The completed derived de Rham complex of a complete intersection

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

Let $B=Q/I$ be a quotient of an algebra $Q$ whose cotangent complex is its flat module of differentials, by a weakly regular ideal. The Čech nerve of $Q$ over the base, completed along the kernel of its map to $B$, is an algebraic model of the infinitesimal neighbourhoods of $\operatorname{Spec}B$ in $\operatorname{Spec}Q$ and its powers. This lesson computes its graded pieces: the first is the two-term complex $I/I^2\to\Omega^1_Q\otimes B$, a model of the cotangent complex, and over a ring containing $\mathbb Q$ the others are its symmetric powers. It deduces that, over such a ring, the completed Čech nerve does not depend on the presentation $B=Q/I$ and is filtered quasi-isomorphic to the Hodge-completed derived de Rham complex of $B$. In any characteristic the same holds for smooth presentations with regular kernels over a noetherian ring, with the divided power envelope of the Čech nerve in place of the Čech nerve and divided powers in place of symmetric powers. The input is the description of the graded ring of the divided power envelope of a regular ideal as a divided power algebra, proved here through the Koszul relations.

Prerequisites. The derived de Rham complex (Lemmas 1.2–1.4, Theorem 2.3, Definition 4.1, Theorem 4.2, Corollaries 5.3–5.4); Cohomology of flat formal categories with divided powers (Proposition 1.1); Formal categories with divided powers and the Čech–Alexander complex (Proposition 1.6, Corollary 3.4); from the AI-integrated Stacks project, divided powers and divided power envelopes [Tags 07GL, 07H8, 07HA, 07HC, 07HE]; Koszul-regular ideals and immersions [Tags 07CV, 062F, 063J, 063K, 067P, 067Q, 067R]; smooth ring maps and local complete intersections [Tags 00T4, 00TD, 00TA, 07D3, 07D0]; resolutions by polynomial algebras of finite type [Tags 08PX, 08QI]; Cotangent complexes of simplicial rings (Proposition 1.3, Definition 2.3, Exercise 5.1); Localization morphisms and the cotangent complex (Proposition 4.1); Weakly regular ideals (Definition 2.4, Theorem 3.3, Proposition 4.2); Weak complete intersections (Definition 2.1, Proposition 2.2, Theorem 4.2); Non-additive functors of complexes in a ringed topos (Example 1.3, Section 2, Proposition 2.1, Lemma 4.1); Koszul complexes and the shift formulas (Conventions); Derived tensor products of simplicial modules (Lemma 1.1, Proposition 2.2).

Conventions

$T$ is a topos and rings are commutative. For a map of rings $A\to B$, $L_{B/A}$ is the cotangent complex, $H_0(L_{B/A})=\Omega^1_{B/A}$ [Stacks, Tag 08UR], and maps $A\to P\to B$ give the transitivity triangle [Tag 08SZ] and its long exact homology sequence, natural in commutative diagrams. For a surjection $P\to B$ with kernel $I$, $H_0(L_{B/P})=0$ and $H_1(L_{B/P})=I/I^2$ naturally; if $I$ is weakly regular, $L_{B/P}\cong(I/I^2)[1]$ (Proposition 2.2 (2) of the lesson on weak complete intersections). For a cosimplicial module $V$, $N(V)$ is its normalized cochain complex; for a functor $F$ of modules and a complex $X$ in degrees $\geq0$, $F(X)=NF\Gamma^\bullet X$ as in Section 2.1 of the lesson on non-additive functors, so that $NF(V)=F(NV)$. Inverse systems of filtered complexes and filtered quasi-isomorphisms are as in Definition 4.1 of the lesson on the derived de Rham complex.

1. Weakly smooth algebras

Definition 1.1. A map of rings $A\to Q$ of $T$ is weakly smooth, and $Q$ is weakly smooth over $A$, if $Q$ is flat over $A$, $L_{Q/A}\to\Omega^1_{Q/A}$ is a quasi-isomorphism and $\Omega^1_{Q/A}$ is a flat $Q$-module.

Proposition 1.2.

A free $A$-algebra, or more generally the symmetric algebra of a flat $A$-module, is weakly smooth. For a smooth morphism of schemes $X\to S$, $\mathcal O_X$ is weakly smooth over $f^{-1}\mathcal O_S$.
If $Q$ is weakly smooth over $A$ and $Q'$ is weakly smooth over $Q$, then $Q'$ is weakly smooth over $A$.
If $Q$ is weakly smooth over $A$, then $Q\otimes_AA'$ is weakly smooth over $A'$ for every $A$-algebra $A'$. If $Q$ and $A'$ are weakly smooth over $A$, so is $Q\otimes_AA'$.

Proof. (1) The first case is Proposition 1.3 of the lesson on cotangent complexes of simplicial rings, with flatness of $\operatorname{Sym}_A(E)$ and $\Omega^1=\operatorname{Sym}_A(E)\otimes_AE$. For a smooth morphism, $\mathcal O_X$ is flat over $f^{-1}\mathcal O_S$, $\Omega^1_{X/S}$ is locally free, and $L_{X/S}\to\Omega^1_{X/S}$ is a quasi-isomorphism (Proposition 5.1 of the lesson The cotangent complex, on affine opens, with Proposition 3.2 of the lesson on localization morphisms).

(2) Flatness is transitive. In the triangle $L_{Q/A}\otimes_QQ'\to L_{Q'/A}\to L_{Q'/Q}\to$, the outer terms are flat modules in degree $0$, so $L_{Q'/A}$ has homology only in degree $0$, and the sequence $\begin{aligned}0\to\Omega^1_{Q/A}\otimes_QQ'&\to\Omega^1_{Q'/A}\\&\to\Omega^1_{Q'/Q}\to0\end{aligned}$ is exact; an extension of flat modules is flat.

(3) $Q$ is flat over $A$, so $\operatorname{Tor}_i^A(Q,A')=0$ for $i>0$, and Proposition 4.1 of the lesson on localization morphisms gives $L_{Q\otimes A'/A'}\cong L_{Q/A}\otimes_QQ\otimes_AA'$, a flat module $\Omega^1_{Q/A}\otimes_AA'$ in degree $0$. Flatness is stable under base change. The second statement follows from the first, with $Q\otimes_AA'$ weakly smooth over $A'$ and $A'$ over $A$, by (2). $\square$

2. The first graded piece

Construction 2.1. Let $A\to Q$ be a map of rings and $Q\to B$ a surjection with kernel $I$. The Čech nerve of $Q$ over $A$ is the cosimplicial $A$-algebra $C=C_{Q/A}$, $C^m=Q^{\otimes_A(m+1)}$, whose cofaces insert $1$ and whose codegeneracies multiply adjacent factors. The total multiplication $\mu\colon C^m\to Q$ is a map of cosimplicial algebras from $C$ to the constant $Q$, and the coface $\iota\colon Q\to C^m$ onto the first factor is a section of it in each degree. Put $\begin{aligned}J&=\ker(C\xrightarrow{\mu}Q),\\K&=\ker(C\to Q\to B),\end{aligned}$ cosimplicial ideals with $J\subseteq K$, $J^0=0$ and $K^0=I$. The completed Čech nerve $\widehat C^K_{Q/A}$ is the inverse system of cosimplicial algebras $(C/K^r)_{r\geq1}$, filtered by the $K^s/K^r$.

