$ be an element of $T_x=(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, the Zariski tangent space to $X$ at $x$. Using $x$ and $\alpha$, I'd like to define a $k$-morphism of schemes Spec $k[\varepsilon]/\varepsilon^2\rightarrow X$.

In the course of the construction, I need to define a local map of local rings $\mathcal{O}_x\rightarrow k[\varepsilon]/\varepsilon^2$. Using $\alpha$, what is a natural way to define such a map?

It might be helpful to note that I have an injection $k\hookrightarrow\mathcal{O}_x$ arising from the $k$-scheme structure on $X$. I think this leads to a decomposition $\mathcal{O}_x/\mathfrak{m}_x^2=k\oplus(\mathfrak{m}_x/\mathfrak{m}_x^2)$ as $k$ vector spaces, from which the desired map would follow, but I can't seem to explicitly see what this decomposition is inside of $\mathcal{O}_x/\mathfrak{m}_x^2$.

This is part of exercise II.2.8 in Hartshorne.

Answer 1

As you note, you have an injection $k \hookrightarrow \mathcal{O}_x$ of rings. Composing with the projection, this produces an injection $k \hookrightarrow \mathcal{O}_x / \mathfrak{m}_x^2$, with the property that further projecting given an isomorphism $k \cong \mathcal{O}_x / \mathfrak{m}_x$ (here you use the assumption that $x$ is $k$-rational). It follows that $\mathcal{O}_x/\mathfrak{m}_x^2 \cong k \bigoplus \mathfrak{m}_x/\mathfrak{m}_x^2$, so that your $\alpha$ gives you the morphism of local rings you need: explicitly, $(x,y) \mapsto x+\alpha(y) \epsilon$.