Defect theory for prime ideals and Dress's induction theorem

Submitted on June 8, 1998,
to appear in {\it Algebras and Representation Theory}

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

Abstract: It was first observed by Dade that defect groups can be defined for 
arbitrary rings with finite group action as long as one replaces points
(resp.~primitve idempotents) with maximal ideals which, in general, are no
longer in bijection. Later The\'evenaz showed that also Puig's theory of
pointed groups carries over to this more general setup. So, in a sense, one
can also define sources of maximal ideals. Moreover, Th\'evenaz demonstrates
that this cannot only be done for rings with finite group action, but also
for the more general notion of a Green functor. The objective of this paper
is to show that the results carry over to a situation where one replaces
maximal ideals by prime ideals. As an application we prove that, for certain
Green functors $A$ for $G$, the Krull dimesion of $A(G)$ can be bounded in
terms of the Krull dimensions of $\overline{A}(P)$, where $P$ ranges over
all primordial subgroups of $G$ for $A$. Moreover, we shows that Puig's 
version of Sylow's first theorem for local pointed groups can be generalized
to the situation we are concerned with in this paper, and we demonstrate that
Dress's induction theorem is a consequence of this result. Beides, this
version of Sylow's first theorem is correlated with a theorem of Dress 
asserting that the primitve idempotents of the integral Burnside ring are 
parametrized by the conjugacy classes of perfect subgroups, and also with 
Brauer's induction theorem for the character ring.