Décalage and the formal Poincaré lemma

Written by Claude Opus 5.5 (Anthropic) on 22 October 2026. Self-checked by the writing AI. Public domain (CC0).

Let $R$ be the nerve of a formal category with divided powers, with objects $A$, arrows $P$ and augmentation ideal $I$. Shifting the nerve by one place, forgetting its first coface and codegeneracy, gives the nerve of a second formal category, whose objects are the arrows $P$ and whose arrows are pairs of composable arrows. Its de Rham complex is $P\otimes_A\Omega^\bullet_R$, a resolution of $A$ in the appropriate sense. This lesson proves the two results behind this. First, when $\Omega^1_R=I/I^{[2]}$ is flat over $A$, the graded ring of the divided power filtration of $P$ is the divided power algebra $\Gamma_A(\Omega^1_R)$. Second, the de Rham complex of the shifted category, augmented by $A$, is acyclic modulo every step of its filtration: the formal Poincaré lemma. For the Čech–Alexander complex of a smooth algebra, the de Rham complex of the shifted category is the linearization of the de Rham complex of the algebra, and the lemma is the formal Poincaré lemma for this linearized de Rham complex.

Prerequisites. The de Rham complex of a formal category with divided powers (its notation throughout: Example 1.3, Proposition 2.4, Definition 3.1, Proposition 3.4, Example 3.5); Formal categories with divided powers and the Čech–Alexander complex (1.4, Proposition 1.6 with (1.4), Lemma 2.3, Propositions 2.6 and 2.7); Koszul complexes and the shift formulas (Proposition 1.1 (2), Theorem 2.1); Non-additive functors of complexes in a ringed topos (Example 1.3: the divided power algebra $\Gamma_A(M)$). Divided power algebras of modules are due to Roby, Lois polynômes et lois formelles en théorie des modules, Ann. Sci. ÉNS 80 (1963) [Numdam].

Conventions

$C$ is a PD formal category with objects $A$ and arrows $P$, given by an affine category or levelwise by a strict one, with nerve $R$, augmentation ideal $I=\ker\varepsilon$ and divided powers $\gamma$ on $I$ for which $\delta$ is a PD morphism. Its PD completion (Proposition 2.7 of the lesson on formal categories) need not be strict; it has the same quotients $P/I^{[n]}$ and $R/J^{[n]}$, through which Corollary 3.1 (3) and Proposition 3.5 (3) concern it. As in the lesson on the de Rham complex of a formal category, $A$ acts on $P$ on the left through $d^1$ and on the right through $d^0$, $\Omega^1_R=I/I^{[2]}$, and $\Omega^\bullet_R=\Lambda_A\Omega^1_R$ carries the exterior derivative $d$ of Proposition 3.4 there. For $m\leq0$, $I^{[m]}=P$. The divided power algebra $\Gamma_A(M)$ of an $A$-module is the algebra generated by symbols $\gamma_m(x)$, $x\in M$, $m\geq0$, with the relations $\gamma_0(x)=1$, $\gamma_m(ax)=a^m\gamma_m(x)$, $\gamma_m(x+y)=\sum_i\gamma_i(x)\gamma_{m-i}(y)$, $\gamma_i(x)\gamma_j(x)=\binom{i+j}{i}\gamma_{i+j}(x)$, sheafified (Roby). By construction, an $A$-algebra $B$ with maps $g_m\colon M\to B$ satisfying the same relations receives a unique algebra map $\Gamma_A(M)\to B$ with $\gamma_m(x)\mapsto g_m(x)$.

