--- title: Hodge Filtration on E∞ de Rham Cohomology url: https://www.emergentmind.com/topics/hodge-filtration-on-e_-infty-de-rham-cohomology type: topic --- # Hodge Filtration on E∞ de Rham Cohomology 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 $\dR(A/k)$ [2512.15509, 2511.01567, 2405.05153]. 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 [2405.05153]. ## 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 [2512.15509, 2511.01567, 1407.2489]. 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 [2512.15509, 1407.2489, 2511.01567]. | 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 [2512.15509, 1407.2489, 2511.01567]. 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 [2512.15509, 2511.01567, 2405.05153]. ## 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 [2011.11549, 1808.05246, 2510.06057]. 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 [2505.03546]. 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 [1407.2489, 2011.11549]. For quasi-lci rings with bounded torsion, the cofiber $L\Omega^{