Robert Boltje

A generalized Brauer construction and linear source modules

For a complete discrete valuation ring 0 with residue field F, a subgroup H of a finite group G and a homomorphism p: H --* OX, we define a functor V ~-4 V(H, (p) from the category of OG-modules to the category of FNG (H, p)-modules and investigate its behaviour with respect to linear source modules.

On p-permutation equivalences: Between Rickard equivalences and isotypies

Broue and Rickard defined in their landmark papers from 1990 and 1996 the notions of an isotypy and a splendid equivalence between p-blocks of finite groups. Here, we define a notion of equivalence, which we call a p-permutation equivalence, that implies an isotypy and is implied by a splendid equivalence. Moreover, we study properties of p-permutation equivalences.

Computation of locally free class groups

We show that the locally free class group of an order in a semisimple algebra over a number field is isomorphic to a certain ray class group. This description is then used to present an algorithm that computes the locally free class group. The algorithm is implemented in MAGMA for the case where the algebra is a group ring over the rational numbers.

Algebraicisation of explicit Brauer induction

This paper deals with canonical forms for induction theorems in the complex, finite-dimensional representation theory of a finite group, G. The first example of an explicit form for an induction theorem is due to R. Brauer [Br] who, in 1951, gave a canonical form for Artin’s induction theorem. In 1946, R. Brauer had also proved that each virtual character of G may be expressed as an integral linear combination of characters which are induced from linear (i.e., one-dimensional) characters of subgroups of G. One may reformulate Brauer’s induction theorem in the following manner. Define R+(G) to be the free abelian group on the G-conjugacy classes, (H 5 S’),, of linear characters of subgroups. Define 6, : R + (G) + R(G) by b,(( H-% S’),) = Indz(rp). Brauer’s theorem states that 6, is onto. By an explicit Brauer induction formula we will mean a section for the map, 6,. Given such a section one may use it to obtain a presentation for R(G) in terms of generators, (H-% S’)c; this problem is mentioned in [Ser, p. 71, footnote]. In 1986 Snaith produced such a section, t,, by means of the algebraic topology of smooth group actions related to G-representations. Details appear in [Sl] (see also [S2; S3; S4]). In his thesis Boltje algebraically constructed another such section, a,. In fact it is possible to give a topological construction of a,, similar to that of [Sl] for t,; this construction is given in (2.26) (see Theorem 2.27). In this paper we derive algebraic formulae for t, (and its topological ancestor, ro) in a manner similar to that used for a, in [Bl]. These results appear in Theorems 3.23 and 3.29. In Section 4 we derive a formula which relates ac and t, directly, rather than by means of the algebraic formulae of Section 3. The prerequisites for the understanding of a,, t,, and L, (the explicit

Conductors in the Non-Separable Residue Field Case

Using the technique of Explicit Brauer Induction an integer-valued conductor homomorphism is constructed for Galois representations of complete, discrete valuation fields. In the special case in which the residue field extension is separable the new conductor coincides with the classical Swan conductor. In the one-dimensional case the new conductor coincides with the abelian conductor of K.Kato. In the non-separable residue field case the problem of making such a conductor was posed by J-P.Serre in 1960, motivated by the need for a generalisation of the Swan representation of a curve to higher-dimensional varieties in characteristic p.

ALPERIN'S WEIGHT CONJECTURE AND CHAIN COMPLEXES

We show that Alperin’s weight conjecture is equivalent to the existence of contractible chain complexes whose entries have the right dimension coming from some of the alternating sum formulations. We conjecture that also for the other formulations and for Dade’s ordinary conjecture there exist such contractible chain complexes.

On p-monomial modules over local domains

Abstract We consider direct summands of monomial modules for group algebras over local rings of positive residue characteristic p. If the base ring is a local domain that contains a root of unity of sufficiently large order, the Krull–Schmidt theorem holds for these modules and their representation rings over two such base rings are canonically isomorphic.