1. The shifted category

Construction 1.1. The décalage $\operatorname{Dec}(R)$ is the cosimplicial algebra with $\operatorname{Dec}(R)^n=R^{n+1}$, cofaces $d^{i+1}$ and codegeneracies $s^{i+1}$, $i\geq0$: $R$ with its first coface $d^0$ and first codegeneracy $s^0$ forgotten. It is the nerve of the affine category $C'$ with objects $P$, arrows $P\otimes_AP$, and $\begin{aligned}d^{0\prime}&=\delta,\qquad d^{1\prime}(x)=x\otimes1,\\ \varepsilon'(x\otimes y)&=x\,d^0(\varepsilon(y)),\\ \delta'(x\otimes y)&=x\otimes\delta(y).\end{aligned}$ Indeed $\operatorname{Ner}^n(C')=$ $(P\otimes_AP)\otimes_P\cdots\otimes_P(P\otimes_AP)$, which is $P\otimes_AP^{\otimes n}=R^{n+1}$, and the formulas (1.2) of the lesson on formal categories for $C'$ are those of $R$ shifted by one. Its augmentation ideal is $I'=\ker\varepsilon'=P\otimes_AI$: by the decomposition (1.4) of that lesson, $\varepsilon'=s^1$ kills $P\otimes I$ and is injective on $P\otimes A$.

Lemma 1.2. $C'$ is a PD formal category for the structure induced from $J^2$ on $I'=\iota_2(I)R^2$, and $\Omega^1_{C'}=P\otimes_A\Omega^1_R,$ a $P$-module through the first factor. The algebra map $d^1\colon A\to P$ makes every term of $\operatorname{Dec}(R)$ an $A$-algebra through the first factor, and all cofaces and codegeneracies of $\operatorname{Dec}(R)$ are $A$-linear for it.

Proof. $I'$ is the ideal $I_2$ of (1.3) in that lesson, a sub-PD-ideal of $J^2$; the components of the augmentation ideal of $\operatorname{Dec}(R)$ are sub-PD-ideals of those of $J$, and the cofaces and codegeneracies of $\operatorname{Dec}(R)$ are among those of $R$, hence PD morphisms. The divided powers of $I'$ are generated by $\iota_2(I^{[r]})$, so $I'^{[2]}$ is the image of $P\otimes_AI^{[2]}$, and right exactness of $P\otimes_A-$ gives $\Omega^1_{C'}=P\otimes_AI/I^{[2]}$. The cofaces $d^{i+1}$ and codegeneracies $s^{i+1}$ leave the first factor untouched, except $d^1$, which applies $\delta$ to it, and $\delta(d^1(a)x)=\iota_1(d^1a)\,\delta(x)$ by (1.1) of that lesson. $\square$

Write $(\Omega^\bullet_{C'},D)$ for the de Rham complex of $C'$ (Proposition 3.4 of the lesson on the de Rham complex of a formal category, for $C'$): $\Omega^\bullet_{C'}=\Lambda_P(P\otimes_A\Omega^1_R)=P\otimes_A\Omega^\bullet_R$.

Lemma 1.3. Let $d^+=d^1-d^2\colon P\to P\otimes_AP$, the cochain differential of $\operatorname{Dec}(R)$ in degree $0$.

For $f\in P$, $Df$ is the class of $d^+f=\delta f-f\otimes1$ in $P\otimes_A\Omega^1_R$. $D$ vanishes on $d^1(A)$ and is $A$-linear.
$D(f\otimes\omega)=Df\wedge\omega+f\otimes d\omega$ for local sections $f$ of $P$ and $\omega$ of $\Omega^\bullet_R$.
$D\bigl(I^{[p]}\otimes_A\Omega^q_R\bigr)\subseteq$ $\operatorname{Im}\bigl(I^{[p-1]}\otimes_A\Omega^{q+1}_R\bigr)$ for all $p,q\geq0$.

Proof. (1) By Example 1.3 of the lesson on the de Rham complex of a formal category, applied to $C'$, the differential on objects is $d^{0\prime}-d^{1\prime}=\delta-(\,\cdot\otimes1)$, and $D$ is its class (Proposition 3.4 there). For $a\in A$, $d^+(d^1a)=\iota_1(d^1a)-d^1a\otimes1=0$. An antiderivation killing $A$ is $A$-linear.