The ideal $J$ is the augmentation ideal of the affine category $C$ in the sense of the lessons on formal categories with divided powers, and $J^1/(J^1)^2=\Omega^1_{Q/A}$, the class of $d^0(q)-d^1(q)$ corresponding to $dq$, as in those lessons.

Lemma 2.2. There is an exact sequence of cosimplicial $B$-modules $\begin{aligned}0\to(J/J^2)\otimes_QB&\longrightarrow K/K^2\\&\longrightarrow I/I^2\to0,\end{aligned}$ with $I/I^2$ constant, the second map induced by $\mu$, and $J/J^2$ a $Q$-module through $C/J=Q$.

Proof. In each degree, $C^m=\iota(Q)\oplus J^m$ as $Q$-modules through $\iota$, and $K^m=\iota(I)\oplus J^m$. So $(K^m)^2=\iota(I^2)+\iota(I)J^m+(J^m)^2$, with $\iota(I^2)\subseteq\iota(Q)$ and the other terms in $J^m$. The map $K^m/(K^m)^2\to I/I^2$ induced by $\mu$ is surjective, and its kernel is the image of $J^m$, which is $J^m/(\iota(I)J^m+(J^m)^2)$, that is, $(J^m/(J^m)^2)\otimes_QQ/I$. The maps are compatible with the cosimplicial structure because $\mu$ is a map of cosimplicial algebras. $\square$

Proposition 2.3. Suppose $\Omega^1_{Q/A}$ is a flat $Q$-module. Then the normalized cochain complex of $K/K^2$ is the complex $N(K/K^2)=\big(I/I^2\xrightarrow{\ d\ }\Omega^1_{Q/A}\otimes_QB\big)$ in degrees $0$ and $1$, with $d(\bar f)=df\otimes1$.

Proof. Let $\bar C$ be the divided power envelope of the affine category $C$, with augmentation ideal $\bar J$ (Corollary 3.4 of the lesson on formal categories with divided powers). In each degree $m$, the map $C^m/(J^m)^2\to\bar C^m/(\bar J^m)^{[2]}$ is an isomorphism. Indeed, the ideal $J^m/(J^m)^2$ of $C^m/(J^m)^2$ has square zero, so $\gamma_1=\mathrm{id}$ and $\gamma_n=0$ for $n\geq2$ define divided powers on it, the axioms of [Tag 07GL] reducing to identities between products of at least two of its elements. The universal property of the envelope [Tag 07H8] gives a map $\bar C^m\to C^m/(J^m)^2$ of $C^m$-algebras compatible with the divided powers, which kills $(\bar J^m)^{[2]}$; and $\bar C^m$ is generated over $C^m$ by the divided powers $\gamma_n(f)$, $f\in J^m$ [Tag 07HA], which lie in $(\bar J^m)^{[2]}$ for $n\geq2$. So $J/J^2=\bar J/\bar J^{[2]}$ as cosimplicial modules, and the module of differentials of $\bar C$ is $J^1/(J^1)^2=\Omega^1_{Q/A}$, which is flat. By Proposition 1.1 (1) of the lesson on cohomology of flat formal categories, applied to $\bar C$, $N(J/J^2)$ is $\Omega^1_{Q/A}$ placed in degree $1$; tensoring with $B$ over $Q$, which commutes with normalization, $N((J/J^2)\otimes_QB)$ is $\Omega^1_{Q/A}\otimes_QB$ in degree $1$. The normalization of the constant $I/I^2$ is $I/I^2$ in degree $0$. Normalization is exact, so Lemma 2.2 gives $N^0(K/K^2)=K^0/(K^0)^2=I/I^2$, $N^1(K/K^2)=\Omega^1_{Q/A}\otimes_QB$ and $N^m=0$ for $m\geq2$. The differential is the coboundary $d^0-d^1\colon K^0\to K^1$, which lands in $J^1$, followed by the class in $J^1/(J^1)^2\otimes_QB=\Omega^1_{Q/A}\otimes_QB$; by the identification of Construction 2.1 this class is $df\otimes1$. $\square$

Corollary 2.4. In the situation of Proposition 2.3, suppose moreover that $Q$ is weakly smooth over $A$. Then the cohomology of $N(K/K^2)$ is $H_1(L_{B/A})$ in degree $0$ and $\Omega^1_{B/A}$ in degree $1$, and a map of such presentations $Q\to Q'$, over $A$ and $B$, induces a quasi-isomorphism of these complexes. If $I$ is weakly regular, $N(K/K^2)$ is quasi-isomorphic to $L_{B/A}[-1]$.

Proof. In the long exact sequence of the triangle of $A\to Q\to B$, $H_1(L_{Q/A}\otimes_QB)=0$ because $L_{Q/A}$ is a flat module in degree $0$, so $\begin{aligned}0&\to H_1(L_{B/A})\to H_1(L_{B/Q})=I/I^2\\&\to\Omega^1_{Q/A}\otimes_QB\to\Omega^1_{B/A}\to0\end{aligned}$ is exact. The middle map is $d$ (Lemma 4.1 of the lesson on weak complete intersections). This exact sequence is natural for maps of presentations, which therefore induce the identity on $H_1(L_{B/A})$ and on $\Omega^1_{B/A}$, hence a quasi-isomorphism. If $I$ is weakly regular, $L_{B/A}$ is isomorphic to the complex $I/I^2\to\Omega^1_{Q/A}\otimes_QB$ in degrees $-1$, $0$ by Theorem 4.2 of the lesson on weak complete intersections, which is $N(K/K^2)[1]$. $\square$

Proposition 2.5 (arbitrary affine categories). Let $C$ be an affine $A$-category, identified with its nerve, a cosimplicial $A$-algebra, let $J$ be its augmentation ideal, the kernel of the augmentation $\mu\colon C\to C^0$ induced by the maps $[m]\to[0]$, and put $U=J^1$. Let $C\to B$ be a surjective map of cosimplicial $A$-algebras onto a constant $B$, with kernel $K$, and $I=K^0$.

