🔢 mathematics

Torsion graded pieces of Nyggard filtration for crystalline representation

This paper investigates the torsion properties of the graded pieces of the Nyggard filtration on the crystalline representation's associated module, demonstrating that nontrivial $p$ -torsion occurs exclusively at indices determined by the Hodge-Tate weights and positive integer multiples of $p$ .

Original authors: Tong Liu

Published 2026-03-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tong Liu

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

The Big Picture: Sorting a Messy Library

Imagine you are a librarian trying to organize a very special, magical library. This library doesn't contain books about history or fiction; it contains mathematical structures that describe how numbers behave in a specific, strange universe called a " $p$ -adic field" (think of it as a world where numbers wrap around in a circle based on a prime number $p$ ).

The author, Tong Liu, is studying a specific type of "book" in this library called a Crystalline Representation. These are complex objects that hold deep secrets about the shape of numbers.

To understand these objects, mathematicians use a tool called a Filtration.

The Analogy: Imagine a set of nested Russian dolls. The biggest doll is the whole object. Inside it is a slightly smaller doll, then a smaller one, and so on.
The Goal: Liu wants to look at the "layers" between these dolls. He wants to know: "If I take the big doll and remove the smaller one, what is left?"

The Problem: The "Ghost" Torsion

In a perfect world, when you peel back these layers, you get a clean, solid piece of math. But in this specific mathematical universe, sometimes the layers contain "torsion."

The Analogy: Imagine you are peeling an orange. Usually, you get a nice segment of fruit. But sometimes, you find a weird, invisible "ghost" segment that is made of pure static electricity. It's there, but it's not solid fruit. In math, this "ghost" is called $p$ -torsion. It's a glitch that happens when you divide by the number $p$ .

Liu's paper asks a very specific question: When do these "ghosts" (torsion) appear?

The Main Discovery: The "Magic Formula"

Liu proves that these ghosts don't just appear randomly. They only show up at very specific "floors" of the Russian doll tower.

He discovers a strict rule (Theorem 1.1):

A ghost (torsion) can only exist at floor number $i$ if $i$ follows this pattern:
$i = \text{a special number} + (\text{a multiple of } p)$

The "Special Numbers": These are pre-determined by the shape of the object itself (called Hodge-Tate weights). Let's call them the "Seed Numbers."
The "Multiple of $p$ ": This is the magical wrapping of the universe.

In plain English: If you are looking at floor 5, and your seed numbers are 0 and 2, and $p=3$ , you will only find ghosts on floors like $0+3=3$ , $0+6=6$ , $2+3=5$ , etc. If you look at floor 4, there will be no ghosts. It's perfectly clean.

The "Adapted Basis": Finding the Perfect Map

The paper also talks about something called an "Adapted Basis."

The Analogy: Imagine you are trying to navigate a maze. You have a map (a basis).
- A bad map might say, "Go 5 steps, then turn left," but the walls keep shifting, so you get stuck.
- An adapted basis is a perfect, magical map that aligns perfectly with the shifting walls. It tells you exactly how to move so you never get stuck in a "ghost" zone.

Liu shows that if the "ghosts" are absent in the lower floors (specifically, if the height of the tower is less than or equal to $p$ ), then you can always find this perfect map. This is a huge deal because it means the math becomes predictable and easy to work with.

Why Does This Matter?

You might ask, "Who cares about ghosts in a Russian doll tower?"

Predictability: Before this paper, mathematicians knew ghosts could exist, but they didn't know exactly where. Now they have a precise rule. It's like knowing exactly which days of the month it will rain, so you can plan your picnic.
Deformation Rings: The author mentions this will help in future work on "crystalline deformation rings."
- The Analogy: Imagine you have a clay sculpture. You want to know how much you can squish or stretch it before it breaks. This paper gives you the blueprint for the clay's internal structure, helping mathematicians understand the limits of how these number-shapes can change.
Connecting Two Worlds: The paper connects two different ways of looking at the same problem (one using "stacks" and another using "operators"). It's like realizing that two different languages are actually describing the same story, just with different words.

The "Secret Weapon": The Griffith Transversality

How did Liu solve this? He used a technique called Griffith Transversality.

The Analogy: Imagine you are walking down a staircase. Usually, you step down one step at a time. But in this math world, there's a rule that says, "If you are on step $i$ , you can only look at step $i-1$ or lower."
Liu used this rule to calculate exactly how the "layers" of the Russian dolls interact. By carefully tracking how the "ghosts" move down the stairs, he proved they can't appear unless the stairs are built according to his specific formula.

Summary

Tong Liu's paper is a detective story in the world of abstract numbers.

The Mystery: Where do the invisible "ghosts" (torsion) hide in our mathematical structures?
The Clue: They only hide on floors that are a "seed number" plus a multiple of $p$ .
The Solution: If the building isn't too tall (height $\le p$ ), the ghosts don't exist at all, and we can draw a perfect map (Adapted Basis) of the whole structure.

This work provides a clearer, more precise map for mathematicians navigating the complex, twisting landscape of $p$ -adic numbers.

1. Problem Statement and Context

The paper investigates the integral structure of crystalline $p$ -adic Galois representations. Specifically, it addresses the behavior of the Nygaard filtration on the Kisin module associated with a crystalline representation $T$ of the absolute Galois group $G_K$ of an unramified $p$ -adic field $K$ .