(2) It suffices to take $\omega$ of degree $1$, $D$ and $d$ being antiderivations. The cochain differential of $R$ on $R^1\to R^2$ is $d_R=d^0-d^+$, and on $R^2$ the differential of $\operatorname{Dec}(R)$ is $d^1-d^2+d^3=d^0-d_R$. For $f\in P$ and $g\in I$, the Leibniz rule (1.1) of that lesson gives $d_R(f\otimes g)=d_Rf\otimes g-f\otimes d_Rg$, and $d^0(f\otimes g)=d^0f\otimes g$. So the differential of $\operatorname{Dec}(R)$ sends $f\otimes g$ to $\begin{aligned}&(d^0f-d_Rf)\otimes g+f\otimes d_Rg\\&\quad=d^+f\otimes g+f\otimes d_Rg.\end{aligned}$ The map $p'$ of $C'$ sends $P\otimes I\otimes I$ to $P\otimes\Omega^2_R$ by $f\otimes u\otimes v\mapsto f\otimes\bar u\wedge\bar v$, so passing to classes gives $D(f\otimes\bar g)=Df\wedge\bar g+f\otimes d\bar g$, where $d\bar g=p(d_Rg)$.

(3) By (2) and $I^{[p]}\subseteq I^{[p-1]}$, it suffices to show $D(I^{[p]})\subseteq\operatorname{Im}(I^{[p-1]}\otimes\Omega^1_R)$. As $\delta$ is a PD morphism and $J^2=I_1+I_2$, Lemma 2.3 (2) of the lesson on formal categories gives $\delta(I^{[p]})\subseteq\sum_{a+b=p}\operatorname{Im}I^{[a]}\otimes I^{[b]}$. Decompose $P\otimes_AP=(P\otimes A)\oplus(P\otimes I)$ through the second factor. Then $d^+f\in P\otimes I$ for every $f$, since $(\mathrm{id}\otimes\varepsilon)\delta=\mathrm{id}$; so for $f\in I^{[p]}$, $d^+f$ lies in $\sum_{b\geq1}\operatorname{Im}I^{[p-b]}\otimes I^{[b]}$, the terms with $b=0$ contributing only through their component in $P\otimes I$, which lies in $\operatorname{Im}I^{[p]}\otimes I$. Modulo $P\otimes I^{[2]}$ only $b=1$ remains. $\square$

2. The filtration and its graded complex

From here on $\Omega^1_R$ is a flat $A$-module; so is every $\Omega^q_R$, as an exterior power of a flat module.

2.1. For $n\geq0$ let $F^n\subseteq\Omega^\bullet_{C'}$ be the subcomplex $\begin{aligned}F^n&=\bigl(I^{[n]}\to I^{[n-1]}\otimes\Omega^1_R\to\cdots\\&\qquad\to I^{[n-q]}\otimes\Omega^q_R\to\cdots\bigr),\end{aligned}$ a subcomplex by Lemma 1.3 (3); the tensor products are over $A$, and $I^{[n-q]}\otimes\Omega^q_R\to P\otimes\Omega^q_R$ is injective by flatness. Put $P^n=P/I^{[n+1]}$ and $\operatorname{gr}^nP=I^{[n]}/I^{[n+1]}$. By flatness, $\begin{aligned}&\Omega^\bullet_{C'}/F^{n+1}\\&=\bigl(P^n\to P^{n-1}\otimes\Omega^1_R\\&\qquad\to\cdots\to P^0\otimes\Omega^n_R\to0\bigr),\\ &F^n/F^{n+1}\\&=\bigl(\operatorname{gr}^nP\to\operatorname{gr}^{n-1}P\otimes\Omega^1_R\\&\qquad\to\cdots\bigr),\end{aligned}$ and $\operatorname{gr}\Omega^\bullet_{C'}=\operatorname{gr}P\otimes_A\Omega^\bullet_R$, bigraded by $(m,q)$ for $\operatorname{gr}^mP\otimes\Omega^q_R$, with the differential $\bar D$ of bidegree $(-1,+1)$ induced by $D$. On $\operatorname{gr}P$ the left and right actions of $A$ agree, as $d^0a-d^1a\in I$. The augmentation $d^1\colon A\to P$ makes $A\to\Omega^\bullet_{C'}$ and its quotients augmented complexes.

