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

Abstract

We prove that étale and de Rham cohomology algebras of a smooth proper rigid-analytic space over a finite extension of ${\mathbf{Q}}_p$ 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.

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