What exactly is $\underset{n}{\varprojlim} \ \mu_{p^n}$? Let $k$ be a field of characteristic $p> 0$.
Let $\mu_{p^n}$ denote the group scheme $\mu_{p^n}(R) =\{ x \in R: x^{p^n}=1 \}$. Then there are natural maps from $\mu_{p^n}$ to $\mu_{p^{n-1}}$. I'm interested in understanding what is the projective limit of this system of finite group schemes $\underset{n}{\varprojlim} \ \mu_{p^n}$? Is there any computation where this profinite group scheme appears? Or is this profinite group scheme well-known?
 A: Let me just do the simple thing and calculate the limit. Perhaps this is not what you are looking for, but then we can perhaps refine the question a bit in that case.
Write $\mu_{p^n} = \mathrm{Spec}(k[x_n]/(x_n^{p^n} - 1))$. Then the natural map $\mu_{p^n} \to \mu_{p^{n-1}}$ that I believe you are thinking about is the one induced by the $k$-algebra map
\begin{align}
\varphi_n \colon k[x_{n-1}]/(x_{n-1}^{p^{n-1}} - 1) & \to k[x_n]/(x_n^{p^n} - 1) \\
x_{n-1} & \mapsto x_n^p.
\end{align}
For our purposes, note that this is actually a morphism of Hopf $k$-algebras, which amounts to saying that this actually induces a morphism of the corresponding finite group schemes.
Now let's compute: since $\mathrm{Spec}$ yields an anti-equivalence between the category of affine (group) schemes over $k$ and the category of commutative $k$-algebras (perhaps Hopf algebras...), we have
\begin{align}
  \mathbf{Z}_p(1)
    & :=  \mathrm{lim}_{\varphi_n}\,\mu_{p^n} \\
    & \:= \mathrm{lim}_{\varphi_n}\, \mathrm{Spec}(k[x_n]/(x_n^{p^n} - 1)) \\
    & \:= \mathrm{Spec}(\mathrm{colim}_{\varphi_n}\,k[x_n]/(x_n^{p^n} - 1)).
\end{align}
Now you can verify that
$$
  \mathrm{colim}_{\varphi_n}\,k[x_n]/(x_n^{p^n} - 1) \cong k[t^{1/p^{\infty}}]/(t - 1)
$$
the natural map from the colimit being induced by the maps
\begin{align}
  k[x_n]/(x_n^{p^n} - 1) & \to k[t^{1/p^\infty}]/(t - 1) \\
  x_n & \mapsto t^{1/p^n}.
\end{align}
So $\mathbf{Z}_p(1) = \mathrm{Spec}(k[t^{1/p^\infty}]/(t - 1))$.
It may be useful to describe it's functor of points: given a $k$-algebra $R$, we have
\begin{align}
  \mathbf{Z}_p(1)(R)
    & = \mathrm{Hom}(\mathrm{Spec}(R),\mathbf{Z}_p(1)) \\
    & = \mathrm{Hom}(k[t^{1/p^\infty}]/(t - 1), R) \\
    & = \big\{(r_1,r_2,\ldots) \in \prod\nolimits_{i = 1}^\infty R \mid r_1^p = 1 \;\text{and}\;r_{n+1}^p = r_n\;\text{for}\;n\geq 1\big\},
\end{align}
i.e. $\mathbf{Z}_p(1)(R)$ consists of sequences of compatible $p$-power roots of unity in $R$, as should be expected.
