Ramification theory from homotopical point of view, I

TL;DR

This paper proves a conjecture relating pushforward operations of étale sheaves and their characteristic cycles using homotopical methods and $$-categories, advancing ramification theory in algebraic geometry.

Contribution

It introduces a homotopical approach to characteristic cycles, confirming Saito's conjecture with the use of $$-categories.

Findings

Established compatibility of pushforward and characteristic cycle in étale cohomology.

Revised the construction of characteristic cycles using homotopical methods.

Applied $$-categories to ramification theory.

Abstract

We prove the compatibility of pushforward along a proper morphism of an \'{e}tale constructible sheaf and the pushforward of its characteristic cycle up to $p$-torsion. This was conjectured by Takeshi Saito. For this, we revisit the construction of the characteristic cycle, due to Saito and Beilinson, from more homotopical point of view. In particular, the language of $\infty$-categories is indispensable to carry this out.