Étale tame vanishing cycles over [\mathbbA¹_S/\mathbbG_m,S]

2022/09/27 by Denis-Charles Cisinski, Cisinski, Denis-Charles, Massimo Pippi +1
Mathematics · #14B07 #14F20 (primary) #32S30 (secondary) #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2209.13381

openalex publication_date 2022/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop a theory of tame vanishing cycles for schemes over [\mathbbA¹_S/\mathbbG_m,S] in the context of étale sheaves. We show some desired properties of this formalism, among which: a compatibility with tame vanishing cycles over a (strctly) henselian trait, a compatibility with the theory of tame vanishing cycles over \mathbbA¹_S, a compatibility with tensor product and with duality. In the last section, we prove that monodromy-invariant vanishing cycles, introduced by the second named author, are the homotopy fixed points with respect to a canonical continuous action of μ_∞ of tame vanishing cycles over [\mathbbA¹_S/\mathbbG_m,S].

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