Slopes and weights of $\ell$-adic cohomology of rigid spaces

TL;DR

The paper proves that Frobenius eigenvalues on $ ext{ell}$-adic (intersection) cohomology of rigid spaces over $p$-adic local fields are algebraic integers with explicit $q$-slope bounds, and uses this to derive Weil-weight bounds and uniform quasi-unipotence indices that address conjectures of Bhatt, Hansen, and Zavyalov. A general, robust framework is developed for stability of integral (and $ ext{id}$-inverse integral) complexes under perverse truncations, enabling the key step of transferring integrality through nearby cycles. It then establishes integrality properties for both algebraic and analytic nearby cycles, leading to slope and weight bounds for $ ext{IH}^*$ and $H^*$-type cohomology, and proves a continuity property for compactly supported cohomology. Finally, the paper constructs counterexamples showing that monodromy-pure perverse sheaves can have non-monodromy-pure cohomology, thereby answering questions in the negative and highlighting limits of monodromy-purity preservation under pushforward.

Abstract

We prove that Frobenius eigenvalues of $\ell$-adic cohomology and $\ell$-adic intersection cohomology of rigid spaces over $p$-adic local fields are algebraic integers and we give bounds for their $p$-adic valuations. As an application, we deduce bounds for their weights, proving conjectures of Bhatt, Hansen, and Zavyalov. We also give examples of monodromy-pure perverse sheaves on projective curves with non monodromy-pure cohomology, answering a question of Hansen and Zavyalov.

Theorems & Definitions (48)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Remark 1.4
• Corollary 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.9
• Conjecture 1.10
• Definition 2.1