The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

23.2 Divided powers

In this section we collect some results on divided power rings. We will use the convention $0! = 1$ (as empty products should give $1$).

Definition 23.2.1. Let $A$ be a ring. Let $I$ be an ideal of $A$. A collection of maps $\gamma _ n : I \to I$, $n > 0$ is called a divided power structure on $I$ if for all $n \geq 0$, $m > 0$, $x, y \in I$, and $a \in A$ we have

  1. $\gamma _1(x) = x$, we also set $\gamma _0(x) = 1$,

  2. $\gamma _ n(x)\gamma _ m(x) = \frac{(n + m)!}{n! m!} \gamma _{n + m}(x)$,

  3. $\gamma _ n(ax) = a^ n \gamma _ n(x)$,

  4. $\gamma _ n(x + y) = \sum _{i = 0, \ldots , n} \gamma _ i(x)\gamma _{n - i}(y)$,

  5. $\gamma _ n(\gamma _ m(x)) = \frac{(nm)!}{n! (m!)^ n} \gamma _{nm}(x)$.

Note that the rational numbers $\frac{(n + m)!}{n! m!}$ and $\frac{(nm)!}{n! (m!)^ n}$ occurring in the definition are in fact integers; the first is the number of ways to choose $n$ out of $n + m$ and the second counts the number of ways to divide a group of $nm$ objects into $n$ groups of $m$. We make some remarks about the definition which show that $\gamma _ n(x)$ is a replacement for $x^ n/n!$ in $I$.

Lemma 23.2.2. Let $A$ be a ring. Let $I$ be an ideal of $A$.

  1. If $\gamma $ is a divided power structure1 on $I$, then $n! \gamma _ n(x) = x^ n$ for $n \geq 1$, $x \in I$.

Assume $A$ is torsion free as a $\mathbf{Z}$-module.

  1. A divided power structure on $I$, if it exists, is unique.

  2. If $\gamma _ n : I \to I$ are maps then

    \[ \gamma \text{ is a divided power structure} \Leftrightarrow n! \gamma _ n(x) = x^ n\ \forall x \in I, n \geq 1. \]
  3. The ideal $I$ has a divided power structure if and only if there exists a set of generators $x_ i$ of $I$ as an ideal such that for all $n \geq 1$ we have $x_ i^ n \in (n!)I$.

Proof. Proof of (1). If $\gamma $ is a divided power structure, then condition (2) (applied to $1$ and $n-1$ instead of $n$ and $m$) implies that $n \gamma _ n(x) = \gamma _1(x)\gamma _{n - 1}(x)$. Hence by induction and condition (1) we get $n! \gamma _ n(x) = x^ n$.

Assume $A$ is torsion free as a $\mathbf{Z}$-module. Proof of (2). This is clear from (1).

Proof of (3). Assume that $n! \gamma _ n(x) = x^ n$ for all $x \in I$ and $n \geq 1$. Since $A \subset A \otimes _{\mathbf{Z}} \mathbf{Q}$ it suffices to prove the axioms (1) – (5) of Definition 23.2.1 in case $A$ is a $\mathbf{Q}$-algebra. In this case $\gamma _ n(x) = x^ n/n!$ and it is straightforward to verify (1) – (5); for example, (4) corresponds to the binomial formula

\[ (x + y)^ n = \sum _{i = 0, \ldots , n} \frac{n!}{i!(n - i)!} x^ iy^{n - i} \]

We encourage the reader to do the verifications to make sure that we have the coefficients correct.

Proof of (4). Assume we have generators $x_ i$ of $I$ as an ideal such that $x_ i^ n \in (n!)I$ for all $n \geq 1$. We claim that for all $x \in I$ we have $x^ n \in (n!)I$. If the claim holds then we can set $\gamma _ n(x) = x^ n/n!$ which is a divided power structure by (3). To prove the claim we note that it holds for $x = ax_ i$. Hence we see that the claim holds 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 the claim for $x + y$ if it holds for $x$ and $y$. This follows immediately from the binomial theorem. $\square$

Example 23.2.3. 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. Then $I = pA$ has a canonical divided power structure. Namely, given $x = pa \in I$ we set

\[ \gamma _ n(x) = \frac{p^ n}{n!} a^ n \]

The reader verifies immediately that $p^ n/n! \in p\mathbf{Z}_{(p)}$ for $n \geq 1$ (for instance, this can be derived from the fact that the exponent of $p$ in the prime factorization of $n!$ is $\left\lfloor n/p \right\rfloor + \left\lfloor n/p^2 \right\rfloor + \left\lfloor n/p^3 \right\rfloor + \ldots $), so that the definition makes sense and gives us a sequence of maps $\gamma _ n : I \to I$. It is a straightforward exercise to verify that conditions (1) – (5) of Definition 23.2.1 are satisfied. Alternatively, it is clear that the definition works for $A_0 = \mathbf{Z}_{(p)}$ and then the result follows from Lemma 23.4.2.

