These are the notes for a course taught at the University of Michigan in F92 as Math 676. They are available at www.math.lsa.umich.edu/∼jmilne/. Please send comments and corrections to me at [email protected]. v2.01 (August 14, 1996.) First version on the web. v2.10 (August 31, 1998.) Fixed many minor errors; added exercises and index.

2 Arithmetic in an algebraic number field 30 2.1 Finitely generated abelian groups . . . . . . . . . . . . . . . . . . 30 2.2 The splitting field and the discriminant . . . . . . . . . . . . . . 31 2.3 Number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.4 The ring of integers . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.5 Ideals . . . . . . . . . . . . . . ....

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) 7→ kk∗(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel’s homotopy algebraic K-theory, KH....

SCHWARZ THEORY 27 Algorithm 4.1 (Abstract Multiplicative Schwarz Method { Implementation Form) un+1 = MS(un; f) where the operation uNEW = MS(uOLD; f) is de ned as: Do k = 1; : : : ; J rk = IT k (f AuOLD) ek = Rkrk uNEW = uOLD + Ikek uOLD = uNEW End do. Note that the rst step through the loop in MS( ; ) gives: uNEW = uOLD + I1e1 = uOLD + I1R1IT 1 (f AuOLD) = (I I1R1IT 1 A)uOLD + I1R1IT 1 f: Con...

Algebraic quantum field theory provides a general, mathematically precise description of the structure of quantum field theories, and then draws out consequences of this structure by means of various mathematical tools — the theory of operator algebras, category theory, etc.. Given the rigor and generality of AQFT, it is a particularly apt tool for studying the foundations of QFT. This paper is...

Acknowledgements Firstly, I would like to thank my supervisor, Dr David R. Kohel whose many helpful comments and patient guidance made this thesis possible. Secondly, I would like to thank David Gruenwald for volunteering his time to proofread this thesis. Almost all of his recommendations were taken on board. This thesis is the better for it. To a less extent, my thanks go to the IT school of ...

We show how methods from K-theory of operator algebras can be applied in a completely algebraic setting to define a bivariant, M∞-stable, homotopy-invariant, excisive Ktheory of algebras over a fixed unital ground ring H, (A, B) 7→ kk∗(A, B), which is universal in the sense that it maps uniquely to any other such theory. It turns out kk is related to C. Weibel’s homotopy algebraic K-theory, KH....