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

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

2. $(A, I, \gamma ) \to (A\langle x_1, \ldots , x_ t \rangle , J, \delta )$ 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07H5. Beware of the difference between the letter 'O' and the digit '0'.