The Stacks project

Proof. We will prove the lemma in case of a divided power polynomial algebra in one variable. The result for the general case can be argued in exactly the same way, or by noting that $A\langle x_1, \ldots , x_ t\rangle $ is isomorphic to the ring obtained by adjoining the divided power variables $x_1, \ldots , x_ t$ one by one.

Let $A\langle x \rangle _{+}$ be the ideal generated by $x, x^{[2]}, x^{[3]}, \ldots $. Note that $J = IA\langle x \rangle + A\langle x \rangle _{+}$ and that

Hence by Lemma 23.2.5 it suffices to show that there exist divided power structures on the ideals $IA\langle x \rangle $ and $A\langle x \rangle _{+}$. The existence of the first follows from Lemma 23.4.2 as $A \to A\langle x \rangle $ is flat. For the second, note that if $A$ is torsion free, then we can apply Lemma 23.2.2 (4) to see that $\delta $ exists. Namely, choosing as generators the elements $x^{[m]}$ we see that $(x^{[m]})^ n = \frac{(nm)!}{(m!)^ n} x^{[nm]}$ and $n!$ divides the integer $\frac{(nm)!}{(m!)^ n}$. In general write $A = R/\mathfrak a$ for some torsion free ring $R$ (e.g., a polynomial ring over $\mathbf{Z}$). The kernel of $R\langle x \rangle \to A\langle x \rangle $ is $\bigoplus \mathfrak a x^{[m]}$. Applying criterion (2)(c) of Lemma 23.4.3 we see that the divided power structure on $R\langle x \rangle _{+}$ extends to $A\langle x \rangle $ as desired.

Proof of the universal property. Given a homomorphism $\varphi : A \to C$ of divided power rings and $k_1, \ldots , k_ t \in K$ we consider

using $\varphi $ on coefficients. The only thing to check is that this is an $A$-algebra homomorphism (details omitted). The inverse construction is clear. $\square$

Proof. If $n$ is not divisible by $p$, then $\gamma _ n(x) = c x \gamma _{n - 1}(x)$ where $c$ is a unit in $\mathbf{Z}_{(p)}$. Moreover,

where $c$ is a unit in $\mathbf{Z}_{(p)}$. Thus the first assertion is clear. For the second assertion, we can, working backwards, use these equalities to define all $\gamma _ n$. More precisely, if $n = a_0 + a_1p + \ldots + a_ e p^ e$ with $a_ i \in \{ 0, \ldots , p - 1\} $ then we set

for $c_ n \in \mathbf{Z}_{(p)}$ defined by

Now we have to show the axioms (1) – (5) of a divided power structure, see Definition 23.2.1. We observe that (1) and (3) are immediate. Verification of (2) and (5) is by a direct calculation which we omit. Let $x, y \in I$. We claim there is a ring map

which maps $u^{[n]}$ to $\gamma _ n(x)$ and $v^{[n]}$ to $\gamma _ n(y)$. By construction of $\mathbf{Z}_{(p)}\langle u, v \rangle $ this means we have to check that

in $A$ and similarly for $y$. This is true because (2) holds for $\gamma $. Let $\epsilon $ denote the divided power structure on the ideal $\mathbf{Z}_{(p)}\langle u, v\rangle _{+}$ of $\mathbf{Z}_{(p)}\langle u, v\rangle $. Next, we claim that $\varphi (\epsilon _ n(f)) = \gamma _ n(\varphi (f))$ for $f \in \mathbf{Z}_{(p)}\langle u, v\rangle _{+}$ and all $n$. This is clear for $n = 0, 1, \ldots , p - 1$. For $n = p$ it suffices to prove it for a set of generators of the ideal $\mathbf{Z}_{(p)}\langle u, v\rangle _{+}$ because both $\epsilon _ p$ and $\gamma _ p = \delta $ satisfy properties (1) and (3) of the lemma. Hence it suffices to prove that $\gamma _ p(\gamma _ n(x)) = \frac{(pn)!}{p!(n!)^ p}\gamma _{pn}(x)$ and similarly for $y$, which follows as (5) holds for $\gamma $. Now, if $n = a_0 + a_1p + \ldots + a_ e p^ e$ is an arbitrary integer written in $p$-adic expansion as above, then

because $\epsilon $ is a divided power structure. Hence we see that $\varphi (\epsilon _ n(f)) = \gamma _ n(\varphi (f))$ holds for all $n$. Applying this for $f = u + v$ we see that axiom (4) for $\gamma $ follows from the fact that $\epsilon $ is a divided power structure. $\square$