Lemma 2.2. The operations $\bar\gamma_m(\bar x)=\operatorname{cl}\gamma_m(x)\in\operatorname{gr}^mP$, for $x\in I$, are well defined on $\Omega^1_R=\operatorname{gr}^1P$ and define a surjective map of graded $A$-algebras $\begin{aligned}&\phi\colon\Gamma_A(\Omega^1_R)\to\operatorname{gr}P,\\&\gamma_m(\bar x)\mapsto\operatorname{cl}\gamma_m(x).\end{aligned}$

Proof. For $y\in I^{[2]}$, $\gamma_m(x+y)-\gamma_m(x)=$ $\sum_{i<m}\gamma_i(x)\gamma_{m-i}(y)$, and $\gamma_{m-i}(y)\in I^{[2(m-i)]}$ by Lemma 2.3 (1) of the lesson on formal categories, so each term lies in $I^{[i+2(m-i)]}\subseteq I^{[m+1]}$. The relations of the Conventions hold for the $\bar\gamma_m$, because they hold for $\gamma$ in $P$ (with $a$ acting through $d^1$), and the universal property gives $\phi$. It is surjective: $I^{[m]}$ is generated by products of divided powers of total weight $\geq m$, whose classes in $\operatorname{gr}^mP$ are products of the $\bar\gamma_{m_j}(\bar x_j)$ with $\sum m_j=m$ (those of larger weight vanish in $\operatorname{gr}^m$). $\square$

Lemma 2.3 (the graded differential). $\bar D$ is the unique $A$-linear derivation of bidegree $(-1,+1)$ of $\operatorname{gr}P\otimes\Omega^\bullet_R$ with $\begin{aligned}\bar D\bigl(\operatorname{cl}\gamma_nx\bigr)&=\operatorname{cl}\gamma_{n-1}x\otimes\bar x,\\ \bar D(1\otimes y)&=0\end{aligned}$ for local sections $x$ of $I$, $y$ of $\Omega^1_R$ and $n\geq1$. Consequently $\phi\otimes\mathrm{id}\colon\operatorname{Kos}^\bullet(\mathrm{id}_{\Omega^1_R})\to\operatorname{gr}\Omega^\bullet_{C'}$ is a map of complexes, where $\operatorname{Kos}^\bullet(\mathrm{id}_{\Omega^1_R})=\Gamma_A\Omega^1_R\otimes\Lambda_A\Omega^1_R$ is the Koszul complex of Proposition 1.1 (2) of the lesson on Koszul complexes.

Proof. $\bar D$ is a derivation, $D$ being one, and $A$-linear by Lemma 1.3 (1). By Lemma 1.3 (2), $\bar D(\bar f\otimes\omega)=\bar D\bar f\wedge\omega$, since $f\otimes d\omega\in I^{[m]}\otimes\Omega^{q+1}_R\subseteq F^{m+q+1}$ for $f\in I^{[m]}$ and $\omega$ of degree $q$; in particular $\bar D(1\otimes y)=0$. By Lemma 1.3 (1), $D(\gamma_nx)$ is the class of $\delta\gamma_nx-\gamma_nx\otimes1$. Write $\delta x=x\otimes1+1\otimes x+w$ with $w\in I\otimes I$ (the differential $1\otimes x-\delta x+x\otimes1$ lies in $I\otimes I$). As $\delta$ is a PD morphism, $\begin{aligned}\delta(\gamma_nx)&=\gamma_n(\delta x)\\&=\textstyle\sum\gamma_i(x\otimes1)\gamma_j(1\otimes x)\gamma_k(w),\end{aligned}$ over $i+j+k=n$. The term $(n,0,0)$ is $\gamma_nx\otimes1$. Modulo $E_n=\operatorname{Im}I^{[n]}\otimes I+P\otimes I^{[2]}$: the terms with $j\geq2$ contain $1\otimes\gamma_jx\in P\otimes I^{[2]}$; for a pure tensor $u\otimes v\in I\otimes I$, $\gamma_k(u\otimes v)=\gamma_k(u)\otimes v^k$, so $\gamma_k(w)$ lies in $\operatorname{Im}I^{[k]}\otimes I^k$, and the terms with $k\geq1$ lie in $\operatorname{Im}I^{[i+k]}\otimes I^{j+k}$, which is in $E_n$ both for $j=0$ (then $i+k=n$) and for $j=1$ (then $I^{1+k}\subseteq I^{[2]}$). What remains is the term $(n-1,1,0)$, $\gamma_{n-1}x\otimes x$. Hence $D(\gamma_nx)\equiv\gamma_{n-1}x\otimes\bar x$ modulo $\operatorname{Im}I^{[n]}\otimes\Omega^1_R$, the first formula. Uniqueness: $\operatorname{gr}P\otimes\Omega^\bullet_R$ is generated by $\operatorname{gr}P$ and $1\otimes\Omega^1_R$, and $\operatorname{gr}P$ by the $\operatorname{cl}\gamma_nx$ (Lemma 2.2). The Koszul differential is the derivation with $\gamma_k(x)\mapsto\gamma_{k-1}(x)\otimes x$ and $1\otimes z\mapsto0$, so $(\phi\otimes\mathrm{id})$ intertwines the two differentials on generators, hence everywhere, both being derivations. $\square$

