On presentations of universal rings of deformations

Question

Let $k$ be a finite field of characteristic $p$, and $R$ a complete local noetherian algebra with residue field $k$. It is well known that $R$ has a natural structure of an algebra over the ring Witt vectors $W(k)$. Define $n:= dim_{k}(\mathfrak{m}/(p,\mathfrak{m}^{2}))$ to be the dimension of the $mod-p$ Zariski tangent space. In theorem 2.4 in A local-to-global principle for deformations of Galois representations it is stated that there is a surjection $W(k)[[t_{1},\cdots,t_{n}]]\rightarrow R$. I would like to understand this surjection. In the context of this paper, $R$ would be a universal deformation ring that pro-represents the functor of deformations of a pro-finite group acting continuously on a finite-dimensional vector space over $k$.

Answer 1

Doesn't this kind of prove itself? Pick some elements $\alpha_1, \dots, \alpha_n \in \mathfrak{m}$ which represent $\mathfrak{m} / (p, \mathfrak{m}^2)$. Clearly sending $t_i$ to $\alpha_i$ defines a map $W(k)[[[t_1, \dots, t_n]] \to R$ and it suffices to show that it is surjective modulo $\mathfrak{m}^r$ for every $r$. This is obvious for $r = 0$, and the assumption on the $\alpha_i$'s shows precisely that $(p, \alpha_1, \dots, \alpha_n)$ generate $\mathfrak{m} / \mathfrak{m}^2$ so they also generate $\mathfrak{m}^r / \mathfrak{m}^{r+1}$ for every $r$, and you are done by induction on $r$.