$p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$.

Question

Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the elements of the $p$-adic completions of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$?

Edit: The completion of $\mathbb{Z}[X]$ contains $\mathbb{Z}_p[X]$ of course. I think it should also contain the series $\sum_{i=0}^\infty p^iX^i$, so the completion is strictly bigger than $\mathbb{Z}_p[X]$.

Answer 1

The completion of $\mathbb{Z}[X]$ is isomorphic to the ring of formal power series with coefficients in $\mathbb{Z}_p$ that tend to $0$ $p$-adically. Indeed, it is, by definition, the ring of equivalence classes of Cauchy sequences $(f_k)$ of polynomials $f_k\in \mathbb{Z}[X]$. To be a Cauchy sequence means that eventually, the polynomials become congruent modulo higher and higher powers of $p$, i.e. all the coefficients of these polynomials must eventually be congruent modulo high powers of $p$. Suppose that for some $k_0$, $f_m$ is congruent to $f_{k_0}$ modulo $p^{100}$ for all $m\geq k_0$. If $d_0$ is the degree of $f_{k_0}$, then this implies that in all these $f_m$, the coefficients of $x^d$ for all $d>d_0$ are divisible by $p^{100}$. By applying this for higher and higher values of $100$, you see that the coefficients in the resulting power series tend to $0$ $p$-adically.

These kinds of rings are very important in rigid analytic geometry.

The completion of $\mathbb{Z}[[X]]$ is just $\mathbb{Z}_p[[X]]$. That's easy to see by just looking at your Cauchy sequence of power series term by term.

Answer 2

The $p$-adic completion of $\mathbf Z[[X]]$ is $\mathbf Z_p[[X]]$. To prove it, it suffices to prove that the map $\mathbf Z[[X]] \to\mathbf Z_p[[X]]$ induces an isomorphism on the $p$-adic completions. This is obvious because $$\mathbf Z[[X]]/p^n\mathbf Z[[X]] \to\mathbf Z_p[[X]]/p^n\mathbf Z_p[[X]]$$ is an isomorphism, both sides identifying with $(\mathbf Z/p^n\mathbf Z)[[X]]$ in a natural way.

However, the $p$-adic completion of $\mathbf Z[X]$ is the ring $\mathbf Z_p\left<X\right>$, the subring of $\mathbf Z_p[[X]]$ consisting of the formal power series $\sum_{n\geq 0} a_n T^n$ such that $|a_n| \to 0$ in $\mathbf Z_p$. It is a nice exercise in limits to prove it.

Such rings are known as Tate algebras and their study forms the starting point of rigid analytic geometry (which is to algebraic geometry over $p$-adic fields what complex analytic geometry is to algebraic geometry over $\mathbf C$).