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?



What exactly is $\underset{n}{\varprojlim} \ \mu_{p^n}$?
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.


