Hodge Filtration on E∞ de Rham Cohomology

• The Hodge filtration is a canonical, decreasing, and multiplicative filtration on the derived de Rham complex of an E∞-algebra, encoding algebraic forms in derived geometry.
• Its associated graded pieces are equivalent to shifted exterior powers of the cotangent complex, leading to a convergent Hodge–de Rham spectral sequence that clarifies graded structures.
• Connections to cyclic homology, prismatic cohomology, and arithmetic geometry underscore its role in understanding special Zeta-values and the transition from infinitesimal to crystalline contexts.

The Hodge filtration on $E_\infty$ de Rham cohomology is a canonical decreasing, exhaustive, and multiplicative filtration on the derived de Rham complex of an $E_\infty$-ring or algebra, encoding the structure of algebraic forms in derived algebraic geometry and controlling the passage from infinitesimal to derived ("crystalline") contexts. The filtration arises functorially, is universal among filtered $E_\infty$–algebras with prescribed degree-zero piece, and its associated graded pieces are shifted exterior powers of the cotangent complex. This theory has deep connections to cyclic homology, prismatic cohomology, arithmetic geometry, and the functional equations of Zeta-functions.

1. Definition and Universal Properties of the Hodge Filtration

Let $A$ be an $E_\infty$–algebra over a fixed ground $E_\infty$–ring $k$. The derived de Rham complex $\dR(A/k)$ is defined as the initial filtered $E_\infty$–$k$–algebra equipped with an identification of its graded-zero piece

$\gr^0 \bigl( \dR(A/k) \bigr) \simeq A$

so that every filtered $E_\infty$–$k$–algebra $R^\star$ with $\gr^0 R^\star \simeq A$ receives a unique map from $\dR(A/k)$ (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Magidson, 2024). This construction is the left adjoint to the functor which extracts the weight-zero graded piece, making $\dR(-/k)$ a left Kan extension from polynomial algebras. In the formalism of derived divided power algebras, $\dR(A/k)$ is the largest filtered divided power thickening of $A$. The definition is functorial in both $A$ and the base ring $k$ and commutes with base change (Magidson, 2024).

2. Structure of the Filtration and Its Associated Graded

The derived de Rham complex $\dR(A/k) \in \widehat{F\!Mod}_k$ is equipped with a canonical multiplicative decreasing filtration

$\cdots \supset F^2 \dR(A/k) \supset F^1 \dR(A/k) \supset F^0 \dR(A/k) = \dR(A/k) \supset F^{-1} = 0$

called the Hodge filtration. Each stage sits in a fiber sequence

$F^{i+1} \dR(A/k) \longrightarrow F^i \dR(A/k) \longrightarrow \gr^i_F \dR(A/k)$

and the complex can alternately be realized as a totalization of its graded pieces shifted appropriately. The $i$th associated graded is canonically equivalent to the $i$th exterior power of the $E_\infty$ cotangent complex, shifted in degree: $\gr^i_F \dR(A/k) \simeq \Lambda^i_A (L_{A/k}) [-i]$ where $L_{A/k}$ is the $E_\infty$–cotangent complex (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Morin, 2014). This is compatible with descent and base change.

3. The Hodge–de Rham Spectral Sequence

The filtration yields a first-quadrant spectral sequence

$E_1^{i,j} = H^{i+j} (\gr^i_F \dR(A/k)) \Longrightarrow H^{i+j} (\dR(A/k))$

which, via the identification of graded pieces, reads

$E_1^{i,j} = H^j (\Lambda^i_A L_{A/k}) \Longrightarrow H^{i+j} (\dR(A/k))$

The spectral sequence converges whenever the filtration is complete and exhaustive, as follows from connectivity estimates in the connective case ($F^i \dR(A/k)$ is $i$–connective if $k\to A$ is $1$–connective). In the classical setting for a smooth algebra over a field of characteristic zero, the spectral sequence degenerates at $E_1$, reflecting the Hodge decomposition (Antieau, 17 Dec 2025, Morin, 2014, Antieau, 3 Nov 2025).

