[Paper Review] Dualité de Cartier et modules de Breuil

Xavier Caruso|arXiv (Cornell University)|Nov 16, 2005

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper explicitly describes Cartier duality on the category of Breuil modules $(\text{Mod}/S)$ via the anti-equivalence established by Breuil between finite flat $\mathcal{O}_K$-group schemes killed by a power of $p$ and these linear algebra objects. It constructs a dualizing functor on $(\text{Mod}/S)$ that corresponds precisely to Cartier duality under Breuil's equivalence, proving a canonical isomorphism $\text{Mod}(\mathcal{G})^\vee \simeq \text{Mod}(\mathcal{G}^\vee)$ for $p$-divisible groups $\mathcal{G}$, compatible with filtration and Frobenius structure.

ABSTRACT

Let O\_K be a complete discrete valuation ring. Denote by K its fractions field and by k its residue field. Assume that k is of characteristic p>0 and perfect. Breuil gives an anti-equivalence between the category of finite flat O\_K-group schemes killed by a power of p and a category of linear algebra objects which is called (Mod/S). The aim of this article is to make explicit the Cartier duality on the category (Mod/S).

Motivation & Objective

• To explicitly describe the Cartier duality functor on the category of Breuil modules $(\text{Mod}/S)$ using only linear algebra.
• To establish that this dualizing functor corresponds precisely to the classical Cartier duality on finite flat $\mathcal{O}_K$-group schemes under Breuil's anti-equivalence.
• To prove that the duality on $(\text{Mod}/S)$ preserves the key structures: filtration $\text{Fil}^1$ and Frobenius map $\phi_1$, ensuring compatibility with the linear algebra framework.
• To extend the duality to the category of strongly divisible modules, showing the isomorphism $\text{Mod}(\mathcal{G})^\vee \simeq \text{Mod}(\mathcal{G}^\vee)$ for $p$-divisible groups $\mathcal{G}$.

Proposed method

• Define the category $(\text{Mod}/S)$ as $S$-modules $\mathcal{M}$ with a filtration $\text{Fil}^1\mathcal{M}$ and a $\phi_1$-linear map $\phi_1: \text{Fil}^1\mathcal{M} \to \mathcal{M}$ satisfying a specific compatibility condition with the Frobenius action.
• Construct a dual object $\mathcal{M}^\vee$ in $(\text{Mod}/S)$ via the dual $S$-module $\mathcal{M}^\vee$, with $\text{Fil}^1\mathcal{M}^\vee$ defined as the annihilator of $\text{Fil}^1\mathcal{M}$, and define $\phi_1^\vee$ via duality.
• Prove that the dual functor $\mathcal{M} \mapsto \mathcal{M}^\vee$ is well-defined on $(\text{Mod}/S)$, preserving exact sequences and the required compatibility conditions.
• Use the anti-equivalence of categories from Breuil's work to lift the classical Cartier duality on group schemes to a duality on $(\text{Mod}/S)$, showing that the dual of $\text{Mod}(\mathcal{G})$ is isomorphic to $\text{Mod}(\mathcal{G}^\vee)$.
• Verify compatibility of the duality with the filtration $\text{Fil}^1$ and the Frobenius map $\phi_1$ by constructing commutative diagrams and using injectivity and surjectivity arguments.
• Apply a dévissage argument to reduce the general case to the case of $p$-torsion groups, where surjectivity of the dual map is automatic.

Experimental results

Research questions

• RQ1 How can Cartier duality be explicitly realized in the category of Breuil modules $(\text{Mod}/S)$ using only linear algebra?
• RQ2 Does the dual of a Breuil module $\text{Mod}(\mathcal{G})$ under the anti-equivalence correspond to $\text{Mod}(\mathcal{G}^\vee)$ for a $p$-divisible group $\mathcal{G}$?
• RQ3 Is the duality functor on $(\text{Mod}/S)$ compatible with the filtration $\text{Fil}^1$ and the Frobenius map $\phi_1$?
• RQ4 Can the duality on $(\text{Mod}/S)$ be extended to the category of strongly divisible modules, and is it compatible with the Galois representation functors?

Key findings

• The paper constructs a canonical dualizing functor on the category $(\text{Mod}/S)$ that corresponds to Cartier duality under Breuil's anti-equivalence of categories.
• The dual of a Breuil module $\mathcal{M}$ is explicitly given by $\mathcal{M}^\vee = \text{Hom}_S(\mathcal{M}, S)$, with $\text{Fil}^1\mathcal{M}^\vee$ defined as the annihilator of $\text{Fil}^1\mathcal{M}$, and $\phi_1^\vee$ defined via duality.
• The duality functor preserves the structure of $(\text{Mod}/S)$: it respects exact sequences, the filtration, and the Frobenius map $\phi_1$, ensuring compatibility with the linear algebra framework.
• For any $p$-divisible group $\mathcal{G}$ over $\mathcal{O}_K$, there is a canonical isomorphism $\text{Mod}(\mathcal{G})^\vee \simeq \text{Mod}(\mathcal{G}^\vee)$ in the category of strongly divisible modules.
• The compatibility of the duality with $\text{Fil}^1$ and $\phi_1$ is proven via diagram chasing in a commutative cube, using injectivity and surjectivity of relevant maps.
• The result is extended to the general case of $p$-divisible groups by dévissage, reducing to the $p$-torsion case where surjectivity of the dual map is automatic.