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A note on weight filtrations at the characteristic

This paper establishes a canonical weight filtration for $\kgl$ -linear cohomology theories on resolvable motives over affine Dedekind schemes without inverting residual characteristics, demonstrating that this filtration generalizes Deligne's results to positive and mixed characteristic, renders weight-filtered cohomology an invariant of the open part of projective sncd pairs, and reproves the independence of the dual complex's singular cohomology from the chosen compactification.

Original authors: Toni Annala, Piotr Pstrągowski

Published 2026-06-30

📖 6 min read🧠 Deep dive

CC0 1.0

Original authors: Toni Annala, Piotr Pstrągowski

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Sorting a Messy Pile of Objects

Imagine you are an archivist trying to organize a massive, chaotic library. This library contains books (mathematical objects) from different eras and styles. Some books are pristine and simple; others are damaged, torn, or have pages glued together in complex ways.

In the world of algebraic geometry, mathematicians study shapes (schemes) and the "cohomology" of those shapes. Think of cohomology as a way to count the holes, loops, and twists in these shapes to understand their structure. However, when these shapes have "singularities" (kinks, corners, or self-intersections) or when we are working in specific number systems (like modular arithmetic), the data we get is often a messy, unsorted pile.

The Problem:
For a long time, mathematicians had a great way to sort these books for "nice" shapes in "nice" number systems (characteristic 0). They could assign a "weight" to every piece of data, organizing it from simple to complex. This sorting system is called a Weight Filtration.

However, when things get messy (singular shapes) or when we work in "at-the-characteristic" scenarios (where the numbers behave differently, like in modular arithmetic), this sorting system often broke down or didn't exist at all.

The Solution:
Annala and Pstrągowski have built a new, universal machine that can sort these messy piles of data without needing to throw away any information or change the rules of the number system. They call this the Canonical Weight Filtration.

Key Concepts Explained with Analogies

1. The "Resolvable Motives" (The Sortable Pile)

The authors focus on a specific category of mathematical objects they call resolvable motives.

Analogy: Imagine a pile of LEGO bricks. Some are simple, single bricks (smooth shapes). Others are complex structures built from those bricks. A "resolvable motive" is any structure that can be built, taken apart, or reassembled using a specific set of rules involving those simple bricks.
Why it matters: The authors prove that for any object in this "resolvable" pile, you can apply their sorting machine. Even if the object is complicated, it's built from simple parts that can be sorted.

2. The "Weight Filtration" (The Sorting Machine)

A weight filtration is like a set of nested boxes.

Box 1 (Low Weight): Contains the simplest, most fundamental parts of the data (like the basic LEGO bricks).
Box 2 (Medium Weight): Contains slightly more complex combinations.
Box 3 (High Weight): Contains the most intricate, tangled structures.
The Breakthrough: The paper shows that for a huge variety of mathematical theories (including those used in $p$ -adic Hodge theory, which is crucial for modern number theory), this sorting machine works perfectly, even when the numbers involved are "sticky" (residual characteristics are not inverted).

3. The "Open Part" vs. The "Boundary" (The Invariance Surprise)

One of the most surprising findings is about invariance.

The Setup: Imagine you have a garden (a shape $X$ ) with a fence made of hedges (a divisor $D$ ). The "open part" is the grass inside the fence ( $U = X - D$ ).
The Discovery: The authors show that the "sorted data" (the weight filtration) of the whole garden with the fence is actually determined entirely by the grass inside the fence.
Analogy: It's as if you could take a photo of the empty grass field, run it through their machine, and get the exact same "sorted report" as if you had analyzed the whole garden with the fence, the flowers, and the trees. The complexity of the boundary doesn't change the fundamental "weight" of the open space. This means the weight filtration is a true property of the open space itself.

4. The "Pole-Order Filtration" (The Old Way vs. The New Way)

In the world of calculus on shapes (de Rham cohomology), mathematicians have used a sorting method called the "pole-order filtration" for decades. It sorts data based on how "badly" a function blows up near the boundary.

