Filtration on the de Rham–Witt Complex

• Filtration on the de Rham–Witt complex is a framework that encodes key arithmetic and geometric data in p-adic and positive characteristic settings.
• Different filtrations—canonical, p-adic, Nygaard, and logarithmic—offer explicit tools for analyzing ramification, cohomological dualities, and class field theory.
• The compatibility with Witt complex operators and functorial properties underpins robust applications in higher displays, p-adic Hodge theory, and ramification control.

Filtrations on the de Rham–Witt complex are a central tool for analyzing ramification phenomena, duality, and canonical structures in $p$-adic and positive characteristic geometry. These filtrations, ranging from canonical and $p$-adic variants to log and Nygaard filtrations, encode subtle arithmetic and geometric data. They are used to define ramification filtrations (in the sense of Kato, Brylinski, Matsuda, and Abbes–Saito), to express cohomological dualities, and to realize structures such as higher displays and strongly divisible lattices. The main types of filtrations on the de Rham-Witt complex—especially those controlling ramification along divisors—play a foundational role in modern $p$-adic cohomology and arithmetic geometry.

1. The de Rham–Witt Complex and its Canonical Filtrations

Let $X$ be a regular scheme over a perfect field $k$ of characteristic $p>0$, and $W_m\Omega_X^\bullet$ the $p$-typical de Rham–Witt complex. This complex arises as a universal Witt complex with operators: Frobenius $F$, Verschiebung $V$, restriction $R$, and the de Rham differential $d$; their relations codify the underlying $p$-typical structure (Davis, 2017).

Three principal internal filtrations appear for $W_m\Omega_X^\bullet$ when $A=W(k)$:

• Canonical ("Hodge") filtration: $$\mathrm{Fil}_{\mathrm{can}}^r W_n\Omega^d_A = \ker(R^r: W_n\Omega^d_A \to W_{n-r}\Omega^d_A)$$ or equivalently as a sum of $V^i$ and $d V^i$-generated terms with $i\geq r$.
• $p$-adic filtration: $\mathrm{Fil}^m_p W_n\Omega^*_A = p^m W_n\Omega^*_A$, a standard $p$-adic valuation decreasing filtration.
• Nygaard filtration: $$N^r W_n\Omega^*_A = \{x \mid F(x) \in p^r W_{n-1}\Omega^*_A\}$$, yielding a decreasing sequence $N^0 \supset N^1 \supset \ldots$ (Davis, 2017, Gregory et al., 2017).

Each filtration is preserved, up to predictable changes in level, by the Witt complex operators—$F$, $V$, $R$, and $d$—which is essential for their arithmetic utility.

2. Logarithmic de Rham–Witt Sheaves and the log-Witt Filtration

In the presence of a divisor with normal crossings $D$ on a semistable $X$, the log-structure induces the logarithmic de Rham–Witt sheaf $W_n\Omega_{X,\log}^r$. Locally on the étale site, this sheaf is generated by "pure" logarithmic forms $d\log[x_1]_n \wedge \cdots \wedge d\log[x_r]_n$ where $[x]_n$ is the Teichmüller representative in $W_n(\mathcal{O}_X)$ (Zhao, 2016).

For $D = \sum m_i D_i$, the "relative" log-Witt sheaf $W_n\Omega^r_{X|mD,\log}$ is defined as the subsheaf of $j_*\bigl.W_n\Omega^r_{U,\log}\bigr.$ generated by forms whose residues along each $D_i$ have pole order at most $m_i$. This construction induces a filtration on $H^1(U,\mathbb{Z}/p^n)$ by images of cohomology with coefficients in $W_n\Omega^0_{X|mD,\log}$: $$\mathrm{Fil}^m H^1(U, \mathbb{Z}/p^n) := \operatorname{Im}\Big(H^1(X, W_n\Omega^0_{X|mD,\log}) \to H^1(U, \mathbb{Z}/p^n)\Big).$$ On the abelianized étale fundamental group, a corresponding decreasing filtration arises, measuring wild ramification, and in relative dimension zero recovers the Matsuda–Brylinski–Kato filtration (Zhao, 2016).

3. Filtrations Controlling Ramification: Kato, Brylinski, Matsuda

For $X$ regular, $E \subset X$ a simple normal crossings divisor, and $D=\sum n_i E_i$ an effective modulus, the filtered de Rham–Witt complex refines ramification theory (Krishna et al., 17 Jan 2026, Majumder, 1 Jan 2025):

• Krishna–Majumder filtration: The subcomplex $_D W_m\Omega^\bullet_U$ is generated by Witt vectors with prescribed pole orders along $E$, and its components $_D W_m U$, $_D W_m\Omega^q_U$ have explicit local and expansion-theoretic descriptions.
• For $X$ the spectrum of a henselian DVR and $D=n\cdot(\pi)$, the filtration on $W_m(K)$ coincides with Brylinski's filtration: $$_n W_m(K) = \{(a_{m-1},\dots,a_0) \mid a_i \cdot \pi^{-\lfloor n/p^i \rfloor} \in A\}$$.
• The Kato ramification filtration on étale cohomology groups $H^q(K,\mathbb{Q}_p/\mathbb{Z}_p(q-1))$ admits a cohomological description via the filtered de Rham–Witt complex, and the structure of the filtration aligns with the jump in Swan conductors (Krishna et al., 17 Jan 2026, Majumder, 1 Jan 2025).