We notice that $\gamma _ n\left(0\right) = 0$ for any ideal $I$ of $A$ and any divided power structure $\gamma $ on $I$. (This follows from axiom (3) in Definition 23.2.1, applied to $a=0$.)

Lemma 23.2.4. Let $A$ be a ring. Let $I$ be an ideal of $A$. Let $\gamma _ n : I \to I$, $n \geq 1$ be a sequence of maps. Assume

  1. (1), (3), and (4) of Definition 23.2.1 hold for all $x, y \in I$, and

  2. properties (2) and (5) hold for $x$ in some set of generators of $I$ as an ideal.

Then $\gamma $ is a divided power structure on $I$.

Proof. The numbers (1), (2), (3), (4), (5) in this proof refer to the conditions listed in Definition 23.2.1. Applying (3) we see that if (2) and (5) hold for $x$ then (2) and (5) hold for $ax$ for all $a \in A$. Hence we see (b) implies (2) and (5) hold for a set of generators of $I$ as an abelian group. Hence, by induction of the length of an expression in terms of these it suffices to prove that, given $x, y \in I$ such that (2) and (5) hold for $x$ and $y$, then (2) and (5) hold for $x + y$.

Proof of (2) for $x + y$. By (4) we have

\[ \gamma _ n(x + y)\gamma _ m(x + y) = \sum \nolimits _{i + j = n,\ k + l = m} \gamma _ i(x)\gamma _ k(x)\gamma _ j(y)\gamma _ l(y) \]

Using (2) for $x$ and $y$ this equals

\[ \sum \frac{(i + k)!}{i!k!}\frac{(j + l)!}{j!l!} \gamma _{i + k}(x)\gamma _{j + l}(y) \]

Comparing this with the expansion

\[ \gamma _{n + m}(x + y) = \sum \gamma _ a(x)\gamma _ b(y) \]

we see that we have to prove that given $a + b = n + m$ we have

\[ \sum \nolimits _{i + k = a,\ j + l = b,\ i + j = n,\ k + l = m} \frac{(i + k)!}{i!k!}\frac{(j + l)!}{j!l!} = \frac{(n + m)!}{n!m!}. \]

Instead of arguing this directly, we note that the result is true for the ideal $I = (x, y)$ in the polynomial ring $\mathbf{Q}[x, y]$ because $\gamma _ n(f) = f^ n/n!$, $f \in I$ defines a divided power structure on $I$. Hence the equality of rational numbers above is true.

Proof of (5) for $x + y$ given that (1) – (4) hold and that (5) holds for $x$ and $y$. We will again reduce the proof to an equality of rational numbers. Namely, using (4) we can write $\gamma _ n(\gamma _ m(x + y)) = \gamma _ n(\sum \gamma _ i(x)\gamma _ j(y))$. Using (4) we can write $\gamma _ n(\gamma _ m(x + y))$ as a sum of terms which are products of factors of the form $\gamma _ k(\gamma _ i(x)\gamma _ j(y))$. If $i > 0$ then

\begin{align*} \gamma _ k(\gamma _ i(x)\gamma _ j(y)) & = \gamma _ j(y)^ k\gamma _ k(\gamma _ i(x)) \\ & = \frac{(ki)!}{k!(i!)^ k} \gamma _ j(y)^ k \gamma _{ki}(x) \\ & = \frac{(ki)!}{k!(i!)^ k} \frac{(kj)!}{(j!)^ k} \gamma _{ki}(x) \gamma _{kj}(y) \end{align*}

using (3) in the first equality, (5) for $x$ in the second, and (2) exactly $k$ times in the third. Using (5) for $y$ we see the same equality holds when $i = 0$. Continuing like this using all axioms but (5) we see that we can write

\[ \gamma _ n(\gamma _ m(x + y)) = \sum \nolimits _{i + j = nm} c_{ij}\gamma _ i(x)\gamma _ j(y) \]

for certain universal constants $c_{ij} \in \mathbf{Z}$. Again the fact that the equality is valid in the polynomial ring $\mathbf{Q}[x, y]$ implies that the coefficients $c_{ij}$ are all equal to $(nm)!/n!(m!)^ n$ as desired. $\square$

Lemma 23.2.5. Let $A$ be a ring with two ideals $I, J \subset A$. Let $\gamma $ be a divided power structure on $I$ and let $\delta $ be a divided power structure on $J$. Then

  1. $\gamma $ and $\delta $ agree on $IJ$,

  2. if $\gamma $ and $\delta $ agree on $I \cap J$ then they are the restriction of a unique divided power structure $\epsilon $ on $I + J$.