The Comparison: The authors proved that their new, abstract "Weight Filtration" is actually the same thing as this old "Pole-Order Filtration," just viewed through a different lens (specifically, a "décalage" or shifting of the boxes).
Why this is cool: It connects a very modern, abstract theory (motives) with a classical, concrete calculation (differential forms). It confirms that the "mixed Hodge theory" (a fancy way of describing mixed complexity) works even in positive and mixed characteristic, not just in the "clean" world of characteristic 0.

5. Singular Schemes (The Broken Shapes)

What if the shape isn't just a garden with a fence, but a crumpled piece of paper?

The Application: The authors show that their method can be extended to "singular" shapes (shapes with kinks or breaks) by using a technique called cdh-descent.
Analogy: If you have a crumpled paper, you can't analyze it directly. But if you know how to "unfold" it into a smooth sheet (a resolution), analyze that, and then "refold" it, you can still get the correct sorted data. The paper proves that the result doesn't depend on how you unfolded it, as long as you followed the rules. This opens the door to studying singularities in new number systems.

What They Actually Computed (The Examples)

The paper doesn't just build the machine; they run it on specific examples to show it works:

Dual Complexes: They showed that the "singular cohomology" (a measure of shape) of the "dual complex" (a skeleton of the boundary) is an invariant. This re-proves a known result but in a more general setting.
Cones: They calculated the weight filtration for "cones" over smooth shapes. They found that the "weight zero" part of the data depends on the geometry of the base shape, while higher weights depend on the specific way the cone is built.

Summary of the "Takeaway"

This paper is a bridge. It takes a powerful, abstract tool (motivic homotopy theory) and uses it to organize messy data in number theory and geometry.

Before: You could only sort data cleanly if the numbers were "nice" or the shapes were "perfect."
Now: You can sort data for almost any shape and any number system, provided the shape can be built from "resolvable" parts.
The Result: The "weight" of the data is a fundamental property of the open space, independent of how you choose to compactify or bound it. This suggests that "Mixed Hodge Theory" is a robust, universal tool for understanding shapes in all mathematical universes, not just the clean ones.

Note on Limitations: The authors are careful to note that this specific machine only works for theories that are "kgl-linear" (a specific type of algebraic structure). It does not work for every possible mathematical theory (like algebraic cobordism), but it covers the vast majority of the ones currently used in practice.

Technical Summary: Weight Filtrations at the Characteristic

Problem Statement
The paper addresses the construction of canonical weight filtrations for various cohomology theories defined over affine Dedekind schemes $S$ , specifically in contexts where residual characteristics are not inverted (i.e., "at the characteristic"). While weight filtrations are well-established for $\ell$ -adic cohomology and in characteristic zero (via Deligne's mixed Hodge theory), their existence for $kgl$-linear cohomology theories (where $kgl$ is the effective K-theory spectrum) in positive and mixed characteristic has been an open problem. A specific gap exists for logarithmic cohomology theories associated with projective strict normal crossing divisor (sncd) pairs $(X, D)$ , where no general weight filtration was previously constructed for theories such as logarithmic crystalline, prismatic, or syntomic cohomology. Furthermore, it is unclear whether these filtrations yield invariants of the open part $U = X - D$ independent of the compactification.

Methodology
The authors employ a motivic approach, generalizing the work of Bondarko on weight structures in the category of Voevodsky motives over a field. The core methodology relies on the following steps:

