A remark on the Puig correspondence and Th\'evenaz's lifting theorem

To appear in Mathematische Zeitschrift

Author: Hubert Fottner
        Friedrich-List-Str.~7
        D-86153 Augsburg
        Germany


Abstract: In B.~K\"ulshammer and the author's joint paper `On indecomposable and 
imprimitive modules for finite groups -- a $G$-algebra approach' (1997, to 
appear in {\it J.~London Math.~Soc.}) it is shown that Th\'evenaz's lifting 
theorem (J.~Th\'evenaz, `Lifting idempotents and Clifford theory', {\it 
Comment.~Math. Helv.~}{\bf 58} (1983), 86--95) is a direct consequence of a 
general ring theoretical fact. The objective of the current paper is to
demonstrate that this ring theoretical lemma can also be applied to the
Puig correspondence, which allows to prove a lifting theorem \`a la
Th\'evenaz in this context (Theorem (2.4)). A consequence of this result is 
that the Green correspondence commutes wth induction. Loosely speaking, the 
resulting theorem on the Green correspondence asserts, in its module 
theoretic form, that if $M$ is an indecomposable module of some modular group 
algebra $O G$ with vertex $P$, $U$ is the Green correspondent of $M$ on the 
level of $O N_G(P)$, $H$ is a subgroup of $G$, $L$ is an indecomposable
$O H$-module with vertex $P$, and $V$ is the Green correspondent of
$L$ on the level of $O N_H(P)$ then $U \cong Ind_{N_H(P)}^{N_G(P)} V$
provided that $M\cong Ind_H^G L$. Moreover, it is shown that the converse
holds too provided that there is a normal subgroup $U$ of $G$ containing $P$
such that $\gamma_\infty(U) \subseteq H$, where $\gamma_\infty(U)$ denotes the
last term in the lower central series $U = \gamma_1(U) \supseteq\gamma_2(U)
\supseteq \gamma_3(U)\supseteq \ldots$ of $U$. 

Keywords: $G$-algebras, pointed groups, Puig correspondence, 
Green correspondence, Clifford theory.

Classification: 20C05, 20C20