## 23.5 Divided power polynomial algebras

A very useful example is the *divided power polynomial algebra*. Let $A$ be a ring. Let $t \geq 1$. We will denote $A\langle x_1, \ldots , x_ t \rangle $ the following $A$-algebra: As an $A$-module we set

\[ A\langle x_1, \ldots , x_ t \rangle = \bigoplus \nolimits _{n_1, \ldots , n_ t \geq 0} A x_1^{[n_1]} \ldots x_ t^{[n_ t]} \]

with multiplication given by

\[ x_ i^{[n]}x_ i^{[m]} = \frac{(n + m)!}{n!m!}x_ i^{[n + m]}. \]

We also set $x_ i = x_ i^{[1]}$. Note that $1 = x_1^{[0]} \ldots x_ t^{[0]}$. There is a similar construction which gives the divided power polynomial algebra in infinitely many variables. There is an canonical $A$-algebra map $A\langle x_1, \ldots , x_ t \rangle \to A$ sending $x_ i^{[n]}$ to zero for $n > 0$. The kernel of this map is denoted $A\langle x_1, \ldots , x_ t \rangle _{+}$.

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.
$\square$

The following lemma can be found in [BO].

Lemma 23.5.3. Let $p$ be a prime number. Let $A$ be a ring such that every integer $n$ not divisible by $p$ is invertible, i.e., $A$ is a $\mathbf{Z}_{(p)}$-algebra. Let $I \subset A$ be an ideal. Two divided power structures $\gamma , \gamma '$ on $I$ are equal if and only if $\gamma _ p = \gamma '_ p$. Moreover, given a map $\delta : I \to I$ such that

$p!\delta (x) = x^ p$ for all $x \in I$,

$\delta (ax) = a^ p\delta (x)$ for all $a \in A$, $x \in I$, and

$\delta (x + y) = \delta (x) + \sum \nolimits _{i + j = p, i,j \geq 1} \frac{1}{i!j!} x^ i y^ j + \delta (y)$ for all $x, y \in I$,

then there exists a unique divided power structure $\gamma $ on $I$ such that $\gamma _ p = \delta $.

**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,

\[ \gamma _{pm}(x) = c \gamma _ m(\gamma _ p(x)) \]

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

\[ \gamma _ n(x) = c_ n x^{a_0} \delta (x)^{a_1} \ldots \delta ^ e(x)^{a_ e} \]

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

\[ c_ n = {(p!)^{a_1 + a_2(1 + p) + \ldots + a_ e(1 + \ldots + p^{e - 1})}}/{n!}. \]

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

\[ \varphi : \mathbf{Z}_{(p)}\langle u, v \rangle \longrightarrow A \]

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

\[ \gamma _ n(x)\gamma _ m(x) = \frac{(n + m)!}{n!m!} \gamma _{n + m}(x) \]

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

\[ \epsilon _ n(f) = c_ n f^{a_0} \gamma _ p(f)^{a_1} \ldots \gamma _ p^ e(f)^{a_ e} \]

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.
$\square$

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