Weak crossed product orders arising from partitions of a finite group

TL;DR

This paper explores the structure of weak crossed product orders in unramified Galois extensions, using semilinear maps to classify and analyze their properties, including criteria for inflating idempotent cocycles.

Contribution

It introduces a novel approach using the monoid Sl(G) to construct and analyze crossed product orders in unramified extensions, providing new classification criteria.

Findings

Constructed specific crossed product orders using Sl(G)

Established criteria for inflating idempotent 2-cocycles

Linked algebraic structures to semilinear maps

Abstract

Let R be a complete discrete valuation ring with quotient field K, L a finite Galois extension of K with Galois group G and S the integral closure of R in L. In this article, using elements of the monoid Sl(G), the set of semilinear maps of G on the set of natural numbers introduced in [8], we construct certain crossed product orders in the case that the extension S/R is unramified. We associate an element of Sl(G) to every crossed product order and give a criterion for inflating idempotent 2-cocycles.