🔢 mathematics

Brauer group of varieties over local fields of finite characteristic

This paper establishes that the nonlogarithmic Kato ramification filtration on the Brauer group of a regular scheme over a henselian discrete valuation field of positive characteristic coincides with the evaluation filtration, thereby extending recent results by Bright and Newton and generalizing several findings by Ieronymou, Saito, Sato, and Kai to the positive characteristic setting.

Original authors: Amalendu Krishna, Subhadip Majumder

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Amalendu Krishna, Subhadip Majumder

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the landscape of modern mathematics, there is a deep and enduring effort to understand the hidden structures of geometric shapes defined by equations. These shapes, known as varieties, exist over different types of number systems. One particularly rich setting is the "local field," a system that behaves like a complete number line but is built from a specific kind of prime number arithmetic. Within these fields, mathematicians study two fundamental invariants: the group of zero-cycles, which tracks how points on a shape can be combined and moved, and the Brauer group, a collection of algebraic objects that encode subtle obstructions to solving equations. The interaction between these two groups is governed by a pairing, a mathematical handshake that reveals whether a shape has points everywhere locally but fails to have a global solution. For decades, this relationship was well understood when the underlying number system had a characteristic of zero, much like the familiar real numbers. However, when the number system has a positive characteristic, meaning it is built from a finite prime number, the rules change, and many of the known tools break down, leaving a significant gap in our understanding.

The researchers Amalendu Krishna and Subhadip Majumder have stepped into this gap to extend the known laws of arithmetic geometry from the zero characteristic world into the positive characteristic realm. Their work focuses on a specific type of geometric object: a smooth, projective variety defined over a local field of positive characteristic. They sought to prove that the deep connections between the Brauer group and the geometry of these shapes, which were previously only known to hold in characteristic zero, actually hold true here as well. To do this, they had to navigate a landscape where standard techniques fail because the geometry behaves differently when the underlying numbers are finite. They constructed a new bridge between two different ways of measuring how "wild" or "ramified" a Brauer class can be. One measurement, known as the Kato filtration, looks at the algebraic complexity of the object, while the other, the evaluation filtration, looks at how the object behaves when tested against specific points on the shape.

The central achievement of their work is a proof that these two distinct measurements are, in fact, the same thing. They demonstrated that for these geometric shapes, the set of Brauer classes that remain constant on small neighborhoods of points is exactly the same as the set of classes with a specific level of algebraic ramification. This equivalence is not merely a technical coincidence; it is a powerful tool that unlocks the ability to apply results from the well-trodden path of characteristic zero to this more difficult territory. By establishing this identity, the authors were able to confirm several long-standing predictions about the behavior of these shapes. They proved that for certain types of varieties, such as those that are "rationally connected" or specific types of surfaces known as Enriques surfaces, the Brauer group does not create any obstruction to finding points; the evaluation map is constant, meaning the algebraic obstructions vanish.

Furthermore, the paper resolves a major question regarding the pairing between zero-cycles and the Brauer group. In the zero characteristic world, it was known that this pairing is perfect, meaning every non-trivial algebraic obstruction corresponds to a unique geometric cycle, and vice versa. The authors proved that this perfect pairing holds true in positive characteristic as well, provided the variety has a specific type of reduction. This result confirms a prediction made by other mathematicians and settles a problem that had remained open. Additionally, they showed that the "cokernel" of the Albanese map—a measure of how far the map from zero-cycles to the shape's associated abelian variety is from being surjective—is a finite group. This finiteness was a known fact in characteristic zero but was previously unproven in the positive characteristic setting.

The path to these results was not a simple extension of old methods. The researchers had to develop new machinery to handle the unique difficulties of positive characteristic, where the geometry of the shapes can be more singular and the behavior of points more erratic. They utilized a sophisticated tool called the Kato complex, which organizes cohomological data in a way that allows for precise tracking of ramification. By combining this with a refined understanding of how the Swan conductor—a measure of wild ramification—behaves when restricted to curves within the larger shape, they were able to reduce high-dimensional problems to manageable one-dimensional cases. They also relied on a version of Bertini's theorem, a classical result that guarantees the existence of smooth cross-sections, adapted to work in this specific, singular context.

The implications of this work are profound for the field of arithmetic geometry. By proving that the evaluation filtration and the Kato filtration coincide, the authors have provided a unified framework that allows mathematicians to translate problems about the arithmetic of points into problems about the algebraic structure of the Brauer group, and back again. This unification confirms that the fundamental principles governing the arithmetic of varieties are robust, persisting even when the underlying number system shifts from the infinite to the finite. The results extend theorems by Ieronymou, Saito, Sato, and Kai, bringing their insights into the positive characteristic world. Ultimately, the paper demonstrates that the deep duality between geometry and arithmetic, which was once thought to be fragile in positive characteristic, is in fact as strong and reliable as it is in the characteristic zero world, opening the door for further exploration of zero-cycles and Brauer groups in these complex settings.

Technical Summary: Brauer Group of Varieties over Local Fields of Finite Characteristic

Problem Statement
The paper addresses fundamental questions regarding the arithmetic of smooth projective varieties over local fields of positive characteristic ( $p > 0$ ). Specifically, it seeks to establish results concerning the Brauer group and the Chow group of zero-cycles that are known in characteristic zero but remain open or unproven in positive characteristic. The central challenge lies in extending the known coincidence between the evaluation filtration (defined by Bright–Newton for smooth schemes) and Kato's ramification filtration to the positive characteristic setting, where smoothness does not automatically follow from regularity. This coincidence is crucial for understanding the Brauer–Manin obstruction to the Hasse principle and weak approximation. Furthermore, the behavior of the Albanese map's cokernel and the structure of the Brauer–Manin pairing in higher dimensions over positive characteristic fields were previously unresolved.