There is an exact sequence of cosimplicial $B$-modules $0\to(J/J^2)\otimes_{C^0}B\to$ $K/K^2\to I/I^2\to0$, with $I/I^2$ constant.
If $U/U^2$ is a flat $C^0$-module, $N(K/K^2)$ is the complex $I/I^2\xrightarrow{\ d\ }(U/U^2)\otimes_{C^0}B$ in degrees $0$ and $1$, where $d$ is induced by $d^0-d^1\colon C^0\to U$.

No condition on $I$ is needed.

Proof. The map $C\to B$ is $\mu$ followed by its component $C^0\to B$ of degree $0$: it is compatible with $C(t)=\mu^m$ for $t\colon[m]\to[0]$. So $J\subseteq K$ and $K^0=I$. For a map $v\colon[0]\to[m]$, $\iota=C(v)$ is a section of $\mu^m$, so $C^m=\iota(C^0)\oplus J^m$ as $C^0$-modules through $\iota$; on $J/J^2$ the $C^0$-module structures through the various $C(v)$ agree, since $C(v)(a)-C(v')(a)$ lies in $J^m$. With $\mu$, $\iota$, $C^0$ and $U/U^2$ in place of the total multiplication, the first coface, $Q$ and $\Omega^1_{Q/A}$, the proofs of Lemma 2.2 and Proposition 2.3 apply word for word. $\square$

3. Characteristic zero: the graded pieces

In this section and the next, $A$ is a ring of $T$ in which every nonzero integer is invertible, $Q$ is weakly smooth over $A$, and $I$ is weakly regular. Then $B$ is a weak complete intersection over $A$ (Theorem 4.2 of the lesson on weak complete intersections). Over such a ring every ideal has a unique divided power structure, $x\mapsto x^n/n!$, so the Čech nerve is its own divided power envelope along $J$, divided powers of ideals are powers, and $\Gamma=\operatorname{Sym}$ as functors of modules.

Proposition 3.1.

For every $m$, $K^m$ is a weakly regular ideal of $C^m$, and the multiplication from $\operatorname{Sym}^r_B(K^m/(K^m)^2)$ to $(K^m)^r/(K^m)^{r+1}$ is an isomorphism for every $r$.
The complex $N(K^r/K^{r+1})$ is $\operatorname{Sym}^r$ of the complex $I/I^2\xrightarrow{d}\Omega^1_{Q/A}\otimes_QB$, and it is quasi-isomorphic to $\Lambda^rL_{B/A}[-r]$; it vanishes in degrees $>r$.

Proof. (1) $C^m$ is weakly smooth over $A$ by Proposition 1.2 (3) and (2), and $C^m\to B$ is surjective with kernel $K^m$. Since $B$ is a weak complete intersection over $A$, Theorem 4.2 of the lesson on weak complete intersections shows that $K^m$ is weakly regular, and Theorem 3.3 (1) of the lesson on weakly regular ideals gives the isomorphism.

(2) By (1), $K^r/K^{r+1}=\operatorname{Sym}^r_B(K/K^2)$ as cosimplicial modules, so $N(K^r/K^{r+1})$ is $N\operatorname{Sym}^r(K/K^2)=\operatorname{Sym}^r(N(K/K^2))$, the latter in the sense of Section 2.1 of the lesson on non-additive functors, and $N(K/K^2)$ is the complex of Proposition 2.3, of flat modules: $I/I^2$ by weak regularity, $\Omega^1_{Q/A}\otimes_QB$ by base change. As $\operatorname{Sym}^r=\Gamma^r$, Corollary 5.3 of the lesson on the derived de Rham complex identifies it with $\operatorname{Kos}^\bullet(d)_r$, and Corollary 5.4 there with $\Lambda^rL_{B/A}[-r]$. The vanishing is the dual of Proposition 2.1 of the lesson on non-additive functors, $\operatorname{Sym}^r$ having degree $r$ and $N(K/K^2)$ living in degrees $[0,1]$. $\square$

Corollary 3.2. Let $Q'\to B$ and $Q''\to B$ be two such presentations, with kernels $I'$, $I''$. There is a canonical chain of filtered quasi-isomorphisms between $\widehat C^{K'}_{Q'/A}$ and $\widehat C^{K''}_{Q''/A}$, after normalization.

Proof. $Q=Q'\otimes_AQ''$ is weakly smooth over $A$ by Proposition 1.2 (3), maps onto $B$, and its kernel is weakly regular by Theorem 4.2 of the lesson on weak complete intersections. The maps $Q'\to Q\leftarrow Q''$ over $B$ induce maps of the completed Čech nerves, compatible with the filtrations. On the first graded piece they are quasi-isomorphisms after normalization by Corollary 2.4. On the $r$-th graded piece they are $\operatorname{Sym}^r$ of these, by Proposition 3.1, which are quasi-isomorphisms by Lemma 4.1 of the lesson on non-additive functors, the complexes being bounded and flat. By induction on the length of the filtration, the maps are quasi-isomorphisms on every $N(K^s/K^r)$. $\square$

4. The comparison theorem

Theorem 4.1. Let $A$ be a ring in which every nonzero integer is invertible, $Q$ a weakly smooth $A$-algebra and $B=Q/I$ with $I$ weakly regular. There is a canonical chain of filtered quasi-isomorphisms between the Hodge-completed derived de Rham complex $\widehat{L\Omega}^\bullet_{B/A}$ and the normalized completed Čech nerve $N(\widehat C^K_{Q/A})$, filtered by the powers of $K$.

Proof. By Corollary 3.2 we may choose the presentation. Let $P=P_A(B)$ be the standard resolution and take $Q=P_0=A[B]$, free over $A$. For every $n$, $P_n$ is free over $A$, hence weakly smooth, and $P_n\to B$ is surjective with kernel $I_n$, weakly regular because $B$ is a weak complete intersection. The Čech nerves $C_{P_n/A}$ form a mixed object, simplicial in $n$ and cosimplicial in $m$, with cosimplicial ideals $J_n\subseteq K_n$ as in Construction 2.1. All realizations below are realizations in the direction $n$ (Definition 1.1 of the lesson on the derived de Rham complex) of normalized cochains in the direction $m$, taken on the quotients by the filtrations.

The Hodge side. By Theorem 4.2 of the lesson on the derived de Rham complex, the realization of $N(C_{P/A}/J^r)$, filtered by the $J^s$, is filtered quasi-isomorphic to $L\Omega^\bullet_{B/A}/F^r$; here the divided power envelopes are the Čech nerves themselves.

