Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture

Alexander Petrov; Bogdan Zavyalov

algebraic geometry

Formality for rigid-analytic spaces satisfying the weight-monodromy conjecture

arXiv:2607.14517

summary

The paper proves that the étale and de Rham cohomology algebras of a smooth proper rigid‑analytic space over a p‑adic field are formal whenever the space satisfies the weight‑monodromy conjecture, and provides examples where formality fails.

Abstract

We prove that étale and de Rham cohomology algebras of a smooth proper rigid-analytic space over a finite extension of $Q_{p}$ are formal if the rigid-analytic space satisfies the weight-monodromy conjecture. This is achieved by showing that the underlying $E_{\infty}$ -algebra of a monodromy-pure $E_{\infty}$ -algebra in Weil--Deligne representations is formal. We give examples of smooth proper rigid-analytic surfaces whose cohomology algebras are not formal.

Minor Updates. Comments are very welcome!

Topics & keywords

#rigid-analytic spaces #weight-monodromy conjecture #étale cohomology #de rham cohomology #formality #smooth proper varietiesformal cohomology algebrap-adic geometryweight-monodromyrigid analyticétale cohomologyde Rham cohomology