Theorem 2.4. If $\Omega^1_R$ is flat over $A$, then $\phi\colon\Gamma_A(\Omega^1_R)\to\operatorname{gr}P$ is an isomorphism of graded algebras, and $\operatorname{Kos}^\bullet(\mathrm{id}_{\Omega^1_R})\cong\operatorname{gr}\Omega^\bullet_{C'}$.

Proof. We show by induction on $n$ that $\phi_n\colon\Gamma^n_A\Omega^1_R\to\operatorname{gr}^nP$ is injective; it is surjective by Lemma 2.2. For $n=0$ and $n=1$, $\phi_n$ is the identity of $A$, resp. of $\Omega^1_R$. Let $n\geq2$, and let $\xi$ be a local section of $\Gamma^n\Omega^1_R$ with $\phi(\xi)=0$. By Lemma 2.3, $(\phi\otimes\mathrm{id})(\kappa\xi)=\bar D\phi(\xi)=0,$ where $\kappa\colon\Gamma^n\Omega^1_R\to\Gamma^{n-1}\Omega^1_R\otimes\Omega^1_R$ is the Koszul differential. By induction $\phi_{n-1}$ is an isomorphism, hence so is $\phi_{n-1}\otimes\mathrm{id}$, and $\kappa\xi=0$. By Theorem 2.1 of the lesson on Koszul complexes, applied to the exact sequence $0\to\Omega^1_R\to\Omega^1_R\to0\to0$ of flat modules, the Koszul sequence $0\to\Gamma^n\Omega^1_R\xrightarrow{\kappa}$ $\Gamma^{n-1}\Omega^1_R\otimes\Omega^1_R\to\cdots\to$ $\Lambda^n\Omega^1_R\to0$ is exact for $n\geq1$; so $\kappa$ is injective and $\xi=0$. The second assertion follows by tensoring with the flat $\Omega^q_R$. $\square$

3. The formal Poincaré lemma

Corollary 3.1. Let $\Omega^1_R$ be flat over $A$.

The augmented complex $A\to\operatorname{gr}\Omega^\bullet_{C'}$ is acyclic, where $A$ maps isomorphically onto $\operatorname{gr}^0P\otimes\Omega^0_R$.
For every $n\geq0$ the augmented complex $A\to\Omega^\bullet_{C'}/F^{n+1}$ of 2.1 is acyclic.
The augmented complex of pro-modules $A\to\text{“}\varprojlim_n\text{”}\,\Omega^\bullet_{C'}/F^{n+1}$ is acyclic at every level, and it is the same for the PD completion of $C$. If $C$ is PD-adic, it is $A\to\Omega^\bullet_{C'}$: this is the formal Poincaré lemma.

