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. $\square$