ECrys · Edged Crystalline Cohomology
Horizon Europe — Marie Skłodowska-Curie Actions
- Duration
- 2023-09-01 → 2025-08-31
- EU contribution
- €195,915
- Participants
- 2
- Scheme
- HORIZON-TMA-MSCA-PF-EF
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Results in brief
Edged Crystalline Cohomology
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.
Data: CORDIS, © European Union
Project objective
This proposed project aims at opening new horizons in Grothendieck and Berthelot's theories of crystalline and rigid cohomology. These are p-adic cohomology theories that are used to study algebraic varieties in positive characteristic. In the last years, the subject has seen an incredible development. Recent important achievements have been, for example, Kedlaya's new proof of the Riemann Hypothesis in positive characteristic and Abe's construction of a p-adic Langlands correspondence for overconvergent F-isocrystals. On the other hand, there are still some fundamental open questions. The main weakness of the theory of rigid cohomology is the difficulty of performing classical geometric operations. For example, it is not known whether the direct image functors have all the desirable propreties (Berthelot's conjecture). This is mainly due to the fact that the definitions rely on differential forms, which need smoothness assumptions to be defined. The Applicant D'Addezio wants to use the edged crystalline site, a new site that he has recently constructed, to solve this issue. In particular, he wants to show that the edged crystalline site gives an alternative new definition of rigid cohomology and overconvergent isocrystals and then use this to prove Berthelot's conjecture. For this second step, he will exploit the fact that the definition of the edged crystalline site is completely algebraic. Other applications that will be developped include the construction of an integral structure for rigid cohomology and the construction of the category of F-isocrystals with log-decay for smooth varieties of arbitrary dimension (extending the results of Kramer--Miller).
Original text from CORDIS.
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Data: CORDIS, © European Union