HKR Filtration in Derived Algebraic Geometry

• HKR Filtration is a canonical structure in algebraic and homotopical contexts that filters Hochschild homology into graded pieces equivalent to exterior powers of the cotangent complex.
• It systematically relates derived de Rham theory to spectral sequences and motivic cohomological invariants, providing a bridge between algebraic and topological methods.
• The filtration’s functorial and multiplicative properties enable practical applications in smooth, logarithmic, and mixed characteristic settings within modern algebraic geometry.

The Hochschild–Kostant–Rosenberg (HKR) filtration is a fundamental structure in algebraic and homotopical contexts, providing a canonical filtration on Hochschild and topological Hochschild homology (HH, THH), as well as their logarithmic and motivic generalizations. This filtration canonically relates invariants of algebraic or ring-theoretic objects to the cotangent complex and to de Rham-type or motivic cohomological information. The HKR filtration is central to comparison theorems, spectral sequences, and structural decompositions relating Hochschild (co)homology to derived de Rham theory, and underlies the deep connections between cycles, forms, and loop spaces in both classical and spectral algebraic geometry.

1. Definition and Universal Properties

The HKR filtration on Hochschild homology $HH(R/k)$ of a (possibly derived) $k$-algebra $R$ is a canonical increasing filtration whose $i$th graded piece is naturally equivalent to the $i$th exterior power of the cotangent complex, suspended by $[i]$: $$\operatorname{gr}_i HH(R/k) \simeq (\Lambda^i L_{R/k})[i]$$ This construction is functorial in $R$ and upgrades $HH(R/k)$ to a filtered commutative algebra in $k$-modules, even as an $S^1$-equivariant object through its interpretation in terms of loop spaces. Explicitly, the filtration is given by images of

$$F_i HH(R/k) := \text{Im}\big(R \otimes^L_{R\otimes_k R} (\tau_{\le i} (R\otimes_k R)) \to HH(R/k)\big)$$

with $\text{gr}_i = F_i/F_{i-1}$ (Robalo, 2023).

For a map of $E_\infty$-rings $A\to R$, the HKR filtration generalizes to a universal decreasing multiplicative filtration on topological Hochschild homology $\THH(R/A) = R \otimes_{R \otimes_A R} R$: $\operatorname{gr}^s_{\HKR}\,\THH(R/A) \simeq \operatorname{Sym}^s_R(L_{R/A}[1]) \simeq \Lambda^s_{R/A}[s]\,,\qquad s \ge 0$ The HKR tower is initial among decreasing multiplicative filtrations with specified properties: it interpolates between $\THH(R/A)$ and the cotangent complex, with the $0$th graded piece canonically $R$ (Antieau, 17 Dec 2025).

2. Explicit Formulas, Graded Pieces, and Associated Spectral Sequences

The graded pieces of the HKR filtration on (topological) Hochschild homology recover, as $E_\infty$-modules, the symmetric (or exterior) powers of the cotangent complex $L_{R/A}$ shifted by homological degree. In the discrete smooth case, these coincide with differential forms: $HH_n(A) \cong \bigoplus_{i=0}^n \Omega^i_{A/k}[-i]$ and the filtration $F^p HH_n(A)$ equals the image of forms in degree at least $p$ mapping into $HH_n(A)$. The associated spectral sequence has $E^1$-page

$E^1_{s,t} = \pi_{s+t}(\operatorname{Sym}^{-s}_R(L_{R/A}[1])) \implies \pi_{s+t}\big(\THH(R/A)\big)$

which converges strongly under mild finiteness assumptions (Antieau, 17 Dec 2025, Moulinos et al., 2019).

3. Splittings, Formal Groups, and Derived Geometry

Multiplicative splittings of the HKR filtration, compatible with the $S^1$-action and de Rham differential, are classified (in characteristic zero) by formal exponentials $\widehat{\mathbb{G}_a}\to \widehat{\mathbb{G}_m}$. There is a natural bijection between such splittings and elements of $k^\ast$ when the base field $k$ has characteristic zero, and none in positive characteristic. The filtration is encoded by a “filtered circle” group stack, and splittings correspond to group-stack isomorphisms of filtered structures, classified via Cartier duality. The compatibility with de Rham differentials requires the circle action on the loop space to intertwine with the de Rham complex (Robalo, 2023).

