[Paper Review] On Beilinson's equivalence for $p$-adic cohomology

Tomoyuki Abe, Daniel Caro|arXiv (Cornell University)|Sep 18, 2013

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes a $p$-adic analogue of Beilinson's equivalence between the derived category of holonomic $\mathcal{D}$-modules and the derived category of complexes with holonomic cohomology, by constructing a unipotent nearby cycle functor for overholonomic $\mathcal{D}^\dagger$-modules with Frobenius structure using Kedlaya's semistable reduction theorem. The key contribution is a triangulated category of overholonomic complexes closed under the six operations, enabling the $p$-adic version of Beilinson's equivalence.

ABSTRACT

In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.

Motivation & Objective

• To overcome the lack of a nearby cycle functor in $p$-adic cohomology, a major technical obstacle in the theory.
• To define a suitable category of holonomic $\mathcal{D}^\dagger$-modules in the $p$-adic setting that is closed under the Grothendieck six operations.
• To establish a $p$-adic analogue of Beilinson's equivalence between $D^b(\mathrm{Hol}(X))$ and $D^b_{\mathrm{hol}}(X)$ for smooth varieties over $\mathbb{C}$.
• To provide foundational tools for the theory of arithmetic $\mathcal{D}$-modules and applications to rigid cohomology and trace formulas.

Proposed method

• Construct a unipotent nearby cycle functor via devissage to modules with Frobenius structure, inspired by Beilinson's original approach.
• Use Kedlaya's semistable reduction theorem for overconvergent isocrystals with Frobenius structure to bypass the problematic definition of $b$-functions in the $p$-adic context.
• Define a smallest triangulated subcategory of overholonomic complexes containing Frobenius-structured modules, ensuring closure under the six operations.
• Employ the pro-ind category to resolve issues in the diagrammatic proof of the Key Lemma, replacing an incorrect isomorphism with a homomorphism that suffices for the argument.
• Apply the functor $\Phi_f$ defined via a short exact sequence involving $j_!$, $j_+$, and $\Xi_f$ to construct the nearby cycle functor.
• Use the functors $\Pi^{a,b}_{!+}$ to interpolate between $j_!$ and $j_+$, and show their stability under shift and Frobenius twist.

Experimental results

Research questions

• RQ1 Can a unipotent nearby cycle functor be constructed in the $p$-adic context where $b$-functions are ill-defined?
• RQ2 What is the correct definition of holonomic $\mathcal{D}^\dagger$-modules in $p$-adic cohomology that is closed under the six operations?
• RQ3 Does Beilinson's equivalence between $D^b(\mathrm{Hol}(X))$ and $D^b_{\mathrm{hol}}(X)$ admit a $p$-adic analogue?
• RQ4 Can the theory of arithmetic $\mathcal{D}$-modules be unified with rigid cohomology using these constructions?

Key findings

• A unipotent nearby cycle functor is constructed for overholonomic $\mathcal{D}^\dagger$-modules with Frobenius structure using Kedlaya's theorem as a substitute for $b$-functions.
• The category $\mathrm{Hol}_F(X/K)$ of holonomic $\mathcal{D}^\dagger$-modules with Frobenius structure is shown to be closed under the Grothendieck six operations via a triangulated subcategory construction.
• The functor $\Phi_f$ is exact and satisfies $\mathcal{E} \cong \Phi_f(\mathcal{E})$ for $\mathcal{E} \in \mathrm{Hol}_F(Z/K)$, confirming its role as a nearby cycle functor.
• The key diagram in the proof of the Key Lemma is corrected by replacing an isomorphism with a homomorphism $\#'$ in the pro-ind category, which suffices for the argument.
• The projective system $\mathrm{coker}(s^i \alpha^{a,b}(i))$ stabilizes for large $b$, showing that $\Pi^{a,a+i}_{!+}$ is well-defined and isomorphic to the limit.
• The construction provides foundational tools for the theory of arithmetic $\mathcal{D}$-modules and applications such as a $p$-adic analogue of Fujiwara's trace formula.