Let $k$ be a finite field, let $\mathbf{CR}$ be the category of complete local Noetherian rings, and let $\mathbf{CR}_{/k}$ be the over category. Let $\mathcal{C}$ be the full subcategory of $\mathbf{CR}_{/k}$ of rings that are complete, Noetherian, local rings $R$ with surjective augmentation $R \to k$. I would like to show that for $R \in \mathcal{C}$, there is an injection
$$
\operatorname{Hom}_\mathcal{C}(R, k[\varepsilon]) \to \operatorname{Hom}_k(\mathfrak{m}_R/\mathfrak{m}_R^2, k),
$$
where $\mathfrak{m}_R$ is the maximal ideal of $R$ and $k[\varepsilon] = k[x]/(x^2)$ is the ring of dual numbers. This is a special case of Lemma 2.6 of these notes.
First of all, I know that the injection is given by the composition
$$
\mathfrak{m}_R/\mathfrak{m}_R^2 \to k[\varepsilon] \to k,
$$
where the first map is induced by $\varphi$ and the second map is $a + b\varepsilon \mapsto b$. My difficulty is showing that the map $\mathfrak{m}_R/\mathfrak{m}_R^2 \to k$ determines $\varphi$.
This differs from the usual statement about Zariski tangent spaces (see for example this question), because $R$ is not a $k$-algebra. As I understand it, the usual argument is to construct a $W(k)$-algebra structure on $R$ and $k[\varepsilon]$, where $W(k)$ is the ring of Witt vectors over $k$. We then show that the $W(k)$-algebra structures are compatible with the augmentations $R\to k$ and $k[\varepsilon] \to k$, and finally that $R$ is generated as a ring by $W(k)$ and $\mathfrak{m}_R$, from which the result follows. However, the Witt vector machinery seems like a lot of theory for such a simple result, and I would like to avoid using it if I can. Can anybody see an elementary way of proving that the map is injective without invoking the Witt vector-algebra structure?
Thank you!
Best Answer
We want to show that for any morphism $f:R\to k[\varepsilon]$ in $\mathcal C$, we can recover $f$ from the corresponding map $g:\mathfrak m_R\to k$. For and any $r\in R$ let us write $f(r)=a+b\varepsilon$. Then we have $$f(r^q)=f(r)^q=(a+b\varepsilon)^q=a^q=a,$$ from which we see $r-r^q\in \mathfrak m_R$ and $b=g(r^q-r)$. We can recover $a$ as the value of the map $R\to k$ at $r$, and thus we can recover the value $a+b\varepsilon$.