The question on a previous final was
"Consider a scheme $X$, for any point $x \in X$, show that the quotient of the maximal ideal at the point $x$, $\mathcal{M}_x/\mathcal{M}^2_x$ is a $k(x)$-vector space. We defined the tangent space $T_x$ to $X$ at $x$ as the dual of the vector space $\mathcal{M}_x/\mathcal{M}^2_x$. Consider the scheme $Spec \, k[x]/(X^2)$. Describe it and $Spec \, k[X]/(X^2)$. Show that to give a morphism of $Spec \, k[x]/(x^2) \rightarrow X$ is equivalent to give a point $x$ (which kind of point?) and an element of $T_x$".
Now the only thing that looks remotely similar is on Hartshorne Pg. 32, for Theorem 5.1 which shows some criteria for the non-singularity of a point, but I struggle to understand it.
I know there is a lot here and know that $k[x]/(X^2)$ is the tracing out the curve X^2, and that to find the tangent at a point p we look at the Jacobian matrix at p. Any insights or leads on this would be helpful and any intuitive description of what is going on would be extremely appreciated.
Thanks,
Brian
There is not a huge amount here if you set up the problem correctly—a morphism $\rm{Spec}\ k[\epsilon]/\epsilon^2 \to X$ is determined by a map on topological spaces, together with a map of sheaves. Since the first scheme is a point, we just need to specify its image $x\in X$, and a map of the stalks at $x$, that is, a local morphism $\mathcal{O}_{X,x} \to k[\epsilon]/\epsilon^2$. (Also note that $x\in X$ must be a closed point—why?)
Rephrasing the problem in local language, what we want to show is that, if $(A,\mathfrak{m})$ is a local $k$-algebra, then a local $k$-algebra morphism $A\to k[\epsilon]/\epsilon^2$ is equivalent to a $k$-vector space homomorphism $\mathfrak{m}/\mathfrak{m}^2\to k$.
This is easiest to do by writing down the bijection explicitly. For example, if we start with $\varphi: A\to k[\epsilon]/\epsilon^2$, we can define $\psi: \mathfrak{m}/\mathfrak{m}^2\to k$ by $\varphi(a) = \psi(a) \cdot \epsilon$, and check that this is well-defined.