Lemma 60.2.4. Let $(B, I, \gamma )$ be a divided power algebra. Let $I \subset J \subset B$ be an ideal. Let $(D, \bar J, \bar\gamma )$ be the divided power envelope of $J$ relative to $\gamma $. Choose elements $f_ t \in J$, $t \in T$ such that $J = I + (f_ t)$. Then there exists a surjection
\[ \Psi : B\langle x_ t \rangle \longrightarrow D \]
of divided power rings mapping $x_ t$ to the image of $f_ t$ in $D$. The kernel of $\Psi $ is generated by the elements $x_ t - f_ t$ and all
\[ \delta _ n\left(\sum r_ t x_ t - r_0\right) \]
whenever $\sum r_ t f_ t = r_0$ in $B$ for some $r_ t \in B$, $r_0 \in I$.
Proof.
In the statement of the lemma we think of $B\langle x_ t \rangle $ as a divided power ring with ideal $J' = IB\langle x_ t \rangle + B\langle x_ t \rangle _{+}$, see Divided Power Algebra, Remark 23.5.2. The existence of $\Psi $ follows from the universal property of divided power polynomial rings. Surjectivity of $\Psi $ follows from the fact that its image is a divided power subring of $D$, hence equal to $D$ by the universal property of $D$. It is clear that $x_ t - f_ t$ is in the kernel. Set
\[ \mathcal{R} = \{ (r_0, r_ t) \in I \oplus \bigoplus \nolimits _{t \in T} B \mid \sum r_ t f_ t = r_0 \text{ in }B\} \]
If $(r_0, r_ t) \in \mathcal{R}$ then it is clear that $\sum r_ t x_ t - r_0$ is in the kernel. As $\Psi $ is a homomorphism of divided power rings and $\sum r_ tx_ t - r_0 \in J'$ it follows that $\delta _ n(\sum r_ t x_ t - r_0)$ is in the kernel as well. Let $K \subset B\langle x_ t \rangle $ be the ideal generated by $x_ t - f_ t$ and the elements $\delta _ n(\sum r_ t x_ t - r_0)$ for $(r_0, r_ t) \in \mathcal{R}$. To show that $K = \mathop{\mathrm{Ker}}(\Psi )$ it suffices to show that $\delta $ extends to $B\langle x_ t \rangle /K$. Namely, if so the universal property of $D$ gives a map $D \to B\langle x_ t \rangle /K$ inverse to $\Psi $. Hence we have to show that $K \cap J'$ is preserved by $\delta _ n$, see Divided Power Algebra, Lemma 23.4.3. Let $K' \subset B\langle x_ t \rangle $ be the ideal generated by the elements
$\delta _ m(\sum r_ t x_ t - r_0)$ where $m > 0$ and $(r_0, r_ t) \in \mathcal{R}$,
$x_{t'}^{[m]}(x_ t - f_ t)$ where $m > 0$ and $t', t \in I$.
We claim that $K' = K \cap J'$. The claim proves that $K \cap J'$ is preserved by $\delta _ n$, $n > 0$ by the criterion of Divided Power Algebra, Lemma 23.4.3 (2)(c) and a computation of $\delta _ n$ of the elements listed which we leave to the reader. To prove the claim note that $K' \subset K \cap J'$. Conversely, if $h \in K \cap J'$ then, modulo $K'$ we can write
\[ h = \sum r_ t (x_ t - f_ t) \]
for some $r_ t \in B$. As $h \in K \cap J' \subset J'$ we see that $r_0 = \sum r_ t f_ t \in I$. Hence $(r_0, r_ t) \in \mathcal{R}$ and we see that
\[ h = \sum r_ t x_ t - r_0 \]
is in $K'$ as desired.
$\square$
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