Injectivity failure in crystalline comparisons

TL;DR

This paper investigates the limitations of injectivity in crystalline comparisons for smooth affine varieties in positive characteristic, revealing slope obstructions and establishing conditions for injectivity, while also exploring related cohomological separation and comparison results.

Contribution

It identifies slope obstructions to injectivity in crystalline comparisons and establishes injectivity results under certain slope conditions, advancing understanding of p-adic cohomology theories.

Findings

Identified slope obstructions to injectivity in crystalline comparison maps.

Proved injectivity for certain subspaces and in degree one cohomology.

Established a new integral comparison theorem and computed non-integral slopes.

Abstract

For smooth affine varieties in positive characteristic, we identify a slope obstruction to the injectivity of the comparison morphism from rigid cohomology to rationalised crystalline cohomology. This yields a negative answer to a question of Esnault--Kisin--Petrov concerning the injectivity of the de Rham-to-crystalline comparison map for smooth affine schemes over the Witt vectors that admit good compactifications. In contrast, we establish injectivity for certain subspaces defined by slope conditions as well as in cohomological degree one. For the latter case, we also prove the result with coefficients in $F$-able overholonomic $D$-modules leveraging a generalisation of Kedlaya's full faithfulness theorem. Beyond injectivity, we obtain various separation results for the affinoid topology on rigid and convergent cohomology. These results allow us to determine integral algebraic de Rham cohomology modulo torsion and to provide a more conceptual explanation for Ertl and Shiho's construction of varieties for which integral Monsky--Washnitzer cohomology modulo torsion is not finitely generated. Along the way, we prove a new integral comparison theorem between Monsky--Washnitzer cohomology and algebraic de Rham cohomology and we define fractional $p$-adic Tate twists, computing non-integral slopes of crystalline cohomology.