The key operational compatibilities with $F$, $V$, and $d$ and the functoriality in $(X,E,D)$ allow for robust ramification-theoretic applications, including generalized higher-dimensional class field theory.

4. Nygaard Filtration, Displays, and Slope Theory

The Nygaard filtration, $N^r W\Omega^*_{X/R}$, is fundamental for encoding Frobenius-divisible structures on crystalline cohomology, leading to "higher displays"—algebro-geometric analogues of strongly divisible lattices in $p$-adic Hodge theory (Gregory et al., 2017). Explicitly, $N^r W_n\Omega^i$ is defined as $\ker(F: W_n\Omega^i \to W_{n-1}\Omega^i)$ for $i<r$, and as $W_n\Omega^i$ otherwise, with graded pieces $Gr^r$. The filtration is strictly compatible with base-change and functorial in $X$, and flows naturally into the construction of displays and crystals associated with lifts or PD-thickenings.

This filtration is quasi-isomorphic to filtrations derived from Fontaine–Messing theory and underlies relative and absolute display structures (as in the work of Langer–Zink), thereby structurally governing the slope pieces in $p$-adic cohomological theories (Gregory et al., 2017).

5. Duality, Cohomological Descriptions, and Applications

Filtrations on the de Rham–Witt complex provide the foundation for new duality theorems for ramified covers and étale sheaves. For proper semistable $X$ over a base, Zhao establishes a canonical perfect pairing between $H^i(X, W_n\Omega^r_{X,\log})$ and $H^{d+2-i}(X, W_n\Omega^{d+1-r}_{X,\log})$, culminating in a trace isomorphism (Zhao, 2016). Throughout, filtered versions of the two-term complexes $W_mF^q_{X|D}$ mediate between geometric, arithmetic, and cohomological viewpoints, manifesting in class field theory correspondences, ramified dualities, and explicit descriptions of Picard and Brauer group filtrations (Majumder, 1 Jan 2025, Krishna et al., 17 Jan 2026).

Concrete applications include:

• Lefschetz hyperplane theorems for ramification subgroups and Brauer groups with modulus,
• Explicit graded descriptions (e.g., for $X=\mathbb{P}_k^1$, divisor $D=n\cdot\infty$, the graded pieces relate to $x^{-n}$ expansions),
• Generalizations of the classical results of Serre, Ekedahl, and Milne to the filtered setting.

6. Functoriality, Structural Properties, and Exact Sequences

The compatibility of the various filtrations on the de Rham–Witt complex with the Witt complex operations is essential for the structural theory. Key properties include:

• Functoriality: The assignments $X \mapsto \mathrm{fil}_D W_m\Omega_X^q$ are contravariantly functorial for snc-pairs, and smooth pull-back and excision are compatible.
• Filtered exact sequences: For example, $$0 \to V(\mathrm{fil}_D W_{m-1}\Omega_X^q) + d V (\mathrm{fil}_D W_{m-1}\Omega_X^{q-1}) \to \mathrm{fil}_D W_m\Omega_X^q \xrightarrow{R} \mathrm{fil}_{\lfloor D/p\rfloor} W_{m-1}\Omega_X^q \to 0$$ (Majumder, 1 Jan 2025).
• Filtered Cartier isomorphisms and devissage: These provide technical tools for cohomological calculations and triangles connecting different levels of the filtration.

7. Context, Comparison, and Outlook

Filtrations on the de Rham–Witt complex—especially in the logarithmic and Nygaard contexts—unify and extend the ramification-theoretic approaches of Kato, Brylinski, and Matsuda. In relative dimension zero, the constructions recover classical Swan conductors and ramification theory; in higher dimensions, they provide a multi-parameter conductor framework and effective tools for higher-dimensional class field theory (Zhao, 2016, Krishna et al., 17 Jan 2026, Majumder, 1 Jan 2025). Possible generalizations include the incorporation of torsion coefficients prime to $p$, the study of mixed characteristic analogues (via syntomic or prismatic theory), and comparison with deeper ramification theories of Abbes, Saito, and Vojta.

Fundamentally, the study of filtrations on the de Rham–Witt complex continues to bridge $p$-adic Hodge theory, ramification, and duality, with explicit computational frameworks and broad applications in modern arithmetic geometry.