Preface

Summary

‘Funny that you should ask. As it happens I have a complete mathematical vision of power. You could call it a programme. Some pieces are in place, some theorems are proved, others await proof and revelation. It is based on the theory of schemes.’

‘Of finite type?’ asked Zhilin, with rapt curiosity.

‘Any type, my friend, any type at all.’

‘Schemes, eh.’ Eli nodded his approval.‘Good. Good.’

‘Schemes are a little like the crania of topology with little light bulbs of algebra, called sheaves, stuck all over them.’

‘Sheaves, very good. Very agriculdural.’

from The Yukiad (Snaith, 1990c)

This volume began as a one-term advanced graduate course in algebra which I gave at the beginning of 1990 at McMaster University. As originally conceived my plan was to give a brief introduction to the representation theory of finite groups in characteristic zero. This sketch was to have been succeeded by an outline of the topological construction of my original Explicit Brauer Induction formula (Snaith, 1988b, 1989b) followed by a description of the behaviour of Explicit Brauer Induction with respect to Adams operations 4.1.6 as originally proved in theorem 2.33 of Snaith (1989a). Equipped with 4.1.6 the course was then to have concluded with a discussion of class-groups of group-rings and a proof of M.J. Taylor's conjecture concerning determinantal congruences 4.3.10 (see also the stronger congruences of 4.3.37).

However, in 1989, I learnt of the work of Robert Boltje (1989, 1990) which axiomatised Explicit Brauer Induction formulae and, entirely algebraically, found a different formula.