[Paper Review] Lattices in crystalline representations and Kisin modules associated with iterate extensions

Yoshiyasu Ozeki|arXiv (Cornell University)|Sep 15, 2016

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper establishes a theory of crystalline $(\varphi,\hat{G})$-modules in the generalized setting of Cais-Liu's $f$-iterate extensions, extending Kisin's framework to classify $G$-stable lattices in crystalline representations via Kisin modules with additional structures. The key result is a full faithfulness theorem for the restriction functor on torsion crystalline representations under the bound $e(r-1) < n_f(p-1)/p$, generalizing earlier results for $f(u) = u^p$ and confirming a conjecture in [CL].

ABSTRACT

Cais and Liu extended the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. Based on their theory, we classify lattices in crystalline representations by Kisin modules with additional structures under a Cais-Liu's setting. Furthermore, we give a geometric interpretation of Kisin modules of height one in terms of Dieudonné crystals of $p$-divisible groups, and show a full faithfulness theorem for a restriction functor on torsion crystalline representations.

Motivation & Objective

• To extend the theory of $(\varphi,\hat{G})$-modules to crystalline representations in the generalized setting of $f$-iterate extensions as defined by Cais and Liu.
• To classify $G$-stable lattices in crystalline $\mathbb{Q}_p$-representations using Kisin modules with additional structures under the Cais-Liu framework.
• To provide a geometric interpretation of height-one Kisin modules in terms of Dieudonné crystals of $p$-divisible groups.
• To establish a full faithfulness theorem for the restriction functor on torsion crystalline representations, generalizing previous results for $f(u) = u^p$.
• To resolve a conjecture in [CL, Remark 5.2.3 and Section 6.3] regarding the faithfulness of the restriction functor in the case $F = \mathbb{Q}_p$.

Proposed method

• Adapts Liu's theory of $(\varphi,\hat{G})$-modules to the Cais-Liu setting of $f$-iterate extensions $K_{\underline{\pi}}/K$, where $f(u) \in \mathbb{Z}_p[u]$ satisfies $f(u) \equiv u^p \mod p$.
• Introduces and studies $(\varphi,\hat{G})$-modules of height $r$ with an additional condition, establishing an anti-equivalence between such modules and $G$-stable lattices in crystalline representations with Hodge-Tate weights in $[0,r]$.
• Uses the theory of maximal objects in the category $\mathrm{Max}^r_{\mathfrak{S}_\infty}$ to analyze the structure of Kisin modules and their associated Galois representations.
• Applies the theory of Dieudonné crystals to interpret Kisin modules of height one geometrically via $p$-divisible groups.
• Employs the restriction functor $\mathrm{Rep}^{r,\mathrm{cris}}_{\mathrm{tor}}(G) \to \mathrm{Rep}_{\mathrm{tor}}(G_{\underline{\pi}})$ and proves full faithfulness under the condition $e(r-1) < n_f(p-1)/p$, leveraging properties of valuation and module maximality.
• Utilizes the isomorphism $T|_{G_{\underline{\pi}}} \simeq \hat{T}(\hat{\mathfrak{M}}(\mathfrak{n}))|_{G_{\underline{\pi}}}$ and commutative diagram arguments involving Hom sets and forgetful functors to establish bijectivity of Hom spaces.

Experimental results

Research questions

• RQ1 Can the theory of $(\varphi,\hat{G})$-modules be extended to crystalline representations in the generalized $f$-iterate extension setting of Cais and Liu?
• RQ2 What is the relationship between Kisin modules associated with different choices of $f(u)$ and compatible systems $\{\pi_n\}$?
• RQ3 Under what conditions is the restriction functor $\mathrm{Rep}^{r,\mathrm{cris}}_{\mathrm{tor}}(G) \to \mathrm{Rep}_{\mathrm{tor}}(G_{\underline{\pi}})$ fully faithful?
• RQ4 How can Kisin modules of height one be interpreted geometrically in terms of Dieudonné crystals?
• RQ5 Does the full faithfulness result hold under weaker assumptions, particularly when $er < p-1$?

Key findings

• An anti-equivalence is established between the category of $(\varphi,\hat{G})$-modules of height $r$ (with an additional condition) and the category of $G$-stable lattices in crystalline $\mathbb{Q}_p$-representations with Hodge-Tate weights in $[0,r]$.
• The restriction functor $\mathrm{Rep}^{r,\mathrm{cris}}_{\mathrm{tor}}(G) \to \mathrm{Rep}_{\mathrm{tor}}(G_{\underline{\pi}})$ is fully faithful under the condition $e(r-1) < n_f(p-1)/p$, generalizing previous results for $f(u) = u^p$.
• For $f(u) = u^p$, the result recovers Theorem 1.2 of [Oz2], and the paper confirms a conjecture in [CL, Remark 5.2.3 and Section 6.3] in the case $F = \mathbb{Q}_p$.
• Kisin modules of height one are shown to correspond geometrically to Dieudonné crystals of $p$-divisible groups, providing a geometric interpretation.
• Under the additional assumption $er < p-1$, the full faithfulness condition can be improved: the restriction functor is fully faithful if $e(r-1) < n_f(p-1)/p$ and $er < p-1$, with simpler proofs due to the maximality of all torsion Kisin modules of height $r$.
• The proof of full faithfulness relies on a commutative diagram involving Hom sets, where bijectivity of the bottom and right arrows (via Theorem 4.13 and 4.8) implies bijectivity of the top arrow, thus proving full faithfulness.