Hodge Filtration on E∞ Infinitesimal Cohomology

• The paper introduces the Hodge filtration as a universal filtered E∞-algebra framework that recovers classical de Rham results via the derived symmetric algebra on the cotangent complex.
• It employs filtered derived algebra techniques to construct exact fiber sequences and spectral sequences that identify graded pieces and control degeneration phenomena.
• The work establishes deep links with prismatic and crystalline cohomologies, highlighting functoriality, multiplicative structures, and comparison isomorphisms.

The Hodge filtration on $E_\infty$ infinitesimal cohomology is a highly structured, multiplicative, and functorial filtration arising in derived algebraic geometry and homotopy theory. It recovers and generalizes classical Hodge theory, controlling spectral sequences and providing canonical comparisons to de Rham and crystalline topologies. The framework features deep links with prismatic cohomology, the cotangent complex, and filtered $E_\infty$-algebra structures over various base rings.

1. Foundations: $E_\infty$-Infinitesimal Cohomology and Filtration

Let $k$ be a fixed $E_\infty$-ring, and $R$ an $E_\infty$-algebra over $k$. The $E_\infty$ infinitesimal cohomology $F_H^\star(R/k)$ is defined as a filtered $E_\infty$-algebra universal among those whose associated graded in weight $0$ returns $R$ itself. The underlying construction arises from the left adjoint to the grade-zero functor: $\operatorname{gr}^0: FCAlg_k \to CAlg_k,$ where $FCAlg_k$ is the $\infty$-category of complete decreasingly filtered $E_\infty$ $k$-algebras (Antieau, 17 Dec 2025). The functor $F_H^\star$ provides the Hodge-filtered $E_\infty$-infinitesimal cohomology: $F_H^\star(R/k) \in FCAlg_k.$ Explicitly, $F_H^\star(R/k)$ is equipped with a complete decreasing filtration $F^i_{H}R/k$, and the associated graded satisfies

$$\operatorname{gr}^i F_H^\star(R/k) \simeq \operatorname{Sym}_R^i (L_{R/k}^{E_\infty}[-1]) [i],$$

where $L_{R/k}^{E_\infty}$ is the $E_\infty$-cotangent complex of $R$ over $k$ (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Holeman, 2023).

2. Construction via Filtered Derived Algebra and Cotangent Complex

The construction of the Hodge filtration on $E_\infty$-infinitesimal cohomology employs the derived symmetric algebra on the cotangent complex:

• $F^\star_H(R/k)$ is obtained by resolving $R$ via a simplicial free $E_\infty$ algebra and applying a filtered symmetric monad.
• The filtration is determined by the canonical tower built from truncations of the cochain complex

$$R \xrightarrow{d} L_{R/k}^{E_\infty} \xrightarrow{d} \operatorname{Sym}_R^2(L_{R/k}^{E_\infty}[-1])[2] \rightarrow \cdots$$

• The $i$th filtration level $F^i_H(R/k)$ admits an exact fiber sequence

$$F^{i+1}_H(R/k) \to F^i_H(R/k) \to \operatorname{Sym}^i_R(L_{R/k}^{E_\infty}[-1])[i],$$

so that the associated graded is as above (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Holeman, 2023).

In the presence of a base $(A, I)$ as a prism, the Hodge-filtered infinitesimal cohomology functor can be described via filtered derived $\delta$-rings, with transitions to prismatic or crystalline cohomology mediated by universal derived $I$-adic (prismatic) envelopes and comparison theorems (Holeman, 2023).

3. Spectral Sequences and Identification of Graded Pieces

Applying homotopy groups to the filtration tower yields the Hodge (infinitesimal) spectral sequence: $$E_1^{s,t} = H^{s+t}(\operatorname{Sym}_R^s(L_{R/k}^{E_\infty}[-1])) \implies H^{s+t}(F_H^\star(R/k)),$$ with the $E_\infty$-page canonically isomorphic to the associated graded for the Hodge filtration on $H_{\inf}^\ast(R/k)$: $$\operatorname{gr}^s H_{\inf}^{s+t}(R/k) \simeq H^{s+t}(\operatorname{Sym}^s_R(L_{R/k}^{E_\infty}[-1]))$$ (Antieau, 17 Dec 2025, Antieau, 3 Nov 2025, Holeman, 2023). In the case of smooth algebras over a field of characteristic zero, $L_{R/k}^{E_\infty}$ is concentrated in degree zero, and the Hodge filtration recovers the truncated de Rham complex.

