[Paper Review] Images directes II: F-isocristaux convergents

Jean-Yves Étesse|arXiv (Cornell University)|Oct 23, 2009

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes the convergence of direct images of convergent F-isocrystals under proper smooth liftable morphisms of k-schemes, where k is a perfect field of positive characteristic. By leveraging Teichmüller liftings and comparing specialization maps, it proves that the Frobenius morphism on these direct images is an isomorphism, extending classical results from rigid cohomology to the convergent F-isocrystal setting with Frobenius structure.

ABSTRACT

This article is the second one of a series of three articles devoted to direct images of isocrystals: here we consider convergent isocrystals with Frobenius structure. Let V be a complete discrete valuation ring, with residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Firstly we characterize convergent F-isocrystals on a smooth affine k-scheme. Secondly, for perfect k and after a detailed exposition of the Teichmûller liftings, especially for the affine rigid line, we derive the existence of Frobenius isomorphisms on the direct images of convergent F-isocrystals under a proper smooth and liftable k-morphism.

Motivation & Objective

• To characterize convergent F-isocrystals on smooth affine k-schemes.
• To study Teichmüller liftings in the context of rigid analytic geometry, especially for the affine line.
• To establish the convergence of direct images of F-isocristaux convergents under proper smooth liftable morphisms.
• To prove that the Frobenius morphism on such direct images is an isomorphism.
• To lay the foundation for studying L-functions associated with these direct images.

Proposed method

• Characterize convergent F-isocrystals on smooth affine k-schemes using a Monsky-Washnitzer-type description.
• Construct Teichmüller liftings for formal schemes and relate them to specialization and reduction maps.
• Use the fact that Teichmüller liftings are sections of the specialization map to reduce to rigid cohomology.
• Leverage known results on Frobenius isomorphisms in rigid cohomology to deduce the same for direct images.
• Apply Bosch-Güntzer-Remmert's criterion: a morphism is an isomorphism if it is so on fibers.
• Use the compatibility of direct images with base change and fiber functors under the Frobenius structure.

Experimental results

Research questions

• RQ1 Under what conditions is the direct image of a convergent F-isocrystal under a proper smooth morphism again a convergent F-isocrystal?
• RQ2 How do Teichmüller liftings interact with specialization and reduction maps in the rigid analytic setting?
• RQ3 Can the Frobenius morphism on the direct image of a convergent F-isocrystal be shown to be an isomorphism via fiber-wise analysis?
• RQ4 What is the role of the Teichmüller lifting in realizing the fiber functor and commuting with Frobenius?
• RQ5 How does the convergence of direct images relate to the cohomological properties of the morphism and the base scheme?

Key findings

• Convergent F-isocrystals on smooth affine k-schemes are characterized via a Monsky-Washnitzer-type description.
• For perfect k, Teichmüller liftings provide a canonical section of the specialization map and commute with Frobenius.
• The direct image of a convergent F-isocrystal under a proper smooth liftable k-morphism is again a convergent F-isocrystal.
• The Frobenius morphism on such direct images is an isomorphism, proven by reduction to rigid cohomology via Teichmüller liftings.
• The direct image construction commutes with base change and fiber functors, preserving the Frobenius structure.
• The results extend to projective and relative complete intersection morphisms, not just finite étale ones.