[Paper Review] Relative log convergent cohomology and relative rigid cohomology II

Atsushi Shiho|ArXiv.org|Jul 12, 2007

Homotopy and Cohomology in Algebraic Topology4 citations

TL;DR

This paper proves a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families in mixed characteristic, using relative log convergent cohomology of radius λ and hypercovering techniques. It establishes the existence and uniqueness of the q-th rigid cohomology overconvergent F-isocrystal on the base, even without assuming smoothness of the base, under certain conditions, including when the base is smooth over a perfect field of positive characteristic.

ABSTRACT

In this paper, we develop the theory of relative log convergent cohomology of radius $λ$ ($0 < λ\leq 1$), which is a generalization of the notion of relative log convergent cohomology in the previous paper. By comparing this cohomology with relative log crystalline cohomology, relative rigid cohomology and its variants and by using some technique of hypercovering, we prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families.

Motivation & Objective

• To prove a version of Berthelot's conjecture on the overconvergence of relative rigid cohomology for proper smooth families in mixed characteristic.
• To establish the existence and uniqueness of the q-th rigid cohomology overconvergent F-isocrystal on the base scheme.
• To generalize the theory of relative log convergent cohomology of radius λ and relate it to relative rigid and crystalline cohomologies.
• To use hypercovering techniques and alterations to reduce the problem to cases where the base is smooth or the morphism admits a nice log structure.
• To extend the result to the case with Frobenius structure when the base is a formal spectrum of a complete discrete valuation ring.

Proposed method

• Develops relative log convergent cohomology of radius λ as a generalization of the previous theory.
• Compares this cohomology with relative log crystalline and rigid cohomologies via a system of hypercoverings.
• Applies hypercovering techniques to reduce the problem to the case where the base is smooth or the morphism admits a nice log structure.
• Uses the full-faithfulness of restriction functors on overconvergent F-isocrystals for smooth schemes to prove equivalence of categories.
• Employs the theory of alterations and formal schemes to handle singularities and reduce to the smooth case.
• Constructs the overconvergent F-isocrystal on the base by gluing local data via hypercovering descent, ensuring functoriality and uniqueness.

Experimental results

Research questions

• RQ1 Does the higher direct image of an overconvergent F-isocrystal under a proper smooth morphism admit a canonical structure of an overconvergent F-isocrystal on the base?
• RQ2 Can Berthelot's conjecture on overconvergence be proven in the absence of a Frobenius lift on the base scheme?
• RQ3 Is the q-th rigid cohomology overconvergent F-isocrystal uniquely determined by its restriction to certain triples via hypercovering?
• RQ4 Can the result be extended to the case with Frobenius structure when the base is the formal spectrum of a complete discrete valuation ring?
• RQ5 Does the full-faithfulness conjecture of Tsuzuki imply the overconvergence result without assuming smoothness of the base?

Key findings

• Theorem 0.3 establishes the existence and uniqueness of the q-th rigid cohomology overconvergent isocrystal on the base when the base is smooth over a perfect field of positive characteristic, without requiring a Frobenius lift.
• Theorem 0.4 proves the same result under the additional assumption of a Frobenius lift on the base, extending the result to the F-isocrystal case.
• Theorem 5.2 shows that if the morphism admits a nice log structure and the coefficient is locally free, the conjecture holds without any smoothness assumption on the base.
• Theorem 7.4 extends the result to non-smooth morphisms after shrinking the base, again without requiring smoothness of the base.
• Theorem 7.8 proves the result without smoothness of the base under the assumption of Tsuzuki's full-faithfulness conjecture.
• Theorem 7.10 establishes the existence of a canonical Frobenius structure on the q-th rigid cohomology overconvergent F-isocrystal when the base is the formal spectrum of a complete discrete valuation ring and the coefficient has a Frobenius structure.