Proof. (1) By Theorem 2.4, $\operatorname{gr}\Omega^\bullet_{C'}$ is the sum over $n$ of the Koszul sequences of total degree $n$, exact for $n\geq1$, and in total degree $0$ it is $A$. (2) Induction on $n$: for $n=0$ the complex is $A\to P^0=A$, an isomorphism. The sequences $0\to F^n/F^{n+1}\to$ $\Omega^\bullet_{C'}/F^{n+1}\to\Omega^\bullet_{C'}/F^n\to0$ are exact, and $F^n/F^{n+1}$ is acyclic for $n\geq1$ by (1); the long exact sequence gives the step. (3) The first assertion is (2) for every $n$. The quotients $\Omega^\bullet_{C'}/F^{n+1}$ are formed from the $P^k$, from $\Omega^\bullet_R$ and from the maps between them induced by the structure maps, which the PD completion does not change (Proposition 2.7 of the lesson on formal categories). If $C$ is PD-adic, $P=\text{“}\varprojlim\text{”}P^n$ by the same proposition, so $\Omega^\bullet_{C'}$ is the pro-object of the quotients in (2). $\square$

Remark 3.2. Forgetting the last coface and codegeneracy instead gives the décalage on the other side, with objects $P$ through the other factor; replacing $C$ by its opposite category (exchange $d^0$ and $d^1$, and the factors of $\delta$) turns one into the other. So the augmented complex $A\xrightarrow{d^0}P\to\Omega^1_R\otimes P\to\Omega^2_R\otimes P\to\cdots$, which is $A$-linear on the right, is acyclic in the same sense when $C$ is PD-adic and $\Omega^1_R$ is flat.

Example 3.3. Let $A=K[x_1,\ldots,x_r]$ and $C$ the divided power envelope of the chaotic category of $A/K$, an affine PD formal category whose PD completion is the Čech–Alexander complex (Corollary 3.4 and Definition 3.5 of the lesson on formal categories). Then $P=A\langle\tau_1,\ldots,\tau_r\rangle$, with $\tau_k=1\otimes x_k-x_k\otimes1$, and $\Omega^1_R$ is free on the $\bar\tau_k=dx_k$. Here $\delta\tau_k=\tau_k\otimes1+1\otimes\tau_k$, so $d^+\tau^{[n]}_k=\sum_{j\geq1}\tau_k^{[n-j]}\otimes\tau_k^{[j]}$ and $D(\tau_k^{[n]})=\tau_k^{[n-1]}\,dx_k$. So $\Omega^\bullet_{C'}$ is the divided power de Rham complex of $A\langle\tau\rangle$ over $A$, and Corollary 3.1 (3) is the divided power Poincaré lemma for its completion.

Remark 3.4. By Theorem 2.4, $P^n$ is built from $P^{n-1}$ as a fibre product: $\begin{aligned}P^n&=P^{n-1}\times_{P^{n-2}\otimes\Omega^1_R}\\&\quad\ker\bigl(D\colon P^{n-1}\otimes\Omega^1_R\\&\qquad\to P^{n-2}\otimes\Omega^2_R\bigr),\end{aligned}$ the map $P^n\to\ker D$ being $D$, for $n\geq2$ and $\Omega^1_R$ flat. Injectivity: an element of the kernel of $P^n\to P^{n-1}$ lies in $\operatorname{gr}^nP$, and $\bar D$ is injective on $\operatorname{gr}^nP$ by Theorem 2.4 and the injectivity of the Koszul differential. Surjectivity: given $u\in P^{n-1}$ and a cycle $c$ of $P^{n-1}\otimes\Omega^1_R$ with the same image in $P^{n-2}\otimes\Omega^1_R$ as $Du$, let $\tilde u\in P^n$ lift $u$. Then $c-D\tilde u$ is a cycle of $\operatorname{gr}^{n-1}P\otimes\Omega^1_R$, hence equal to $\bar Dg$ for some $g\in\operatorname{gr}^nP$ by Corollary 3.1 (1), and $\tilde u+g$ maps to $(u,c)$.

Proposition 3.5 (the linearized de Rham complex). For $A$-modules $E,F$ and $n\geq0$, call an $A$-linear map $u\colon P^n\otimes_AE\to F$ a differential operator of order $\leq n$ from $E$ to $F$ relative to $C$; here $P^n=P/I^{[n+1]}$, the tensor product uses its right structure (through $d^0$), and $A$ acts on the left (through $d^1$). Its linearization is $\begin{gathered}L(u)\colon P\otimes_AE\to P\otimes_AF,\\ p\otimes e\mapsto\textstyle\sum p_{(1)}\otimes u(\bar p_{(2)}\otimes e),\end{gathered}$ where $\delta(p)=\sum p_{(1)}\otimes p_{(2)}$ in $P\otimes_AP$ and $\bar p_{(2)}$ is the class in $P^n$.

