Definition 23.4.1. Given a divided power ring $(A, I, \gamma )$ and a ring map $A \to B$ we say $\gamma $ *extends* to $B$ if there exists a divided power structure $\bar\gamma $ on $IB$ such that $(A, I, \gamma ) \to (B, IB, \bar\gamma )$ is a homomorphism of divided power rings.

## 23.4 Extending divided powers

Here is the definition.

Lemma 23.4.2. Let $(A, I, \gamma )$ be a divided power ring. Let $A \to B$ be a ring map. If $\gamma $ extends to $B$ then it extends uniquely. Assume (at least) one of the following conditions holds

$IB = 0$,

$I$ is principal, or

$A \to B$ is flat.

Then $\gamma $ extends to $B$.

**Proof.**
Any element of $IB$ can be written as a finite sum $\sum \nolimits _{i=1}^ t b_ ix_ i$ with $b_ i \in B$ and $x_ i \in I$. If $\gamma $ extends to $\bar\gamma $ on $IB$ then $\bar\gamma _ n(x_ i) = \gamma _ n(x_ i)$. Thus, conditions (3) and (4) in Definition 23.2.1 imply that

Thus we see that $\bar\gamma $ is unique if it exists.

If $IB = 0$ then setting $\bar\gamma _ n(0) = 0$ works. If $I = (x)$ then we define $\bar\gamma _ n(bx) = b^ n\gamma _ n(x)$. This is well defined: if $b'x = bx$, i.e., $(b - b')x = 0$ then

because $\gamma _ n(x)$ is divisible by $x$ (since $\gamma _ n(I) \subset I$) and hence annihilated by $b - b'$. Next, we prove conditions (1) – (5) of Definition 23.2.1. Parts (1), (2), (3), (5) are obvious from the construction. For (4) suppose that $y, z \in IB$, say $y = bx$ and $z = cx$. Then $y + z = (b + c)x$ hence

as desired.

Assume $A \to B$ is flat. Suppose that $b_1, \ldots , b_ r \in B$ and $x_1, \ldots , x_ r \in I$. Then

where the sum is over $e_1 + \ldots + e_ r = n$ if $\bar\gamma _ n$ exists. Next suppose that we have $c_1, \ldots , c_ s \in B$ and $a_{ij} \in A$ such that $b_ i = \sum a_{ij}c_ j$. Setting $y_ j = \sum a_{ij}x_ i$ we claim that

in $B$ where on the right hand side we are summing over $d_1 + \ldots + d_ s = n$. Namely, using the axioms of a divided power structure we can expand both sides into a sum with coefficients in $\mathbf{Z}[a_{ij}]$ of terms of the form $c_1^{d_1} \ldots c_ s^{d_ s}\gamma _{e_1}(x_1) \ldots \gamma _{e_ r}(x_ r)$. To see that the coefficients agree we note that the result is true in $\mathbf{Q}[x_1, \ldots , x_ r, c_1, \ldots , c_ s, a_{ij}]$ with $\gamma $ the unique divided power structure on $(x_1, \ldots , x_ r)$. By Lazard's theorem (Algebra, Theorem 10.81.4) we can write $B$ as a directed colimit of finite free $A$-modules. In particular, if $z \in IB$ is written as $z = \sum x_ ib_ i$ and $z = \sum x'_{i'}b'_{i'}$, then we can find $c_1, \ldots , c_ s \in B$ and $a_{ij}, a'_{i'j} \in A$ such that $b_ i = \sum a_{ij}c_ j$ and $b'_{i'} = \sum a'_{i'j}c_ j$ such that $y_ j = \sum x_ ia_{ij} = \sum x'_{i'}a'_{i'j}$ holds^{1}. Hence the procedure above gives a well defined map $\bar\gamma _ n$ on $IB$. By construction $\bar\gamma $ satisfies conditions (1), (3), and (4). Moreover, for $x \in I$ we have $\bar\gamma _ n(x) = \gamma _ n(x)$. Hence it follows from Lemma 23.2.4 that $\bar\gamma $ is a divided power structure on $IB$.
$\square$

Lemma 23.4.3. Let $(A, I, \gamma )$ be a divided power ring.

If $\varphi : (A, I, \gamma ) \to (B, J, \delta )$ is a homomorphism of divided power rings, then $\mathop{\mathrm{Ker}}(\varphi ) \cap I$ is preserved by $\gamma _ n$ for all $n \geq 1$.

