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

摘要

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.