The map (*) from the $J$-adic to the $K$-adic completion, induced by $J\subseteq K$. Fix $m$. By Proposition 1.1 (a) of the lesson on cohomology of flat formal categories, $J^r/J^{r+1}=\operatorname{Sym}^r_{P_n}(J/J^2)$ in degree $m$, and by Proposition 3.1 (1), $K^r/K^{r+1}=\operatorname{Sym}^r_B(K/K^2)$. By Lemma 2.2, $J/J^2\to K/K^2$ factors as $J/J^2\to(J/J^2)\otimes_{P_n}B\to K/K^2$, the second map injective with cokernel $I_n/I_n^2$. As simplicial modules in $n$: the first map is a quasi-isomorphism because $J/J^2$ is flat over $P_n$ and $P\to B$ is a quasi-isomorphism (Proposition 2.2 of the lesson on derived tensor products); and $n\mapsto I_n/I_n^2$ is acyclic. Indeed, the bisimplicial module $L_{B/P}$, with columns $L_{B/P_n}$, has an acyclic diagonal by Exercise 5.1 of the lesson on cotangent complexes of simplicial rings, $P\to B$ being a quasi-isomorphism; its columns are naturally quasi-isomorphic to $(I_n/I_n^2)[1]$, through $L_{B/P_n}\to\tau_{\leq1}L_{B/P_n}\leftarrow H_1(L_{B/P_n})[1]$; so by Lemma 1.2 of the lesson on the derived de Rham complex and the diagonal lemma (Lemma 1.1 of the lesson on derived tensor products), $n\mapsto I_n/I_n^2$ is acyclic. Hence $J/J^2\to K/K^2$ is a quasi-isomorphism of simplicial modules in every degree $m$, between flat $P$- and $B$-modules, and so is $\operatorname{Sym}^r$ of it, through $\begin{aligned}\operatorname{Sym}^r_P(J/J^2)&\to\operatorname{Sym}^r_P(J/J^2)\otimes_PB\\&=\operatorname{Sym}^r_B(J/J^2\otimes_PB)\\&\to\operatorname{Sym}^r_B(K/K^2)\end{aligned}$ (Lemma 4.1 of the lesson on non-additive functors). The normalized cochains of both graded pieces vanish in degrees $m>r$ (Proposition 3.1 (2), and the same argument for $J$, whose $N(J/J^2)$ lives in degree $1$), and the normalized cochains are natural direct summands. Filtering by $m$, a finite filtration, and applying Lemma 1.2 of the lesson on the derived de Rham complex in the direction $n$, (*) is a quasi-isomorphism on every graded piece, hence on every $N(\cdot/F^r)$ restricted to $F^s$.

The map (**) from the constant object. The degeneracies $P_0\to P_n$ form a map of simplicial algebras from the constant $P_0$ to $P$, over $B$, and induce maps of completed Čech nerves. For each $n$, $P_0\to P_n$ is a map of presentations, so by Corollary 2.4 it induces a quasi-isomorphism on the normalized first graded pieces, and, as in Corollary 3.2, on all $N(K^s/K^r)$. By Lemma 1.2 of the lesson on the derived de Rham complex, the realizations are quasi-isomorphic, and by Lemma 1.4 there the realization of the constant object is homotopy equivalent to $N(\widehat C^K_{P_0/A})$.

Composing, $\widehat{L\Omega}^\bullet_{B/A}$ is connected to $N(\widehat C^K_{Q/A})$ by filtered quasi-isomorphisms of inverse systems. $\square$

So, over a ring containing $\mathbb Q$, the Hodge-completed derived de Rham complex of a weak complete intersection $B=Q/I$ is computed by the formal completion of the Čech nerve of any weakly smooth presentation, and its graded pieces $\Lambda^rL_{B/A}[-r]$ are the symmetric powers of the two-term complex $I/I^2\to\Omega^1_{Q/A}\otimes_QB$.

Remark 4.2. For a morphism of schemes $f\colon X\to S$ over $\mathbb Q$ that factors as a closed immersion $X\to Y$, defined by a weakly regular ideal, followed by a smooth $Y\to S$, Theorem 4.1 applied to $A=f^{-1}\mathcal O_S$, $Q$ the inverse image of $\mathcal O_Y$ and $B=\mathcal O_X$, on the topos of $X$, identifies $\widehat{L\Omega}^\bullet_{X/S}$ with the formal completion along $X$ of the Čech nerve of $Y$ over $S$.

5. Divided power envelopes in any characteristic

In this section $T$ is the punctual topos, so that rings are commutative rings, and $A$ is an arbitrary ring. For a ring $R$ and an ideal $\mathfrak a$, $(D_R(\mathfrak a),\bar{\mathfrak a},\gamma)$ is the divided power envelope [Tag 07H8], and $\bar{\mathfrak a}^{[n]}$ is its divided power filtration: the ideal generated by the products $\gamma_{n_1}(y_1)\cdots\gamma_{n_k}(y_k)$ with $y_i\in\bar{\mathfrak a}$ and $n_1+\cdots+n_k\geq n$. A map of rings carrying $\mathfrak a$ into an ideal $\mathfrak a'$ induces a map of envelopes compatible with the divided powers [Tag 07H8], hence with the filtrations. So the envelopes of the $K^m$ form a cosimplicial ring $D_C(K)$, and the divided power completed Čech nerve $\widehat D^K_{Q/A}$ is the inverse system of cosimplicial rings $(D_C(K)/\bar K^{[r]})_{r\geq1}$, filtered by the $\bar K^{[s]}/\bar K^{[r]}$. If every nonzero integer is invertible in $A$, every ideal has the unique divided powers $y\mapsto y^n/n!$, so $D_C(K)=C$, $\bar K^{[r]}=K^r$ and $\widehat D^K_{Q/A}=\widehat C^K_{Q/A}$.

An ideal of a ring is Koszul-regular (resp. regular) if, near every prime containing it, it is generated by a Koszul-regular (resp. regular) sequence [Tag 07CV]; a regular sequence is Koszul-regular [Tag 062F]. A Koszul-regular ideal $I$ of a ring $R$ is weakly regular. Indeed, $I/I^2$ and $\theta_{R,I}$ commute with localization, $\operatorname{Tor}$ commuting with flat base change. On a localization $R_g$ where $I$ is generated by a Koszul-regular sequence, $I_g$ is weakly regular by Proposition 4.2 of the lesson on weakly regular ideals; and at a prime not containing $I$, both sides of $\theta$ vanish. Flatness of $I/I^2$ and bijectivity of $\theta$ are checked at every prime, so they hold.