Let $\mathfrak a \subset A$ be an ideal and set $I' = I \cap \mathfrak a$. The following are equivalent

$I'$ is preserved by $\gamma _ n$ for all $n > 0$,

$\gamma $ extends to $A/\mathfrak a$, and

there exist a set of generators $x_ i$ of $I'$ as an ideal such that $\gamma _ n(x_ i) \in I'$ for all $n > 0$.

**Proof.**
Proof of (1). This is clear. Assume (2)(a). Define $\bar\gamma _ n(x \bmod I') = \gamma _ n(x) \bmod I'$ for $x \in I$. This is well defined since $\gamma _ n(x + y) = \gamma _ n(x) \bmod I'$ for $y \in I'$ by Definition 23.2.1 (4) and the fact that $\gamma _ j(y) \in I'$ by assumption. It is clear that $\bar\gamma $ is a divided power structure as $\gamma $ is one. Hence (2)(b) holds. Also, (2)(b) implies (2)(a) by part (1). It is clear that (2)(a) implies (2)(c). Assume (2)(c). Note that $\gamma _ n(x) = a^ n\gamma _ n(x_ i) \in I'$ for $x = ax_ i$. Hence we see that $\gamma _ n(x) \in I'$ for a set of generators of $I'$ as an abelian group. By induction on the length of an expression in terms of these, it suffices to prove $\forall n : \gamma _ n(x + y) \in I'$ if $\forall n : \gamma _ n(x), \gamma _ n(y) \in I'$. This follows immediately from the fourth axiom of a divided power structure.
$\square$

Lemma 23.4.4. Let $(A, I, \gamma )$ be a divided power ring. Let $E \subset I$ be a subset. Then the smallest ideal $J \subset I$ preserved by $\gamma $ and containing all $f \in E$ is the ideal $J$ generated by $\gamma _ n(f)$, $n \geq 1$, $f \in E$.

**Proof.**
Follows immediately from Lemma 23.4.3.
$\square$

Lemma 23.4.5. Let $(A, I, \gamma )$ be a divided power ring. Let $p$ be a prime. If $p$ is nilpotent in $A/I$, then

the $p$-adic completion $A^\wedge = \mathop{\mathrm{lim}}\nolimits _ e A/p^ eA$ surjects onto $A/I$,

the kernel of this map is the $p$-adic completion $I^\wedge $ of $I$, and

each $\gamma _ n$ is continuous for the $p$-adic topology and extends to $\gamma _ n^\wedge : I^\wedge \to I^\wedge $ defining a divided power structure on $I^\wedge $.

If moreover $A$ is a $\mathbf{Z}_{(p)}$-algebra, then

for $e$ large enough the ideal $p^ eA \subset I$ is preserved by the divided power structure $\gamma $ and

\[ (A^\wedge , I^\wedge , \gamma ^\wedge ) = \mathop{\mathrm{lim}}\nolimits _ e (A/p^ eA, I/p^ eA, \bar\gamma ) \]in the category of divided power rings.

**Proof.**
Let $t \geq 1$ be an integer such that $p^ tA/I = 0$, i.e., $p^ tA \subset I$. The map $A^\wedge \to A/I$ is the composition $A^\wedge \to A/p^ tA \to A/I$ which is surjective (for example by Algebra, Lemma 10.96.1). As $p^ eI \subset p^ eA \cap I \subset p^{e - t}I$ for $e \geq t$ we see that the kernel of the composition $A^\wedge \to A/I$ is the $p$-adic completion of $I$. The map $\gamma _ n$ is continuous because

by the axioms of a divided power structure. It is clear that the axioms for divided power structures are inherited by the maps $\gamma _ n^\wedge $ from the maps $\gamma _ n$. Finally, to see the last statement say $e > t$. Then $p^ eA \subset I$ and $\gamma _1(p^ eA) \subset p^ eA$ and for $n > 1$ we have

as $p^ n/n! \in \mathbf{Z}_{(p)}$ and as $n \geq 2$ and $e \geq 2$ so $n(e - 1) \geq e$. This proves that $\gamma $ extends to $A/p^ eA$, see Lemma 23.4.3. The statement on limits is clear from the construction of limits in the proof of Lemma 23.3.2. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)