The Object of Study: Let $T$ be a crystalline finite free $\mathbb{Z}_p$ -representation of $G_K$ with Hodge-Tate weights $r_1, \dots, r_d$ . Associated with $T$ is a Breuil-Kisin module $M$ and its extension $M^* = S \otimes_{\phi, S} M$ , where $S = W(\kappa)[[u]]$ .
The Filtration: The Nygaard filtration $\text{Fil}^i M^*$ is defined by the condition $(1 \otimes \phi_M)(x) \in E^i M$ , where $E(u) = u - p$ .
The Graded Pieces: The paper focuses on the graded pieces of the induced filtration on the reduction $M = M^*/EM^*$ , denoted as $\text{gr}^i M = \text{ev}_p(\text{Fil}^i M^*) / \text{ev}_p(\text{Fil}^{i+1} M^*)$ .
The Core Question: While it is known that $\text{gr}^i M[1/p]$ is non-zero only when $i$ corresponds to a Hodge-Tate weight, the integral graded pieces $\text{gr}^i M$ can contain $p$ -torsion for other values of $i$ . The paper seeks to determine exactly when this torsion occurs.

2. Methodology

The author employs a constructive approach based on the properties of Breuil modules and the Griffith transversality of the connection $\nabla$ , avoiding the use of the Sen operator (which was used in related recent works by Gee-Kisin and Dat Pham).

Integral Griffith Transversality (Lemma 2.3): A crucial technical step is establishing an integral version of Griffith transversality. The author proves that $\nabla(\text{Fil}^i M^*) \subset S_p \otimes_S \text{Fil}^{i-1} M^*$ , where $S_p$ is a specific subring of $S[1/p]$ . This is derived using the action of the Galois group on the period rings ( $A_{\inf}$ and $A_{\text{cris}}$ ) and the formula for the Galois action involving the connection $\nabla$ .
Inductive Basis Construction (Section 2.3): The core of the proof involves an inductive construction of an $S$ $S$ -basis for $\text{Fil}^i M^*$ $Fil^{i} M^{*}$ .
- The author defines a set $J = \{r_j + mp \mid j=1,\dots,d, m \ge 0\}$ .
- The proof distinguishes between two cases for the index $i$ $i$ :
  1. Case $i \notin J$ : The author constructs elements $\beta_j$ such that $\text{ev}_p(\beta_j)$ remains in the filtration, but the "error terms" can be eliminated using the condition $p \nmid (i-l)$ . This leads to the conclusion that $\text{Fil}^{i+1} M^*$ is generated by $E \cdot \text{Fil}^i M^*$ , implying $\text{gr}^i M = 0$ .
  2. Case $i \in J$ : The author analyzes the structure of the torsion in $\text{gr}^i M$ . By decomposing the basis into torsion-free and torsion parts, and utilizing the invertibility of matrices over $\mathcal{O}_K$ , the author constructs a basis that respects the filtration levels.
Comparison with Breuil Modules: The proof relies heavily on the canonical isomorphism between the Kisin module tensored with $S[1/p]$ and the filtered $\phi$ -module $D_{\text{cris}}$ , allowing the translation of properties between the integral Kisin module and the rational filtered module.

3. Key Contributions and Results

Main Theorem (Theorem 1.1)

The primary result characterizes the indices $i$ for which the graded piece $\text{gr}^i M$ has non-trivial $p$ -torsion:
$\{i \mid (\text{gr}^i M)_{\text{tor}} \neq 0\} \subset \{r_j + mp \mid j = 1, \dots, d, m > 0, r_j + mp \le h\}$
where $h$ is the highest Hodge-Tate weight.

Implication: Torsion can only appear at indices that are congruent to a Hodge-Tate weight modulo $p$ . If $i$ is not of the form $r_j + mp$ , then $\text{gr}^i M$ is torsion-free (and in fact, zero if $i$ is not a Hodge-Tate weight at all).

Corollary on Adapted Bases (Corollary 1.3)

The paper recovers and provides a new proof for a result by Gee and Liu-Savitt (GLS14):

If the highest Hodge-Tate weight $h \le p$ , then the Kisin module $M^*$ admits an adapted basis.
An adapted basis is an $S$ -basis $\{e_1, \dots, e_d\}$ such that the filtration $\text{Fil}^i M^*$ is generated by elements of the form $E^{a_{ij}} e_j$ .
This confirms that for "small" weights, the integral structure is particularly well-behaved.

Connection to Gee-Kisin Theorem (Theorem 1.4)

The author notes that the main theorem and the Gee-Kisin theorem (regarding the diagonal entries of the Frobenius matrix modulo $p$ ) are both consequences of the Sen operator on $M_{\text{HT}}$ . However, Liu's approach provides a more explicit description of the generators of $\text{Fil}^h M^*$ via the connection $\nabla$ .

4. Significance

Explicit Integral Structure: The paper provides a concrete, constructive description of the Nygaard filtration on Kisin modules. This is vital for understanding the integral $p$ -adic Hodge theory, where rational isomorphisms often fail to capture torsion phenomena.
Deformation Theory: The explicit generators of the filtration are expected to be crucial for future work on crystalline deformation rings. Understanding the torsion in the graded pieces helps in analyzing the geometry of these deformation spaces.
Alternative Proof Strategy: By avoiding the stack-theoretic methods and the Sen operator (used by Gee, Kisin, and Pham), this paper offers a more classical, module-theoretic perspective. This makes the results potentially more accessible to researchers working directly with Kisin modules and Breuil modules without requiring the machinery of perfectoid stacks.
Resolution of Torsion Conditions: It precisely delineates the "danger zones" for torsion in the filtration, showing that torsion is strictly controlled by the arithmetic of the Hodge-Tate weights and the prime $p$ .

In summary, Tong Liu's paper refines the understanding of the integral Nygaard filtration, proving that $p$ -torsion in the graded pieces is strictly constrained to indices related to Hodge-Tate weights via the relation $i \equiv r_j \pmod p$ . This result strengthens the bridge between the integral Kisin modules and the filtered $\phi$ -modules, with significant implications for the study of crystalline deformations.