The Steinberg complex of an arbitrary finite group in arbitrary positive characteristic.

Abstract

Given a finite group G and a field R of positive characteristic, one may build the Steinberg complex of G over R , which is a complex of projective RG -modules. This generalizes the Steinberg module for a finite group of Lie type.

We prove that an important theorem of Webb holds even for infinite-dimensional complexes, which allows for the possibility of "Steinberg complex analogues" coming from a new class of CW-complexes. We then consider some infinite-dimensional CW-complexes which appear in the literature.

We explicitly calculate a particular example of a Steinberg complex, whose homology is known to include non-projective RG -modules in some degrees. The result shows in particular that the Steinberg complex need not be a partial tilting complex.

We then exhibit another example of a Steinberg complex with non-projective homology. We show that no group of smaller order divisible by only two primes will share this property.

We close by examining functors, called coefficient systems, which are defined on the category of G -sets and which themselves form an abelian category. These arise in Chapter 3 with the proof of Webb's Theorem and its generalization. We are able to prove that a complex of coefficient systems, closely related to the Steinberg complex, satisfies a "tilting complex" property that the Steinberg complex lacks.