Proof. Let $x \in I$ and $y \in J$. Then

\[ \gamma _ n(xy) = y^ n\gamma _ n(x) = n! \delta _ n(y) \gamma _ n(x) = \delta _ n(y) x^ n = \delta _ n(xy). \]

Hence $\gamma $ and $\delta $ agree on a set of (additive) generators of $IJ$. By property (4) of Definition 23.2.1 it follows that they agree on all of $IJ$.

Assume $\gamma $ and $\delta $ agree on $I \cap J$. Let $z \in I + J$. Write $z = x + y$ with $x \in I$ and $y \in J$. Then we set

\[ \epsilon _ n(z) = \sum \gamma _ i(x)\delta _{n - i}(y) \]

for all $n \geq 1$. To see that this is well defined, suppose that $z = x' + y'$ is another representation with $x' \in I$ and $y' \in J$. Then $w = x - x' = y' - y \in I \cap J$. Hence

\begin{align*} \sum \nolimits _{i + j = n} \gamma _ i(x)\delta _ j(y) & = \sum \nolimits _{i + j = n} \gamma _ i(x' + w)\delta _ j(y) \\ & = \sum \nolimits _{i' + l + j = n} \gamma _{i'}(x')\gamma _ l(w)\delta _ j(y) \\ & = \sum \nolimits _{i' + l + j = n} \gamma _{i'}(x')\delta _ l(w)\delta _ j(y) \\ & = \sum \nolimits _{i' + j' = n} \gamma _{i'}(x')\delta _{j'}(y + w) \\ & = \sum \nolimits _{i' + j' = n} \gamma _{i'}(x')\delta _{j'}(y') \end{align*}

as desired. Hence, we have defined maps $\epsilon _ n : I + J \to I + J$ for all $n \geq 1$; it is easy to see that $\epsilon _ n \mid _{I} = \gamma _ n$ and $\epsilon _ n \mid _{J} = \delta _ n$. Next, we prove conditions (1) – (5) of Definition 23.2.1 for the collection of maps $\epsilon _ n$. Properties (1) and (3) are clear. To see (4), suppose that $z = x + y$ and $z' = x' + y'$ with $x, x' \in I$ and $y, y' \in J$ and compute

\begin{align*} \epsilon _ n(z + z') & = \sum \nolimits _{a + b = n} \gamma _ a(x + x')\delta _ b(y + y') \\ & = \sum \nolimits _{i + i' + j + j' = n} \gamma _ i(x) \gamma _{i'}(x')\delta _ j(y)\delta _{j'}(y') \\ & = \sum \nolimits _{k = 0, \ldots , n} \sum \nolimits _{i+j=k} \gamma _ i(x)\delta _ j(y) \sum \nolimits _{i'+j'=n-k} \gamma _{i'}(x')\delta _{j'}(y') \\ & = \sum \nolimits _{k = 0, \ldots , n}\epsilon _ k(z)\epsilon _{n-k}(z') \end{align*}

as desired. Now we see that it suffices to prove (2) and (5) for elements of $I$ or $J$, see Lemma 23.2.4. This is clear because $\gamma $ and $\delta $ are divided power structures.

The existence of a divided power structure $\epsilon $ on $I+J$ whose restrictions to $I$ and $J$ are $\gamma $ and $\delta $ is thus proven; its uniqueness is rather clear. $\square$

Lemma 23.2.6. Let $p$ be a prime number. Let $A$ be a ring, let $I \subset A$ be an ideal, and let $\gamma $ be a divided power structure on $I$. Assume $p$ is nilpotent in $A/I$. Then $I$ is locally nilpotent if and only if $p$ is nilpotent in $A$.

Proof. If $p^ N = 0$ in $A$, then for $x \in I$ we have $x^{pN} = (pN)!\gamma _{pN}(x) = 0$ because $(pN)!$ is divisible by $p^ N$. Conversely, assume $I$ is locally nilpotent. We've also assumed that $p$ is nilpotent in $A/I$, hence $p^ r \in I$ for some $r$, hence $p^ r$ nilpotent, hence $p$ nilpotent. $\square$

[1] Here and in the following, $\gamma $ stands short for a sequence of maps $\gamma _1, \gamma _2, \gamma _3, \ldots $ from $I$ to $I$.

Comments (0)


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.




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 07GK. Beware of the difference between the letter 'O' and the digit '0'.