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*}

23.3 Divided power rings

There is a category of divided power rings. Here is the definition.

Definition 23.3.1. A divided power ring is a triple $(A, I, \gamma )$ where $A$ is a ring, $I \subset A$ is an ideal, and $\gamma = (\gamma _ n)_{n \geq 1}$ is a divided power structure on $I$. A homomorphism of divided power rings $\varphi : (A, I, \gamma ) \to (B, J, \delta )$ is a ring homomorphism $\varphi : A \to B$ such that $\varphi (I) \subset J$ and such that $\delta _ n(\varphi (x)) = \varphi (\gamma _ n(x))$ for all $x \in I$ and $n \geq 1$.

We sometimes say “let $(B, J, \delta )$ be a divided power algebra over $(A, I, \gamma )$” to indicate that $(B, J, \delta )$ is a divided power ring which comes equipped with a homomorphism of divided power rings $(A, I, \gamma ) \to (B, J, \delta )$.

Lemma 23.3.2. The category of divided power rings has all limits and they agree with limits in the category of rings.

Proof. The empty limit is the zero ring (that's weird but we need it). The product of a collection of divided power rings $(A_ t, I_ t, \gamma _ t)$, $t \in T$ is given by $(\prod A_ t, \prod I_ t, \gamma )$ where $\gamma _ n((x_ t)) = (\gamma _{t, n}(x_ t))$. The equalizer of $\alpha , \beta : (A, I, \gamma ) \to (B, J, \delta )$ is just $C = \{ a \in A \mid \alpha (a) = \beta (a)\} $ with ideal $C \cap I$ and induced divided powers. It follows that all limits exist, see Categories, Lemma 4.14.10. $\square$

The following lemma illustrates a very general category theoretic phenomenon in the case of divided power algebras.

Lemma 23.3.3. Let $\mathcal{C}$ be the category of divided power rings. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that

  1. there exists a cardinal $\kappa $ such that for every $f \in F(A, I, \gamma )$ there exists a morphism $(A', I', \gamma ') \to (A, I, \gamma )$ of $\mathcal{C}$ such that $f$ is the image of $f' \in F(A', I', \gamma ')$ and $|A'| \leq \kappa $, and

  2. $F$ commutes with limits.

Then $F$ is representable, i.e., there exists an object $(B, J, \delta )$ of $\mathcal{C}$ such that

\[ F(A, I, \gamma ) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {C}((B, J, \delta ), (A, I, \gamma )) \]

functorially in $(A, I, \gamma )$.

Proof. This is a special case of Categories, Lemma 4.25.1. $\square$

Lemma 23.3.4. The category of divided power rings has all colimits.

Proof. The empty colimit is $\mathbf{Z}$ with divided power ideal $(0)$. Let's discuss general colimits. Let $\mathcal{C}$ be a category and let $c \mapsto (A_ c, I_ c, \gamma _ c)$ be a diagram. Consider the functor

\[ F(B, J, \delta ) = \mathop{\mathrm{lim}}\nolimits _{c \in \mathcal{C}} Hom((A_ c, I_ c, \gamma _ c), (B, J, \delta )) \]

Note that any $f = (f_ c)_{c \in C} \in F(B, J, \delta )$ has the property that all the images $f_ c(A_ c)$ generate a subring $B'$ of $B$ of bounded cardinality $\kappa $ and that all the images $f_ c(I_ c)$ generate a divided power sub ideal $J'$ of $B'$. And we get a factorization of $f$ as a $f'$ in $F(B')$ followed by the inclusion $B' \to B$. Also, $F$ commutes with limits. Hence we may apply Lemma 23.3.3 to see that $F$ is representable and we win. $\square$

Remark 23.3.5. The forgetful functor $(A, I, \gamma ) \mapsto A$ does not commute with colimits. For example, let

\[ \xymatrix{ (B, J, \delta ) \ar[r] & (B'', J'', \delta '') \\ (A, I, \gamma ) \ar[r] \ar[u] & (B', J', \delta ') \ar[u] } \]

be a pushout in the category of divided power rings. Then in general the map $B \otimes _ A B' \to B''$ isn't an isomorphism. (It is always surjective.) An explicit example is given by $(A, I, \gamma ) = (\mathbf{Z}, (0), \emptyset )$, $(B, J, \delta ) = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta )$, and $(B', J', \delta ') = (\mathbf{Z}/4\mathbf{Z}, 2\mathbf{Z}/4\mathbf{Z}, \delta ')$ where $\delta _2(2) = 2$ and $\delta '_2(2) = 0$ and all higher divided powers equal to zero. Then $(B'', J'', \delta '') = (\mathbf{F}_2, (0), \emptyset )$ which doesn't agree with the tensor product. However, note that it is always true that

\[ B''/J'' = B/J \otimes _{A/I} B'/J' \]

as can be seen from the universal property of the pushout by considering maps into divided power algebras of the form $(C, (0), \emptyset )$.

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