Lemma 23.5.1. Let $(A, I, \gamma )$ be a divided power ring. There exists a unique divided power structure $\delta $ on

\[ J = IA\langle x_1, \ldots , x_ t \rangle + A\langle x_1, \ldots , x_ t \rangle _{+} \]

such that

$\delta _ n(x_ i) = x_ i^{[n]}$, and

$(A, I, \gamma ) \to (A\langle x_1, \ldots , x_ t \rangle , J, \delta )$ is a homomorphism of divided power rings.

Moreover, $(A\langle x_1, \ldots , x_ t \rangle , J, \delta )$ has the following universal property: A homomorphism of divided power rings $\varphi : (A\langle x_1, \ldots , x_ t \rangle , J, \delta ) \to (C, K, \epsilon )$ is the same thing as a homomorphism of divided power rings $A \to C$ and elements $k_1, \ldots , k_ t \in K$.

**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

\[ IA\langle x \rangle \cap A\langle x \rangle _{+} = IA\langle x \rangle \cdot A\langle x \rangle _{+} \]

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

\[ A\langle x_1, \ldots , x_ t \rangle \to C,\quad x_1^{[n_1]} \ldots x_ t^{[n_ t]} \longmapsto \epsilon _{n_1}(k_1) \ldots \epsilon _{n_ t}(k_ t) \]

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

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