\'Etale tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$

TL;DR

This paper develops a formalism for tame vanishing cycles over a quotient stack, establishing compatibility properties and relating monodromy-invariant cycles to homotopy fixed points under a group action.

Contribution

It introduces a theory of tame vanishing cycles over $[A^1_S/G_{m,S}]$ with key compatibility results and links monodromy-invariant cycles to homotopy fixed points.

Findings

Established compatibility with vanishing cycles over henselian traits

Proved tensor product and duality properties of the formalism

Connected monodromy-invariant cycles to homotopy fixed points under $mu_$ action

Abstract

We develop a theory of tame vanishing cycles for schemes over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$ in the context of \'etale 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 $\mathbb{A}^1_{S}$, 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 $\mu_{\infty}$ of tame vanishing cycles over $[\mathbb{A}^1_{S}/\mathbb{G}_{m,S}]$.