Lemma 5.1 (the graded ring of a divided power envelope). Let $R$ be a ring, $\mathfrak a$ an ideal, $\bar R=R/\mathfrak a$, $(D,\bar{\mathfrak a},\gamma)$ the divided power envelope of $(R,\mathfrak a)$, and $\operatorname{gr}D=\bigoplus_n\bar{\mathfrak a}^{[n]}/\bar{\mathfrak a}^{[n+1]}$.

$D/\bar{\mathfrak a}=\bar R$, and there is a unique homomorphism of graded $\bar R$-algebras $\Gamma_{\bar R}(\mathfrak a/\mathfrak a^2)\to\operatorname{gr}D$ sending $\gamma_n(\bar y)$ to the class of $\gamma_n(y)$, for $y\in\mathfrak a$. It is surjective, and bijective in degree $1$.
If $\mathfrak a$ is generated by elements $f_1,\ldots,f_r$ whose Koszul complex has no homology in degree $1$, for instance a Koszul-regular sequence, this homomorphism is an isomorphism. Then $\mathfrak a/\mathfrak a^2$ is free with basis the classes of the $f_i$, and $\operatorname{gr}^nD$ is free with basis the classes of the $\gamma_{a_1}(f_1)\cdots\gamma_{a_r}(f_r)$, $a_1+\cdots+a_r=n$.
If $\mathfrak a$ is Koszul-regular, the homomorphism of (1) is an isomorphism.

Proof. (1) Choose generators $f_t$, $t\in S$, of $\mathfrak a$. By [Tag 07HC], applied to $R$ with its zero ideal and the zero divided powers, there is a surjection $\Psi\colon R\langle x_t\rangle\to D$ of rings with divided powers, from the divided power polynomial algebra, with $\Psi(x_t)=f_t$. Its kernel $\mathfrak k$ is generated by the $x_t-f_t$ and the $\delta_n(\sum_tr_tx_t)$, $n\geq1$, for the families $(r_t)$ with $\sum_tr_tf_t=0$. The span $R\langle x_t\rangle_{\geq n}$ of the monomials $x^{[a]}=\prod_tx_t^{[a_t]}$ with $|a|=\sum_ta_t\geq n$ is the $n$-th divided power ideal of $R\langle x_t\rangle_{\geq1}$. Indeed, $x^{[a]}$ is a product of divided powers of total weight $|a|$, and $\delta_k$ carries homogeneous elements of degree $e\geq1$ to homogeneous elements of degree $ke$. The ring $D/\bar{\mathfrak a}$ is $\bar R$, and the ideal $\bar{\mathfrak a}$ is generated by the $\gamma_n(y)$, $y\in\mathfrak a$, $n\geq1$ [Tag 07HA], and $\gamma_n(\sum_tc_tf_t)=\Psi(\delta_n(\sum_tc_tx_t))$. As $\Psi$ respects divided powers, $\bar{\mathfrak a}=\Psi(R\langle x_t\rangle_{\geq1})$ and $\bar{\mathfrak a}^{[n]}=\Psi(R\langle x_t\rangle_{\geq n})$. So $\operatorname{gr}D$ is an $\bar R$-algebra, and $\operatorname{gr}^nD$ is spanned over $\bar R$ by the classes of the products $\Psi(x^{[a]})=\prod_t\gamma_{a_t}(f_t)$ with $|a|=n$.

For $y\in\mathfrak a$, let $\theta_n(y)\in\operatorname{gr}^nD$ be the class of $\gamma_n(y)$. The axioms of divided powers give $\theta_0(y)=1$, $\theta_n(cy)=c^n\theta_n(y)$ for $c\in R$, $\theta_n(y+z)=\sum_{i+j=n}\theta_i(y)\theta_j(z)$ and $\theta_i(y)\theta_j(y)=\binom{i+j}{i}\theta_{i+j}(y)$. For $z\in\mathfrak a^2\subseteq\bar{\mathfrak a}^{[2]}$, $\gamma_j(z)\in\bar{\mathfrak a}^{[2j]}$, so $\theta_n(y+z)=\theta_n(y)$. So $\theta_n$ is defined on $\mathfrak a/\mathfrak a^2$, and it satisfies the relations defining $\Gamma_{\bar R}(\mathfrak a/\mathfrak a^2)$ (Conventions of the lesson on Koszul complexes). This gives the homomorphism. It is unique because the $\gamma_n(\bar y)$ generate, and surjective by the description of $\operatorname{gr}^nD$. In degree $1$, the ideal $\mathfrak a/\mathfrak a^2$ of $R/\mathfrak a^2$ has square zero, so $\gamma_1=\mathrm{id}$ and $\gamma_n=0$ for $n\geq2$ are divided powers on it. The universal property gives a map $D\to R/\mathfrak a^2$ of $R$-algebras that kills $\bar{\mathfrak a}^{[2]}$. So the composite $\mathfrak a/\mathfrak a^2\to\operatorname{gr}^1D\to\mathfrak a/\mathfrak a^2$ is the identity, and the surjection $\mathfrak a/\mathfrak a^2\to\operatorname{gr}^1D$ is injective.

(2) Take the generators $f_1,\ldots,f_r$. Since $H_1$ of the Koszul complex vanishes, every family $(r_t)$ with $\sum_tr_tf_t=0$ is a combination of the Koszul relations: $r_t=\sum_sc_{ts}f_s$ with $c_{ts}=-c_{st}$. As $\sum_{s,t}c_{ts}x_sx_t=0$, $\begin{gathered}\textstyle\sum_tr_tx_t=\sum_ta_tx_t,\\ a_t=-\textstyle\sum_sc_{ts}(x_s-f_s).\end{gathered}$ For $n\geq1$, $\delta_n(\sum_ta_tx_t)$ is the sum of the products $\prod_ta_t^{n_t}x_t^{[n_t]}$ over $n_1+\cdots+n_r=n$, which lie in the ideal generated by the $x_s-f_s$. So $\mathfrak k=(x_1-f_1,\ldots,x_r-f_r)$.

