Periodic Reporting for period 1 - ECrys (Edged Crystalline Cohomology)
Reporting period: 2023-09-01 to 2025-08-31
The realm of p-adic cohomology theories in algebraic geometry has long been shaped by two foundational frameworks: crystalline cohomology and rigid cohomology. While crystalline cohomology excels in finiteness properties for smooth and proper varieties, it falters for singular or non-proper schemes. Conversely, rigid cohomology, supports powerful tools like Poincaré duality and weight structures, but struggles with coherence and finiteness results—most notably, Berthelot’s conjecture on the coherence of relative rigid cohomology remains unresolved. Beyond these, the study of log-decay F-isocrystals has highlighted the need for cohomological frameworks capable of accommodating logarithmic decay behaviors, as conjectured by Wan and others.
This project introduces tau-edged crystalline cohomology, a novel unification of these theories through a family of ringed sites parameterized by edge-types. Key innovations include:
- Marked schemes: Generalizing modulus pairs, these structures systematically bound poles of functions beyond log-geometry.
- Edged localisation: A unified way to talk about functions with assigned decay.
The project offers a unified lens to tackle longstanding problems:
- Finiteness and coherence: By leveraging crystalline techniques in rigid settings, the framework may resolve Berthelot’s conjecture and strengthen finiteness results for non-proper schemes.
- Log-decay F-isocrystals: The tau-edged formalism provides a cohomological foundation for these objects, enabling progress on p-adic L-function meromorphy and trace formulas.
- Characteristic-zero connections: The theory recovers Deligne’s pole order filtration, linking p-adic and complex geometric phenomena.
Scale and Significance.
The implications span arithmetic geometry and number theory:
- Theoretical advances: A deeper synthesis of cohomology theories could streamline proofs and inspire new advancements.
- Algorithmic applications: Enhanced understanding of F-isocrystals may refine tools for computing L-functions or Galois representations.
- Interdisciplinary reach: While primarily mathematical, the project’s emphasis on algebraic structures and filtrations could improve cryptographic protocols or mirror symmetry studies, where bounded growth conditions are critical.
This project introduces tau-edged crystalline cohomology, a novel unification of these theories through a family of ringed sites parameterized by edge-types. Key innovations include:
- Marked schemes: Generalizing modulus pairs, these structures systematically bound poles of functions beyond log-geometry.
- Edged localisation: A unified way to talk about functions with assigned decay.
The project offers a unified lens to tackle longstanding problems:
- Finiteness and coherence: By leveraging crystalline techniques in rigid settings, the framework may resolve Berthelot’s conjecture and strengthen finiteness results for non-proper schemes.
- Log-decay F-isocrystals: The tau-edged formalism provides a cohomological foundation for these objects, enabling progress on p-adic L-function meromorphy and trace formulas.
- Characteristic-zero connections: The theory recovers Deligne’s pole order filtration, linking p-adic and complex geometric phenomena.
Scale and Significance.
The implications span arithmetic geometry and number theory:
- Theoretical advances: A deeper synthesis of cohomology theories could streamline proofs and inspire new advancements.
- Algorithmic applications: Enhanced understanding of F-isocrystals may refine tools for computing L-functions or Galois representations.
- Interdisciplinary reach: While primarily mathematical, the project’s emphasis on algebraic structures and filtrations could improve cryptographic protocols or mirror symmetry studies, where bounded growth conditions are critical.
The central technical achievement of the project lies in establishing a refined de Rham comparison theorem for tau-edged crystalline cohomology. This comparison connects the edged crystalline cohomology of a marked scheme to its filtered de Rham complex, leveraging the interplay between poles, filtrations, and divided power structures.
The comparison theorem was established using a method inspired by the Bhatt-de Jong approach, which employs Alexander--Čech complexes to compute cohomology. The main ingredients are an affine acyclicity theorem of edged crystals and a filtered Poincaré Lemma.
The comparison theorem was established using a method inspired by the Bhatt-de Jong approach, which employs Alexander--Čech complexes to compute cohomology. The main ingredients are an affine acyclicity theorem of edged crystals and a filtered Poincaré Lemma.
A critical breakthrough lies in the treatment of log-decay F-isocrystals. By encoding logarithmic growth into edge-types, the project provides a cohomological foundation for Wan’s p-adic L-function results, linking analytic meromorphy to structural properties of the cohomology. This resolves a conceptual gap in the field, where such behaviors were previously understood only through ad-hoc coordinate calculations.
The tau-edged formalism also reveals deep ties to topological cyclic homology (TCH). The filtered divided power algebras central to the construction mirror pro-cyclic structures in TCH, suggesting a future synthesis where tau-edged cohomology could refine the cyclotomic trace map or enhance deformation-theoretic analyses of K-theory. Such a synthesis would unify rigid cohomology’s geometric insights with TCH’s computational toolkit, offering new strategies for global duality or trace formulas in arithmetic geometry.
For broader adoption, the field must prioritize standardising edge-type methods and expanding computational tools (e.g. for singular schemes). Collaborative efforts to integrate tau-edged methods with prismatic cohomology or F-gauges could further consolidate its role in modern arithmetic geometry, while workshops would lower barriers for researchers adapting these techniques. By anchoring its advances in classical problems while forging connections to others frameworks, the project redefines the landscape of overconvergent and log-decay theories, offering a unified language to tackle finiteness, comparison, and duality across geometric contexts.
The tau-edged formalism also reveals deep ties to topological cyclic homology (TCH). The filtered divided power algebras central to the construction mirror pro-cyclic structures in TCH, suggesting a future synthesis where tau-edged cohomology could refine the cyclotomic trace map or enhance deformation-theoretic analyses of K-theory. Such a synthesis would unify rigid cohomology’s geometric insights with TCH’s computational toolkit, offering new strategies for global duality or trace formulas in arithmetic geometry.
For broader adoption, the field must prioritize standardising edge-type methods and expanding computational tools (e.g. for singular schemes). Collaborative efforts to integrate tau-edged methods with prismatic cohomology or F-gauges could further consolidate its role in modern arithmetic geometry, while workshops would lower barriers for researchers adapting these techniques. By anchoring its advances in classical problems while forging connections to others frameworks, the project redefines the landscape of overconvergent and log-decay theories, offering a unified language to tackle finiteness, comparison, and duality across geometric contexts.