$L(u)$ is well defined, and the rule $\begin{gathered}u_d(\bar p\otimes\omega)=\varepsilon(p)\,d\omega\\ +[p-d^1\varepsilon(p)]\wedge\omega,\end{gathered}$ with $[\ ]$ the class in $\Omega^1_R=I/I^{[2]}$, defines a differential operator $u_d$ of order $\leq1$ from $\Omega^q_R$ to $\Omega^{q+1}_R$ with $u_d(1\otimes\omega)=d\omega$.
The differential $D$ of $\Omega^\bullet_{C'}=P\otimes_A\Omega^\bullet_R$ is the linearization $L(u_d)$.
Let $R=\hat C_{A/K}$ be the Čech–Alexander complex of a $K$-algebra $A$ with $\Omega^1_{A/K}$ flat, for instance a smooth one, the PD completion of the envelope $C$ of the chaotic category of $A/K$. Then $\Omega^\bullet_R$ is the de Rham complex of $A/K$, $u_d$ is its exterior derivative regarded as a differential operator through the divided power neighbourhoods of the diagonal, and the augmented complex of pro-modules $A\to\text{“}\varprojlim_n\text{”}\,\Omega^\bullet_{C'}/F^{n+1}$, the linearized de Rham complex of $A/K$, is acyclic at every level. If $K$ contains $\mathbf Q$, the divided power neighbourhoods are the infinitesimal ones: $P^n$ is the module of principal parts $(A\otimes_KA)/J^{n+1}$, $J$ the ideal of the diagonal.

Proof. (1) The map $p\otimes e\mapsto\sum p_{(1)}\otimes\bar p_{(2)}\otimes e$ is $\delta\otimes1$ followed by $P\otimes_AP\to P\otimes_AP^n$. For $a\in A$, $\delta(p\,d^0a)=\delta(p)\,\iota_2(d^0a)$ by (1.1) of the lesson on formal categories, so $L(u)(p\,d^0a\otimes e)=$ $\sum p_{(1)}\otimes u(\bar p_{(2)}d^0a\otimes e)=$ $L(u)(p\otimes ae)$. The rule for $u_d$ vanishes for $p\in I^{[2]}$, since then $\varepsilon(p)=0$ and $[p]=0$, and it is additive and left $A$-linear, as $\varepsilon$ and $p\mapsto p-d^1\varepsilon(p)$ are. It is compatible with the right structure: $\varepsilon(p\,d^0a)=\varepsilon(p)a$, and $p\,d^0a-d^1(\varepsilon(p)a)=$ $(p-d^1\varepsilon(p))\,d^0a+$ $d^1\varepsilon(p)\,(d^0a-d^1a)$, whose class is $a[p-d^1\varepsilon(p)]+\varepsilon(p)\,da$, the left and right actions on $\Omega^1_R$ agreeing and $da$ being the class of $d^0a-d^1a$ (Proposition 3.4 of the lesson on the de Rham complex of a formal category). So $u_d(\bar p\,d^0a\otimes\omega)=$ $\varepsilon(p)(a\,d\omega+da\wedge\omega)+$ $a[p-d^1\varepsilon(p)]\wedge\omega=$ $u_d(\bar p\otimes a\omega)$. For $p=1$, $\varepsilon(1)=1$ and $[1-1]=0$.

