The Stacks project

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. $\square$


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