Filtration Level	Associated Graded	Shift
$i$	$\Lambda^i_A L_{A/k}$	$[-i]$

4. Multiplicative, Functorial, and Structural Properties

The Hodge filtration is multiplicative: for all $p, q$,

$F^p \dR(A/k) \otimes F^q \dR(A/k) \longrightarrow F^{p+q} \dR(A/k)$

and each $F^p \dR(A/k)$ is an $E_\infty$–ideal. The graded object inherits the structure of the exterior algebra of $L_{A/k}$, with multiplication given by the wedge product (Antieau, 17 Dec 2025, Morin, 2014, Antieau, 3 Nov 2025). The functoriality is encoded by the universal property: any map $f: A \to B$ of $E_\infty$–algebras induces a map of filtered complexes compatible with the filtration and the exterior power maps on the associated graded.

5. Classical Case: Smooth Algebras and Differential Forms

If $k$ is a $\mathbb{Q}$–algebra and $A = k[x_1, \dots, x_n]$ is smooth, then $L_{A/k} \simeq \Omega^1_{A/k}$ and

$\dR(A/k) \simeq (\Omega^\bullet_{A/k}, d), \qquad F^i \dR(A/k) = \Omega^{\ge i}_{A/k}$

with associated graded

$\gr^i_F \dR(A/k) \simeq \Omega^i_{A/k}[-i] \simeq \Lambda^i_A \Omega^1_{A/k}[-i]$

This recovers the traditional Hodge filtration on the de Rham complex, and the associated spectral sequence is the classical Hodge–to–de Rham spectral sequence, which degenerates in characteristic zero (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Magidson, 2024).

6. Connections to Cyclic Homology, Prismatic Cohomology, and Arithmetic Applications

The Hodge filtration on $\dR$ is mirrored in filtrations on negative and periodic cyclic homology (via the Beilinson $t$–structure), topological Hochschild homology, and prismatic cohomology. The graded pieces correspond to shifts of the Hodge-completed derived de Rham complex, and in the $p$–adic context, the BMS filtration on $TP$ and $TC^-$ recovers the Hodge filtration after $p$–completion (Morin, 2020, Antieau, 2018, Wagner, 7 Oct 2025). In the stacky approach, the Hodge filtration is realized as the pushforward of the structure sheaf of the Hodge-filtered de Rham stack, and admits comparison to the Nygaard filtration on prismatic cohomology via base change (Hauck, 6 May 2025).

In arithmetic, truncated Hodge-filtered derived de Rham complexes encode Euler characteristics relevant to Milne's correcting factor for special values of zeta functions; the spectral sequence provides determinant calculations, and the approach recovers the functional equation phenomena in Zeta-values and the Bloch conductor (Morin, 2014, Morin, 2020). For quasi-lci rings with bounded torsion, the cofiber $L\Omega^{<n}_{\mathcal{X}/\mathbb{S}}$ directly relates to special Zeta-values and etale cohomology with compact support (Morin, 2020).

7. Derived Divided Power Structures and Infinitesimal–Crystalline Transition

The filtered derived de Rham complex admits a description as the largest filtered divided power thickening of a derived commutative algebra, with the explicit cosimplicial model involving divided power envelopes of the "diagonal" ideal. This description generalizes to square-zero extensions and universal derivations, providing a unifying framework for the first Hodge truncation and the construction of derived crystalline cohomology, which is compatible with the classical theory in smooth contexts (Magidson, 2024).

• "Filtrations and cohomology III: cohomology of $E_\infty$ rings" (Antieau, 17 Dec 2025).
• "Filtrations and cohomology I: crystallization" (Antieau, 3 Nov 2025).
• "Divided Powers and Derived De Rham Cohomology" (Magidson, 2024).
• "Milne's correcting factor and derived de Rham cohomology" (Morin, 2014).
• "Topological Hochschild homology and Zeta-values" (Morin, 2020).
• "A stacky comparison of the Hodge and Nygaard filtrations" (Hauck, 6 May 2025).
• "Periodic cyclic homology and derived de Rham cohomology" (Antieau, 2018).
• "$q$-de Rham cohomology and topological Hochschild homology over ku" (Wagner, 7 Oct 2025).
• "A higher Hodge extension of the Feigin-Tsygan Theorem" (Yeung, 2022).