(2) For $f\in P$ and $\omega\in\Omega^q_R$, $\begin{aligned}&L(u_d)(f\otimes\omega)\\&\quad=\textstyle\sum f_{(1)}\varepsilon(f_{(2)})\otimes d\omega\\&\quad+\textstyle\sum f_{(1)}\otimes[f_{(2)}-d^1\varepsilon(f_{(2)})]\wedge\omega.\end{aligned}$ In $P\otimes_AP$, $\sum f_{(1)}\otimes d^1\varepsilon(f_{(2)})=$ $\sum f_{(1)}d^0\varepsilon(f_{(2)})\otimes1=$ $f\otimes1$, and $\sum f_{(1)}\varepsilon(f_{(2)})=f$, both by $(\mathrm{id}\otimes\varepsilon)\delta=\mathrm{id}$. So the second sum is the class of $\delta f-f\otimes1$ in $P\otimes_A\Omega^1_R$, which is $Df$ (Lemma 1.3 (1)), and $L(u_d)(f\otimes\omega)=$ $f\otimes d\omega+Df\wedge\omega=$ $D(f\otimes\omega)$ by Lemma 1.3 (2).

(3) By Example 3.5 of the lesson on the de Rham complex of a formal category, $(\Omega^\bullet_R,d)$ is the de Rham complex $\Omega^\bullet_{A/K}$, and it is that of $C$, which has the same quotients. Corollary 3.1 (3) for $C$ applies, and (2) identifies the quotients $\Omega^\bullet_{C'}/F^{n+1}$ with those of the linearization. If $\mathbf Q\subseteq K$, the ideal $J$ of the diagonal has the divided powers $x\mapsto x^m/m!$, which every divided power structure on $J$ must equal; so $J$ is its own divided power envelope, $I^{[m]}=J^m$, and $P^n=(A\otimes_KA)/J^{n+1}$. $\square$

4. Exercises

Exercise 4.1. In Example 3.3 with $r=1$, write the complexes of Corollary 3.1 (2) for $n=2$ explicitly and check their exactness directly.

Exercise 4.2. Show that for the PD formal category of Exercise 4.2 of the lesson on the de Rham complex of a formal category ($P=A\langle t\rangle$, $d^0=d^1$), $\Omega^\bullet_{C'}$ is $A\langle t\rangle\otimes\Lambda(A\,dt)$ with $D(t^{[n]})=t^{[n-1]}\,dt$, and deduce Theorem 2.4 for it directly.

5. Complete solutions

Solution 4.1. Here $P^2=A\oplus A\tau\oplus A\tau^{[2]}$, $P^1=A\oplus A\tau$, $P^0=A$, $\Omega^1_R=A\,dx$, $\Omega^2_R=0$. The complex is $A\to P^2\to P^1\,dx\to0$, with $A\to P^2$ the inclusion and $D(a+b\tau+c\tau^{[2]})=(b+c\tau)\,dx$ for $a,b,c\in A$ (Lemma 1.3 (1): $A$-linear, $D\tau=dx$, $D\tau^{[2]}=\tau\,dx$). Its kernel is $A$, the image of the augmentation, and it is surjective onto $P^1dx=(A\oplus A\tau)dx$. So the augmented complex is exact.

Solution 4.2. By Proposition 2.4 of the lesson on the de Rham complex of a formal category, $N'R=NR$, and $\Omega^1_R=A\,\bar t$ with $d=0$ (Exercise 4.2 there). By Lemma 1.2, $\Omega^\bullet_{C'}=A\langle t\rangle\otimes\Lambda(A\,\bar t)$. Since $\delta(t^{[n]})=\sum_{i+j=n}t^{[i]}\otimes t^{[j]}$, Lemma 1.3 (1) gives $D(t^{[n]})=t^{[n-1]}\bar t$, and Lemma 1.3 (2) with $d=0$ gives $D(f\otimes\bar t)=Df\wedge\bar t=0$. Here $\operatorname{gr}P=\bigoplus_nA\,t^{[n]}$ and $\Gamma_A(A\bar t)=\bigoplus_nA\,\gamma_n(\bar t)$, and $\phi(\gamma_n\bar t)=t^{[n]}$: an isomorphism.

Sources

The décalage of a formal category with divided powers, the identification of the graded ring of its divided power filtration with the divided power algebra of its differentials when these are flat, and the formal Poincaré lemma are due to Illusie, in his work on the cotangent complex. The reconstruction of a formal category from its de Rham complex is attributed there to Quillen. The linearization of differential operators, and the formal Poincaré lemma for the linearized de Rham complex of a smooth scheme, are due to Grothendieck, in his work on crystals; their identification with the shifted category is due to Illusie.