On a question by Roggenkamp about group algebras

TL;DR

The paper analyzes when the group algebra $bZ_{(p)}G$ is semiperfect by examining liftings of idempotents from the semisimple quotient and linking this to Krull–Schmidt decompositions. In the ordinary case ($p mid |G|$), it provides a complete arithmetic criterion: every complex irreducible character $chi$ must have Schur index $iota(chi)=1$ and $p$ must be inert in $bQ(chi)$ for all $chi$, with concrete illustrations. In the modular case ($p mid |G|$), a conjecture (KG conjecture) is proposed to guide the lifting problem via Grothendieck groups and Cartan maps, and partial results are developed using Brauer–Humphreys filtrations; explicit lifting constructions are given for certain groups such as $A_5$ and semidirect products, highlighting how arithmetic of character fields governs semiperfectness. Overall, the work connects module-lifting questions to number-theoretic data, providing a clear solution in the ordinary case and a structured conjectural framework for the modular case. The findings offer a roadmap for determining semiperfectness of localized group algebras and for constructing explicit liftings in nontrivial examples.

Abstract

We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a conjecture, which extends the criterion.

Theorems & Definitions (26)

• Proposition 5
• Proposition 7
• Theorem 8
• Conjecture 9
• Lemma 1.1
• proof
• Corollary 1.2
• Lemma 1.3
• Proposition 1.4
• proof