4. HKR Filtration in Logarithmic and Motivic Homotopy Theory

In the context of logarithmic motivic homotopy theory, the HKR filtration appears on logarithmic Hochschild homology $HH(X/R)$ for a map $(R,P)\to (A,M)$ of log rings and a log-smooth $X$. There is a complete descending filtration

$\{\,\Fil^{HKR}_i HH(-/R)\}_{i\in \mathbb{Z}} \subset HH(-/R)$

with graded pieces equivalent to external powers of the log cotangent complex $L_{X/R}$ suspended by $[i]$: $$\operatorname{gr}^{HKR}_i HH(-/R) \simeq (\wedge^i L_{X/R})[i]$$ The filtration satisfies a $P^1$-bundle formula reflecting geometric splitting in projective bundles, with natural equivalences

$\Fil^{HKR}_i \,HH(X\times P^1/R) \simeq \Fil^{HKR}_i \,HH(X/R) \oplus \Fil^{HKR}_{i-1} \,HH(X/R)$

and is compatible with Bond periodicity in the log-motivic $P^1$-spectrum (Binda et al., 5 Mar 2024).

The main compatibility theorem establishes that, for smooth $S \to \operatorname{Spec} R$, the HKR filtration aligns with the very-effective slice filtration in log-motivic homotopy theory: filtered pieces above a given degree vanish after truncation, and the motivic slice tower realizes the HKR and Beilinson filtrations.

5. Filtration in Non-Characteristic Zero and Universal Constructions

In positive or mixed characteristic, the HKR filtration requires a refinement: the equivalence between $S^1$-equivariant mixed complexes and derived de Rham theory fails, and the loop space is replaced by a filtered circle, constructed via affine stacks associated to Witt vectors. The filtered HKR filtration interpolates between Hochschild homology and derived de Rham cohomology, providing a universal approach applicable in all characteristics. For a (derived) affine $X = \operatorname{Spec} A$, the mapping stack construction yields

$$\mathrm{HH}^{fil}(A/R):= \Gamma(\mathcal{L}^{fil} X, \mathcal{O})$$

with underlying object $HH(A/R)$ and associated graded $DR(A/R)$. The associated spectral sequences and fixed-point constructions (e.g., filtered homotopy fixed points for $HC^{-\,fil}(X/R)$) connect with Nygaard and conjugate filtrations in $p$-adic contexts, and recover the Antieau and Bhatt–Morrow–Scholze structures in suitable cases (Moulinos et al., 2019).

6. Examples and Applications

• For $R$ a smooth $k$-algebra over a characteristic-zero field, the HKR filtration coincides with the Hodge filtration on forms.
• For $R$ an $E_\infty$-Thom spectrum, the associated graded of the HKR filtration expresses $\THH(R)$ in terms of Thom classes.
• In logarithmic and motivic frameworks, the filtration provides a canonical motivic realization and compatibility with slice and Beilinson filtrations, bringing motivic and spectral approaches into alignment (Binda et al., 5 Mar 2024, Antieau, 17 Dec 2025).
• Formal group analogues and $q$-deformed theories extend the HKR filtration to new contexts (e.g., elliptic and $q$-de Rham–type filtrations) by replacing Witt vectors with appropriate formal groups (Moulinos et al., 2019).

7. Structural and Conceptual Consequences

The HKR filtration is a universal and robust structure, characterized by functoriality, compatibility with multiplicative and equivariant structures, and spectral convergence. Its splittings and incompatibilities encode deep arithmetic and geometric information, including the difference between characteristic-zero and characteristic $p$ phenomena, and the passage from algebraic to homotopical or motivic contexts. The filtration provides the bridge between homological invariants, derived geometry, and motivic and topological filtrations, influencing a wide range of contemporary research in algebraic and derived geometry, $E_\infty$ ring theory, and homotopy theory (Antieau, 17 Dec 2025, Robalo, 2023, Binda et al., 5 Mar 2024, Moulinos et al., 2019).