The formalism specializes to produce, after suitable completion, degenerating spectral sequences identifying Hodge and de Rham cohomologies, spectral sequence collapses (e.g., for Kähler manifolds), and isomorphisms of $E_\infty$-pages to associated gradeds in classic filtered cohomological contexts (Antieau, 17 Dec 2025, Wu, 2022, Robles, 2013).

4. Multiplicative Structure, Functoriality, and Universal Properties

The filtration $F^\star_H(R/k)$ is canonically multiplicative as a filtered $E_\infty$-algebra, with the filtration compatible with cup products and all higher operations. Functoriality is governed by the symmetric monoidal adjunction structure: $(k \to R) \mapsto F^\star_{H\,R/k}$ with base-change compatibility: $$F^\star_{H\,S/k} \simeq F^\star_{H\,R/k} \otimes_{F^\star_{H\,R/k}(R)} F^\star_{H\,S/R}$$ and descent for suitable covers (Antieau, 17 Dec 2025).

Universal properties for derived (filtered) envelopes relate the Hodge filtration on $E_\infty$ infinitesimal cohomology to prismatic cohomology: the prismatic cohomology of an $A/I$-algebra is obtained as the derived prismatic envelope of its (Hodge-filtered) infinitesimal cohomology, establishing equivalence with the Bhatt–Scholze theory (Holeman, 2023).

5. Crystallization, Comparison with de Rham, and Degeneration

A key structural theorem is the "crystallization" isomorphism: the filtered derived commutative algebra of Hodge-filtered de Rham cohomology is the image of the Hodge-filtered infinitesimal cohomology under an explicit left adjoint crystallization functor between filtered monads: $$\divideontimes(F^\star_H(R/k)) \simeq F^\star_H dR_{R/k}$$ (Antieau, 3 Nov 2025). In the smooth case, both filtrations agree and recover the classical Hodge filtration on the de Rham complex.

Degeneration phenomena for the relevant spectral sequences are controlled by the cotangent complex and the functorial filtration; in favorable cases (smooth, proper, Kähler, etc.), the Hodge-to-de Rham or Hodge-Tate spectral sequences degenerate at $E_1$, and the resulting $A_\infty$-structure becomes minimal (Antieau, 17 Dec 2025, Wu, 2022).

6. Explicit and Geometric Examples

In the case of a polynomial ring $R = k[x]$, the $E_\infty$ infinitesimal cohomology filtration and spectral sequence calculationally recovers the classical de Rham complex with its Hodge filtration and the trivializing Poincaré lemma. For Artinian rings like $R = k[x]/(x^2)$, the filtration captures the truncated infinitesimal expansion. For complex analytic or algebraic varieties, the construction generalizes to spaces of forms, degenerating to the classical Hodge structure in appropriate cases (Wu, 2022, Antieau, 3 Nov 2025, Robles, 2013).

7. Advanced Contexts and Recent Research Directions

Recent work exhibits intricate links with prismatic and derived $\delta$-ring theory (Holeman, 2023), filtered homotopy transfer and $A_\infty$-structures on $E_\infty$-pages of spectral sequences (e.g., in the context of Frölicher or Hodge-Tate spectral sequences and characteristic cohomology governed by Griffiths transversality) (Cirici et al., 2020, Robles, 2013). The associated filtered derived categories provide the natural habitat for these comparisons, and the filtration’s structural properties (e.g., torsion-freeness, flag variety interpretation, and graded piece decompositions) are fundamental in both arithmetic and complex-analytic contexts (Wu, 2022, Holeman, 2023).