Twisting by a torsor

TL;DR

This paper formalizes the concept of twisting objects by G-torsors within algebraic geometry and explores applications, notably in the specialization of covers, extending classical tools to a broader, more rigorous framework.

Contribution

It provides a general formalization of twisting by G-torsors in algebraic geometry and applies it to the problem of specialization of covers.

Findings

Formalization of twisting operation in algebraic geometry

Application to specialization of covers problem

Extension of classical torsion operation

Abstract

Twisting by a G-torsor an object endowed with an action of a group G is a classical tool. For instance one finds in the paragraph 5.3 of the book "cohomologie galoisienne" by Serre, the description of the "op\'eration de torsion" in a particular context. The aim of this note is to give a formalization of this twisting operation as general as possible in the algebraic geometric framework and to present a few applications. We will focus in particular to the application to the problem of specialization of covers addressed by P. D\`ebes and al. in a series of papers.