The Stacks Project


Tag 07KD

Chapter 23: Divided Power Algebra > Section 23.4: Extending divided powers

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

  1. the $p$-adic completion $A^\wedge = \mathop{\rm lim}\nolimits_e A/p^eA$ surjects onto $A/I$,
  2. the kernel of this map is the $p$-adic completion $I^\wedge$ of $I$, and
  3. 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

  1. (4)    for $e$ large enough the ideal $p^eA \subset I$ is preserved by the divided power structure $\gamma$ and $$ (A^\wedge, I^\wedge, \gamma^\wedge) = \mathop{\rm lim}\nolimits_e (A/p^eA, I/p^eA, \bar\gamma) $$ 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 10.95.1). 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 $$ \gamma_n(x + p^ey) = \sum\nolimits_{i + j = n} p^{je}\gamma_i(x)\gamma_j(y) = \gamma_n(x) \bmod p^eI $$ 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 $$ \gamma_n(p^ea) = p^n \gamma_n(p^{e - 1}a) = \frac{p^n}{n!} p^{n(e - 1)}a^n \in p^e A $$ 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. $\square$

    The code snippet corresponding to this tag is a part of the file dpa.tex and is located in lines 629–649 (see updates for more information).

    \begin{lemma}
    \label{lemma-extend-to-completion}
    Let $(A, I, \gamma)$ be a divided power ring. Let $p$ be a prime.
    If $p$ is nilpotent in $A/I$, then
    \begin{enumerate}
    \item the $p$-adic completion $A^\wedge = \lim_e A/p^eA$ surjects onto $A/I$,
    \item the kernel of this map is the $p$-adic completion $I^\wedge$ of $I$, and
    \item 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 $I^\wedge$.
    \end{enumerate}
    If moreover $A$ is a $\mathbf{Z}_{(p)}$-algebra, then
    \begin{enumerate}
    \item[(4)] for $e$ large enough the ideal $p^eA \subset I$ is preserved by the
    divided power structure $\gamma$ and
    $$
    (A^\wedge, I^\wedge, \gamma^\wedge) = \lim_e (A/p^eA, I/p^eA, \bar\gamma)
    $$
    in the category of divided power rings.
    \end{enumerate}
    \end{lemma}
    
    \begin{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 \ref{algebra-lemma-completion-generalities}).
    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
    $$
    \gamma_n(x + p^ey) =
    \sum\nolimits_{i + j = n} p^{je}\gamma_i(x)\gamma_j(y) =
    \gamma_n(x) \bmod p^eI
    $$
    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
    $$
    \gamma_n(p^ea) = p^n \gamma_n(p^{e - 1}a) = \frac{p^n}{n!} p^{n(e - 1)}a^n
    \in p^e A
    $$
    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 \ref{lemma-kernel}.
    The statement on limits is clear from the construction of limits in
    the proof of Lemma \ref{lemma-limits}.
    \end{proof}

    Comments (0)

    There are no comments yet for this tag.

    Add a comment on tag 07KD

    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 lower-right corner).

    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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?