Let $n\geq0$ and $h\in\mathfrak k\cap R\langle x\rangle_{\geq n}$. We show that the component of $h$ of degree $n$ lies in $\mathfrak aR\langle x\rangle_n$. Write $h=\sum_t(x_t-f_t)g_t$, and let $m$ be the smallest degree of a nonzero homogeneous component $g_{t,m}$ of the $g_t$. If $m<n$, the component of $h$ of degree $m$, $-\sum_tf_tg_{t,m}$, vanishes. Its coefficients on the monomials of degree $m$ are relations among the $f_t$, so as above $g_{t,m}=\sum_sc_{ts}f_s$ with $c_{ts}=-c_{st}\in R\langle x\rangle_m$. Replacing $g_t$ by $g_t+\sum_sc_{ts}(x_s-f_s)$ does not change $h$, as $\sum_{s,t}c_{ts}(x_t-f_t)(x_s-f_s)=0$, and it kills the components of degree $m$. After at most $n$ steps every $g_t$ lies in $R\langle x\rangle_{\geq n}$, and the component of $h$ of degree $n$ is $-\sum_tf_tg_{t,n}$. Conversely, for $|a|=n$, $(f_t-x_t)x^{[a]}$ lies in $\mathfrak k\cap R\langle x\rangle_{\geq n}$ and has the component $f_tx^{[a]}$ of degree $n$. Since $\bar{\mathfrak a}^{[n]}=\Psi(R\langle x\rangle_{\geq n})$, $\begin{aligned}\operatorname{gr}^nD&=R\langle x\rangle_{\geq n}/\bigl(R\langle x\rangle_{\geq n+1}\\&\qquad+\mathfrak k\cap R\langle x\rangle_{\geq n}\bigr)\\&=R\langle x\rangle_n/\mathfrak aR\langle x\rangle_n,\end{aligned}$ free over $\bar R$ with basis the classes of the $x^{[a]}$, $|a|=n$, that is, of the $\prod_t\gamma_{a_t}(f_t)$. In degree $1$, with (1), the classes of the $f_t$ form a basis of $\mathfrak a/\mathfrak a^2$. The functor $\Gamma$ carries sums to tensor products (Example 1.3 of the lesson on non-additive functors). For a free module $\bar Re$, $\Gamma_{\bar R}(\bar Re)$ is spanned by the $\gamma_a(e)$, by its relations, and the homomorphism to the divided power polynomial algebra $\bar R\langle x\rangle$ with $\gamma_a(e)\mapsto x^{[a]}$ carries them to a basis. So $\Gamma_{\bar R}(\mathfrak a/\mathfrak a^2)$ has the basis $\prod_t\gamma_{a_t}(\bar f_t)$, which the homomorphism of (1) carries to the basis of $\operatorname{gr}D$.

(3) Let $g\in R$. As $R\to R_g$ is flat, the envelope of $(R_g,\mathfrak a_g)$ is $D\otimes_RR_g$ [Tag 07HE], with the ideal $\bar{\mathfrak a}_g$ and the filtration $(\bar{\mathfrak a}^{[n]})_g$, since $\gamma_n(y/g^k)=\gamma_n(y)/g^{kn}$. So the homomorphism of (1) for $(R_g,\mathfrak a_g)$ is the localization of the one for $(R,\mathfrak a)$, $\Gamma$ commuting with localization. Every prime of $R$ lies in some $D(g)$ on which $\mathfrak a_g$ is generated by a Koszul-regular sequence: by definition if the prime contains $\mathfrak a$, and otherwise for some $g\in\mathfrak a$ outside it, since $\mathfrak a_g=R_g$ is generated by $1$. By (2) these localizations are bijective, hence so is the homomorphism. $\square$

Lemma 5.2 (Koszul-regular kernels). Let $A$ be a ring.

If $Q$ is smooth over $A$ and $I$ is Koszul-regular, every $K^m$ is a Koszul-regular ideal of $C^m$, and so is every $J^m$.
Let $Q'\to B$ and $Q''\to B$ be surjections of $A$-algebras with Koszul-regular kernels, $Q'$ and $Q''$ smooth over $A$. Then $Q'\otimes_AQ''$ is smooth over $A$, and the kernel of $Q'\otimes_AQ''\to B$ is Koszul-regular.
If $Q$ is smooth over $A$ and $I$ is Koszul-regular, the kernel of every surjection $P\to B$ of $A$-algebras from a polynomial algebra of finite type is Koszul-regular.

Proof. A closed immersion of affine schemes is a Koszul-regular immersion exactly when its ideal is Koszul-regular [Tag 063J]. Regular immersions are Koszul-regular immersions [Tag 063K], and Koszul-regular immersions are stable under composition [Tag 067Q] and flat base change [Tag 067P]. Smooth ring maps are stable under base change and composition [Tags 00T4, 00TD]. (1) The ideal $J^m$ defines the diagonal $\operatorname{Spec}Q\to\operatorname{Spec}C^m$, given by the multiplication. It is a section of the projection to the first factor, which is smooth as a base change of $\operatorname{Spec}Q^{\otimes m}\to\operatorname{Spec}A$; so it is a regular immersion [Tag 067R]. The ideal $K^m$ defines its composite with $\operatorname{Spec}B\to\operatorname{Spec}Q$. (2) $\operatorname{Spec}B\to\operatorname{Spec}(Q'\otimes_AQ'')$ is the composite of $\operatorname{Spec}B\to\operatorname{Spec}(B\otimes_AQ'')$, a section of the smooth projection to $\operatorname{Spec}B$, hence a regular immersion [Tag 067R], and of $\operatorname{Spec}(B\otimes_AQ'')\to\operatorname{Spec}(Q'\otimes_AQ'')$. The latter is the base change of $\operatorname{Spec}B\to\operatorname{Spec}Q'$ along the flat morphism $\operatorname{Spec}(Q'\otimes_AQ'')\to\operatorname{Spec}Q'$. (3) $Q$ is syntomic over $A$ [Tag 00TA], hence a local complete intersection [Tag 07D3]: the kernel of a presentation $A[x_1,\ldots,x_n]\to Q$ is Koszul-regular [Tag 07D0]. Composing with $\operatorname{Spec}B\to\operatorname{Spec}Q$, $A\to B$ is a local complete intersection, and the kernel of every presentation of $B$ by a polynomial algebra of finite type is Koszul-regular [Tag 07D0]. $\square$

Proposition 5.3. Let $Q$ be smooth over $A$ and $I$ Koszul-regular, and write $\operatorname{gr}^r\widehat D^K_{Q/A}$ for the cosimplicial module $\bar K^{[r]}/\bar K^{[r+1]}$.

For every $m$, the homomorphism $\Gamma_B(K^m/(K^m)^2)\to\operatorname{gr}D_{C^m}(K^m)$ of Lemma 5.1 (1) is an isomorphism, and these isomorphisms are compatible with the cosimplicial maps.
$N\operatorname{gr}^1\widehat D^K_{Q/A}$ is the complex $I/I^2\xrightarrow{d}\Omega^1_{Q/A}\otimes_QB$ of Proposition 2.3. The complex $N\operatorname{gr}^r\widehat D^K_{Q/A}$ is $\Gamma^r$ of it, quasi-isomorphic to $\Lambda^rL_{B/A}[-r]$, and it vanishes in degrees $>r$.

