Decomposition of idempotent 2-cocycles

TL;DR

This paper presents a method to decompose idempotent 2-cocycles in Galois extensions using ideals of associated crossed product algebras, with special cases for semilinear maps.

Contribution

It introduces a novel decomposition technique for idempotent 2-cocycles in Galois cohomology, extending understanding of their structure in algebraic extensions.

Findings

Decomposition of idempotent 2-cocycles via descending ideals

Specialization to cocycles from semilinear maps

Enhanced structural understanding of crossed product algebras

Abstract

Let L be a finite Galois extension of K with Galois group G. We decompose any idempotent 2-cocycle f using finite sequences of descending two-sided ideals of the corresponding weak crossed product algebra A:= (L/k, G, f). We specialise the results in case f is the corresponding idempotent 2-cocycle for some semilinear map from G to a multiplicative monoid with minimum element.