Motivic Framework: The authors work within the category of $kgl$-motives, $DM^{kgl}_S$ , and its subcategory of resolvable motives, $DM^{kgl}_{S, res}$ . Resolvable motives are defined as the subcategory generated by perfect pure motives (extensions and retracts of Tate twists of motives of smooth projective $S$ -schemes) under finite limits, colimits, and retracts.
Weight Structure Construction: Using the vanishing of $kgl$-cohomology above the "Chow line" (Lemma 2.1) and the connectivity of mapping spectra between perfect pure motives (Lemma 2.2), the authors establish a bounded "Chow weight structure" on $DM^{kgl}_{S, res}$ . The heart of this structure is the category of perfect pure motives.
Extension of Functors: Leveraging the Elmanto–Sosnilo recognition principle, the authors show that any additive functor defined on perfect pure motives extends uniquely to an exact functor on resolvable motives. By composing a cohomology theory $E$ with the Whitehead filtration (an increasing filtration on spectra), they construct a unique weight filtration $W_*E$ on resolvable motives.
Logarithmic Motives: The authors identify the logarithmic motive $M^{kgl}(X, D)$ of an sncd pair as a resolvable motive. This is constructed as the total fiber of a hypercube involving duals of Tate-twisted motives of the intersections of the divisor components.
Comparison with Classical Filtrations: For de Rham cohomology, the authors construct motivic spectra representing Hodge-filtered de Rham cohomology. They compare the motivic weight filtration with the classical pole-order filtration (filtered by the order of poles along the divisor) on logarithmic differential forms. This comparison utilizes Poincaré duality for logarithmic forms and the properties of the décalage operation.

Key Contributions and Results

Existence of Weight Filtrations (Theorem 1.1): The paper proves that for any affine Dedekind scheme $S$ and any exact functor $E$ from $kgl$-motives to a stable $\infty$ -category with a $t$ -structure, there exists a unique enhancement to a filtered object $W_*E$ on resolvable motives. This filtration is defined by the condition $(W_*E)(M) \simeq \tau_{\geq -*} (E(M))$ for perfect pure motives $M$ . Crucially, this holds without inverting residual characteristics.
Invariance of the Open Part (Theorem 1.2): The weight-filtered cohomology of a projective sncd pair $(X, D)$ is shown to be an invariant of the open complement $U = X - D$ . This follows from the fact that the logarithmic motive $M^{kgl}(X, D)$ is equivalent to the $A^1$ -invariant $kgl$-motive of $U$ .
Comparison with Pole-Order Filtration (Theorem 3.24): The authors prove that for logarithmic de Rham cohomology, the motivic weight filtration coincides with the décalaged pole-order filtration (the filtration obtained by applying the décalage operation to the filtration by pole order). This result generalizes Deligne's theorem from characteristic 0 to positive and mixed characteristic.
Applications to Singular Schemes (Section 4.2): The framework is extended to singular schemes admitting a "strong compactifiable resolution of singularities." By utilizing $cdh$-descent and the $cdh$-localization of motives, the authors define weight-filtered cohomology theories for singular schemes. These theories recover logarithmic cohomology for smooth open parts and provide new invariants (reminiscent of the du Bois complex) for singularities in positive and mixed characteristic.
Explicit Computations (Section 4.1): The paper computes explicit weight-graded pieces for various examples. Notably, it reproves a result of Thuillier: the singular cohomology of the dual complex associated with the boundary divisor of a good projective compactification is an invariant of the open variety $U$ .

Significance and Claims
The paper claims to provide a unified, motivic mechanism for generating weight filtrations for a broad class of cohomology theories (including motivic, syntomic, de Rham, crystalline, prismatic, and topological cyclic homology) in settings where previous methods (relying on $\ell$ -adic realization or characteristic 0 assumptions) fail.

The significance lies in:

Generalization: Extending the concept of "mixed Hodge theory" to positive and mixed characteristic schemes, suggesting that the spectral sequence associated with the pole-order filtration is an invariant of the open part $U$ from the $E_2$ -page onwards.
Universality: The construction applies to "essentially all known logarithmic cohomology theories" when evaluated on projective sncd pairs, provided they are $kgl$-linear.
New Invariants: The weight-filtered cohomology provides new invariants for singular schemes in positive characteristic, potentially offering tools to study singularities analogous to the Hodge-du Bois numbers used in characteristic zero.

The authors explicitly note limitations: the construction applies only to $kgl$-linear theories (excluding algebraic cobordism $MGL$) and the resulting weight filtration is natural only with respect to $kgl$-linear maps. Furthermore, the comparison with the pole-order filtration is proven to be functorial only for maps that do not change the ambient variety, though a more functorial comparison is conjectured to be possible via a geometric presentation of resolvable motives.