Proof. (1) By Lemma 5.2 (1) and Lemma 5.1 (3). The homomorphisms of Lemma 5.1 (1) are natural for maps of rings carrying the ideals into each other, the maps of envelopes respecting divided powers.

(2) By Lemma 5.1 (1), $\operatorname{gr}^1\widehat D^K_{Q/A}=K/K^2$ as cosimplicial modules, and $\Omega^1_{Q/A}$ is flat (Proposition 1.2 (1)); so Proposition 2.3 applies. By (1), $\operatorname{gr}^r\widehat D^K_{Q/A}=\Gamma^r_B(K/K^2)$, so its normalized cochain complex is $\Gamma^r(N(K/K^2))$ in the sense of the Conventions. The complex $I/I^2\to\Omega^1_{Q/A}\otimes_QB$ consists of flat modules, $I$ being weakly regular. So Corollary 5.3 of the lesson on the derived de Rham complex identifies $\Gamma^r$ of it with $\operatorname{Kos}^\bullet(d)_r$, and Corollary 5.4 there with $\Lambda^rL_{B/A}[-r]$, as $Q$ is weakly smooth (Proposition 1.2 (1)). The vanishing is the dual of Proposition 2.1 of the lesson on non-additive functors, $\Gamma^r$ having degree $r$. $\square$

Corollary 5.4. Let $Q'\to B$ and $Q''\to B$ be surjections of $A$-algebras with Koszul-regular kernels $I'$, $I''$, with $Q'$ and $Q''$ smooth over $A$. There is a canonical chain of filtered quasi-isomorphisms between $\widehat D^{K'}_{Q'/A}$ and $\widehat D^{K''}_{Q''/A}$, after normalization.

Proof. By Lemma 5.2 (2), $Q=Q'\otimes_AQ''$ is smooth over $A$ and maps onto $B$ with a Koszul-regular kernel. The maps $Q'\to Q\leftarrow Q''$ over $B$ induce maps of the divided power completed Čech nerves, compatible with the filtrations. On the first graded pieces they are quasi-isomorphisms after normalization, by Proposition 5.3 (2) and Corollary 2.4, all three algebras being weakly smooth (Proposition 1.2 (1)). On the $r$-th graded pieces they are $\Gamma^r$ of these, by Proposition 5.3 (1) and the naturality of the homomorphisms of Lemma 5.1 (1). These are quasi-isomorphisms by Lemma 4.1 of the lesson on non-additive functors, the complexes being bounded and flat. By induction on the length of the filtration, the maps are quasi-isomorphisms on every $N(\bar K^{[s]}/\bar K^{[r]})$. $\square$

Theorem 5.5. Let $A$ be a noetherian ring, $Q$ a smooth $A$-algebra and $B=Q/I$ with $I$ Koszul-regular, for instance regular. There is a canonical chain of filtered quasi-isomorphisms between the Hodge-completed derived de Rham complex $\widehat{L\Omega}^\bullet_{B/A}$ and $N(\widehat D^K_{Q/A})$, filtered by the divided powers of $\bar K$.

Proof. As $B$ is of finite type over the noetherian ring $A$, there is a simplicial $A$-algebra $P'$ with an augmentation $P'\to B$, each $P'_n$ a polynomial algebra of finite type, such that $P'\to B$ is a trivial Kan fibration of simplicial sets [Tag 08PX]. So every $P'_n\to B$ is surjective and $P'\to B$ is a quasi-isomorphism [Tag 08QI]: $P'$ is a degreewise free resolution of $B$. Every $P'_n$ is smooth over $A$, and its kernel $I'_n$ is Koszul-regular by Lemma 5.2 (3), hence weakly regular. By Corollary 5.4 we may take $Q=P'_0$. The Čech nerves $C_{P'_n/A}$ form a mixed object, simplicial in $n$ and cosimplicial in $m$, with cosimplicial ideals $J\subseteq K$ as in Construction 2.1. Their envelopes $\bar C_{P'_n/A}$ along $J$ and $D_{C_{P'_n/A}}(K)$ along $K$ form mixed objects too, and $J\subseteq K$ induces a map between them compatible with the divided power filtrations. Realizations are taken in the direction $n$, of normalized cochains in the direction $m$, on the quotients by the filtrations.

The Hodge side. By Theorems 2.3 and 4.2 of the lesson on the derived de Rham complex, whose Čech–Alexander complexes are the envelopes $\bar C_{P'_n/A}$, $\widehat{L\Omega}^\bullet_{B/A}$ is connected by filtered quasi-isomorphisms to the realization of $N(\bar C_{P'/A}/\bar J^{[r]})$, filtered by the $\bar J^{[s]}$.

The map (*) from the $J$-adic to the $K$-adic side. Fix $m$. By Lemma 5.2 (1) and Lemma 5.1 (3), $\bar J^{[r]}/\bar J^{[r+1]}=\Gamma^r_{P'_n}(J/J^2)$ in degree $m$, and by Proposition 5.3 (1), $\bar K^{[r]}/\bar K^{[r+1]}=\Gamma^r_B(K/K^2)$. By the naturality of the homomorphisms of Lemma 5.1 (1), the map between them is $\Gamma^r$ of the map $J/J^2\to K/K^2$. As in the proof of Theorem 4.1, this map factors as $J/J^2\to(J/J^2)\otimes_{P'_n}B\to K/K^2$, the second map injective with cokernel $I'_n/I_n'^2$ (Lemma 2.2). As simplicial modules in $n$, the first map is a quasi-isomorphism, because $J/J^2$ is flat over $P'_n$ (Proposition 1.1 (1) of the lesson on cohomology of flat formal categories) and $P'\to B$ is a quasi-isomorphism (Proposition 2.2 of the lesson on derived tensor products). And $n\mapsto I'_n/I_n'^2$ is acyclic, by the argument of the proof of Theorem 4.1, which uses only that $P'\to B$ is a quasi-isomorphism with weakly regular kernels. Hence $J/J^2\to K/K^2$ is a quasi-isomorphism of simplicial modules in every degree $m$, between flat $P'$- and $B$-modules, and so is $\Gamma^r$ of it, through $\begin{aligned}\Gamma^r_{P'}(J/J^2)&\to\Gamma^r_{P'}(J/J^2)\otimes_{P'}B\\&=\Gamma^r_B(J/J^2\otimes_{P'}B)\\&\to\Gamma^r_B(K/K^2)\end{aligned}$ (Lemma 4.1 of the lesson on non-additive functors). The normalized cochains of both graded pieces vanish in degrees $m>r$: for $K$ by Proposition 5.3 (2), and for $J$ because $N(J/J^2)$ lives in degree $1$ (Proposition 1.1 (1) there). Filtering by $m$, a finite filtration, and applying Lemma 1.2 of the lesson on the derived de Rham complex in the direction $n$, (*) is a quasi-isomorphism on every graded piece, hence on every quotient by the filtration, restricted to every step of it.