Methodology
The authors employ a combination of ramification theory, étale cohomology, and the geometry of schemes over discrete valuation rings (DVRs). The core technical machinery involves:

Kato's Filtration and Swan Conductors: The paper utilizes Kato's ramification filtration on the cohomology groups $H^q_{\text{ét}}(X, \mathbb{Q}/\mathbb{Z}(q-1))$ . A critical component is the use of the generalized refined Swan conductor, a tool developed in the authors' prior work [47], which provides a cohomological description of the graded pieces of Kato's filtration.
Non-Logarithmic Filtrations: The authors introduce a "non-logarithmic" version of Kato's filtration (reminiscent of Matsuda's filtration for $H^1$ ) to handle the specific difficulties arising in positive characteristic where smoothness does not automatically follow from regularity.
Specialization of Swan Conductors: A significant portion of the methodology involves proving specialization theorems for the Swan conductor. The authors demonstrate how the Swan conductor behaves when restricting cohomology classes to subschemes (specifically, admissible hypersurface sections) using Bertini-type theorems for semistable schemes. This allows for an inductive reduction from higher-dimensional varieties to curves.
Kato Complex: The Kato complex is used to relate local ramification data to global cohomological properties, particularly in establishing the non-vanishing of certain cup products and boundary maps.
Milne–Tate Duality: The proof of the main results regarding the Albanese cokernel relies on establishing a compatibility between the Brauer–Manin pairing and the Tate–Milne pairing for abelian varieties over local fields of positive characteristic.

Key Contributions and Results

Equivalence of Filtrations (Theorem 1.1): The primary result establishes that for a separated regular scheme $X$ of finite type over a henselian discrete valuation field of positive characteristic with a finite residue field, the evaluation filtration (defined by the constancy of Brauer class evaluations on adic neighborhoods of closed points) coincides with the non-logarithmic Kato filtration. This extends a recent theorem of Bright–Newton, originally proven for characteristic zero, to the positive characteristic setting. Notably, this result holds even when the special fiber is not reduced or irreducible, provided the scheme is regular and the special fiber is a simple normal crossing divisor.
Extensions of Ieronymou's Theorems (Theorems 1.3 & 1.4): Using the filtration equivalence, the authors prove that for smooth projective schemes over such local fields:
- If the special fiber is separably rationally connected, every Brauer class induces a constant evaluation map on the $k$ -points.
- If the generic fiber is an Enriques surface, Brauer classes of odd order induce constant evaluation maps.
  These results generalize previous findings by Ieronymou from characteristic zero to positive characteristic without requiring the field to be complete or the residue characteristic to be odd.
Saito–Sato Theorem Extension (Theorem 1.6): The paper verifies Colliot-Thélène's prediction in positive characteristic regarding the Brauer–Manin pairing. For a smooth projective variety $X$ over a local field admitting a semistable model, provided the generic fiber is smooth over $k$ and the field is complete, the kernel of the map $\theta_X: \text{Br}(X) \to \text{Hom}(\text{CH}_0(X), \mathbb{Q}/\mathbb{Z})$ is exactly the image of $\text{Br}(k)$ , and the dual map $\theta^*_X$ is surjective. This resolves a major open problem that was previously known only in characteristic zero.
Finiteness of the Albanese Cokernel (Theorem 1.10): The authors prove that for a smooth projective geometrically integral variety $X$ over a local field of positive characteristic, assuming the Picard scheme $\text{Pic}(X)$ is smooth over $S$ , the cokernel of the Albanese map $A_0(X) \to \text{Alb}_X(k)$ is a finite $p$ -group. Furthermore, this cokernel is canonically Pontryagin dual to the cokernel of the map $\text{Pic}(X_s)^\Gamma \to \text{NS}(X_s)^\Gamma$ . This provides a higher-dimensional extension of Lichtenbaum's duality theorem in positive characteristic.
Structure of the Relative Brauer Group (Theorem 1.8): The paper establishes an inclusion of the relative Brauer group $\text{Br}(X)/(\text{Br}(X) + \text{Br}_0(X))$ into a direct sum of $\mathbb{Q}_p/\mathbb{Z}_p$ and a finite group, implying the finiteness of the prime-to- $p$ part of this group.

Significance
The paper claims significance by bridging a critical gap between arithmetic geometry in characteristic zero and positive characteristic. By extending the Bright–Newton theorem, the authors provide a robust framework to analyze the Brauer–Manin obstruction in positive characteristic, a setting where previous tools (such as Mattuck's theorem) often fail. The results resolve long-standing questions about the surjectivity of the Brauer–Manin pairing and the finiteness of the Albanese cokernel in higher dimensions over local fields of positive characteristic. The methodology, particularly the use of the generalized refined Swan conductor and the specialization of Swan conductors via Bertini theorems, offers a new toolkit for studying ramification in positive characteristic that does not rely on the smoothness of the special fiber. The work confirms that many structural results known in characteristic zero hold true in positive characteristic, provided one works with the appropriate non-logarithmic filtrations and accounts for the specific behavior of $p$ -torsion.

More like this