Ramification theory from homotopical point of view, II

TL;DR

This paper explores the compatibility of pushforward operations on étale constructible sheaves and their characteristic cycles in algebraic geometry, employing Zariski-Riemann spaces to demonstrate an alteration process.

Contribution

It introduces a new proof technique using Zariski-Riemann spaces to establish the compatibility of pushforward and characteristic cycles for étale sheaves.

Findings

Proves compatibility of pushforward and characteristic cycles up to p-torsion.

Uses Zariski-Riemann spaces to show the termination of an alteration process.

Provides a new method for handling ramification in algebraic geometry.

Abstract

This is the second part of the paper which proves 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. In this second part, we show a result which is postponed from the first part because the technique of the proof is different. Especially, we use Zariski-Riemann spaces to show a certain alteration process terminates.