The map (**) from the constant object. The degeneracies $P'_0\to P'_n$ form a map of simplicial algebras from the constant $P'_0$ to $P'$, over $B$, and induce maps of divided power completed Čech nerves. For each $n$, $P'_0\to P'_n$ is a map of presentations of $B$ by smooth algebras with Koszul-regular kernels, so, as in the proof of Corollary 5.4, it induces quasi-isomorphisms on all $N(\bar K^{[s]}/\bar K^{[r]})$. By Lemma 1.2 of the lesson on the derived de Rham complex, the realizations are quasi-isomorphic, and by Lemma 1.4 there the realization of the constant object is homotopy equivalent to $N(\widehat D^K_{P'_0/A})$.

Composing, $\widehat{L\Omega}^\bullet_{B/A}$ is connected to $N(\widehat D^K_{Q/A})$ by filtered quasi-isomorphisms of inverse systems. $\square$

So, over a noetherian ring, the Hodge-completed derived de Rham complex of $B=Q/I$, with $Q$ smooth and $I$ regular, is computed by the divided power envelope of the Čech nerve of $Q$, completed along the kernel of its map to $B$, and its graded pieces $\Lambda^rL_{B/A}[-r]$ are the divided powers of the two-term complex $I/I^2\to\Omega^1_{Q/A}\otimes_QB$.

6. Exercises

Exercise 6.1. Let $A$ be a ring containing $\mathbb Q$, $f\in A$ a nonzerodivisor and $B=A/fA$. Using the presentation $Q=A$, show that $\widehat{L\Omega}^\bullet_{B/A}$ is filtered quasi-isomorphic to the inverse system $(A/f^rA)_{r\geq1}$, filtered by the $f^sA/f^rA$.

Exercise 6.2. Let $A$ be a ring containing $\mathbb Q$, $Q=A[x]$ and $B=Q/(x^2)$. Write the complex $N(K/K^2)$ of Proposition 2.3 and compute its cohomology. Deduce $H_1(L_{B/A})$ and $\Omega^1_{B/A}$.

Exercise 6.3. Let $D$ be the divided power envelope of $(\mathbb Z,2\mathbb Z)$. Using Lemma 5.1, show that $\operatorname{gr}^nD\cong\mathbb F_2$ for every $n\geq0$, and that $\gamma_2(2)\neq2$ in $D$ although $2(\gamma_2(2)-2)=0$. Conclude that the map $D\to\mathbb Q$ given by the universal property, $\gamma_n(2)\mapsto2^n/n!$, is not injective.

7. Complete solutions

Solution 6.1. $Q=A$ is weakly smooth over $A$, and $I=fA$ is weakly regular (Proposition 4.2 of the lesson on weakly regular ideals). The Čech nerve of $A$ over $A$ is the constant cosimplicial algebra $A$, with $J=0$ and $K=fA$ in every degree. So $\widehat C^K_{A/A}$ is the constant cosimplicial inverse system $(A/f^rA)_r$, whose normalized cochains are $A/f^rA$ in degree $0$, filtered by the $f^sA/f^rA$. Theorem 4.1 gives the claim. In particular $H^0(L\Omega^\bullet_{B/A}/F^r)\cong A/f^rA$, which makes Exercise 6.2 of the lesson on the derived de Rham complex explicit.

Solution 6.2. $I=(x^2)$, $I/I^2$ is free over $B$ on the class of $x^2$, and $\Omega^1_{Q/A}\otimes_QB$ is free on $dx$. The differential sends $\overline{x^2}$ to $2x\,dx$, and $b\,\overline{x^2}\mapsto2bx\,dx$. Since $2$ is invertible, its kernel is $xB\cdot\overline{x^2}\cong A$ and its cokernel is $B\,dx/xB\,dx\cong A$. So $H^0(N(K/K^2))\cong A$, generated by the class of $x^3$, and $H^1\cong A$, generated by $dx$. By Corollary 2.4, $H_1(L_{B/A})\cong A$ and $\Omega^1_{B/A}\cong B\,dx/(2x\,dx)=A\,dx$.

Solution 6.3. The element $2$ is a nonzerodivisor of $\mathbb Z$, so the sequence $(2)$ is Koszul-regular, and by Lemma 5.1 (2) $\operatorname{gr}^nD$ is free over $\mathbb F_2=\mathbb Z/2\mathbb Z$ with basis the class of $\gamma_n(2)$. In $D$, $2\gamma_2(2)=\gamma_1(2)\gamma_1(2)=4$, so $2(\gamma_2(2)-2)=0$. If $\gamma_2(2)=2$, then $\gamma_1(2)=2$ would lie in the second step of the divided power filtration, and its class in $\operatorname{gr}^1D$ would vanish, which is not the case. The ring $\mathbb Q$, with the ideal $\mathbb Q$ and the divided powers $y\mapsto y^n/n!$, receives $(\mathbb Z,2\mathbb Z)$; the universal property of the envelope gives a map $D\to\mathbb Q$ with $\gamma_n(2)\mapsto2^n/n!$, which sends $\gamma_2(2)-2\neq0$ to $4/2-2=0$. So $D$ has $2$-torsion, and it is not the subring of $\mathbb Q$ generated by the $2^n/n!$.

Sources

Weakly smooth algebras, the computation of the first graded piece of the completed Čech nerve of a presentation, its symmetric powers, the independence of the presentation and the comparison with the Hodge-completed derived de Rham complex are due to Illusie, in his work on the cotangent complex; he states Proposition 2.5 (2) for a weakly regular $I$. The comparison in any characteristic of Section 5, for smooth presentations with regular kernels over a noetherian ring, is also due to Illusie. It rests on Berthelot's description of the graded ring of the divided power envelope of a regular ideal, proved in Lemma 5.1 through the Koszul relations, and on André's resolutions by polynomial algebras of finite type; with crystalline cohomology it identifies the derived de Rham complex of a complete intersection with its crystalline cohomology. Bhatt, p-adic derived de Rham cohomology (arXiv:1204.6560), Section 3, and Completions and derived de Rham cohomology (arXiv:1207.6193) extend these comparisons.