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ICTP Lecture Notes Series, Volume 23

(ISBN 92-95003-38-1) - *October 2008*

## Some Recent Developments in Algebraic K-Theory

Editors: *E. Friedlander* (Northwestern University, USA), *A. Kuku* (University of Iowa, USA) and *C. Pedrini* (Univ. degli Studi di Genova, Italy)

**Introduction**

There was a School and Conference on Algebraic K-theory and its applications from May 14 to June 1, 2007. The first two weeks, May 14-25, were devoted to the School while the Conference took place May 28 - June 1. This volume contains the expository lectures given at the School.

The School and Conference was a follow-up to the earlier ones in 1997 and 2002. The Proceedings of the 1997 School and Conference was published by World Scientific in 1999. The 2002 School and Conference was dedicated to Hyman Bass on the occasion of his seventieth birthday. The Proceedings of the School appeared in the ICTP Lecture Notes Series, while the Proceedings of the Conference appeared in a special issue of the journal “K-Theory” in 2003.

These schools and conferences have emphasized the multidisciplinary nature of algebraic K-theory. Indeed, K-theory has become part of the fabric of Algebra, Algebraic Geometry, Analysis, Number Theory, and Topology. K-theory has brought new techniques and structures to these subjects, enabling fundamental contributions.

The lectures presented here were given in order to introduce some of the fundamental aspects of Algebraic K-theory and to present important current connections with topics such as the Bloch-Kato Conjecture, motivic cohomology and quadratic forms. The six topics of the lecture series are as follows: (1) Introduction to K-theory; (2) Linear algebraic groups and K-theory; (3) K-theory and quadratic forms; (4) Motives; (5) Proof of the Bloch-Kato conjecture and (6) K-theory and non-commutative algebraic geometry.

Each lecture series has its own introduction, but we provide in this general introduction a quick listing of some topics discussed in each.

1– E. Friedlander’s “An Introduction to K-theory” presents a brief overview of algebraic K-theory, emphasizing elementary aspects which serve as useful background for the other lecture series. Topics discussed include

*K*of rings, algebraic and topological vector bundles, varieties and schemes, as well as_{0}*K*,_{1}*K*of rings; classifying spaces and higher algebraic K-theory; topological K-theory, higher K-theory of algebraic varieties and schemes, Chow groups, etale K-theory, Beilinson conjectures and a review of some open problems._{2}2– U. Rehmann’s “Linear algebraic groups and K-theory” focuses on results for

*K*,_{0}*K*and_{1}*K*and provides an essential background for understanding quadratic forms and the proof of the Bloch-Kato conjecture. Topics covered include linear algebraic groups with illustrative examples, involving the structure of SL_{2}_{n}over fields; root systems; Chevalley groups and related K-theoretic results; structure and classification of almost simple groups.3– M. Levine’s “Motives” offers an introduction to the theory of pure and mixed motives. Starting from the theory of algebraic cycles and adequate equivalence relations, it covers Grothendieck motives, Voevodsky’s construction of triangulated categories of motives, motivic sheaves and cycle complexes, bivariant cycle cohomology. The final lecture discusses Bloch-Beilinson’s Conjectures on the filtration of the Chow ring, Murre’s Conjecture on Chow-Kunneth decomposition and finite dimensionality of motives.

4– A. Vishik’s “K-theory and quadratic forms” presents the theory of
quadratic forms and its various connections to Milnor K-theory, stable
homotopy groups of spheres and the study of quadrics. These lectures
introduce Chow motives, motives of quadrics and Rost motive. Among more
advanced topics discussed are algebraic cobordism theory, Steenrod
operations and u-invariants of fields.

5– C. Weibel’s “Proof of the Bloch-Kato’s Conjecture” provides an
overview of the proof of this fundamental conjecture due to M. Rost and
V. Voevodsky, and gives important details of the proof unavailable in
the unpublished work of Rost and Voevodsky. The validity of this
conjecture, now with sufficiently detailed proofs to satisfy the
experts, is one of the most important results in K-theory, revealing
remarkable structure in the (Galois) cohomology of fields. This lecture
series uses concepts and techniques presented in other lectures,
revealing the power and utility of the existence of Rost motives and
Steenrod operations in motivic cohomology.

6– A. Rosenberg’s “K-theory and non-commutative algebraic geometry” provides a perspective on non-commutative algebraic geometry and its connections to K-theory. The abstract, categorical approach presented in these lectures encompasses examples such as quantized enveloping algebras, quantum flag varieties and associated quantum D-schemes. The K-theory of these non-commutative objects is based upon the notion of non-abelian, nonadditive homological algebra. These lectures sketch how such a K-theory satisfies some of the important properties, e.g. reduction by resolution and devissage, enjoyed by the K-theory of more familiar commutative objects.

ACKNOWLEDGMENTS: It is our great pleasure to thank all those who have contributed to the success of the School and Conference. We gratefully acknowledge the financial support, excellent facilities and efficient staff of ICTP. In particular, we thank Suzie Radosic for her excellent oversight as secretary to the School and Conference. We thank Loether Goettsche, the local organizer, as well as the staff of the Mathematics section of ICTP for their cooperation. Finally, we thank Dilys Grilli of the Publications section of ICTP for enabling the publication of this volume.

*Eric Friedlander*

*Aderemi Kuku*

*Claudio Pedrini*

April 2008

**Table of Contents** (PostScript and PDF sources available)

Most (La)TeX & PS sources of Volume 23: lns023.tgz (7573813 bytes)

(NB: This tar-compressed file contains LaTeX files, postscript (.ps) and/or encapsulated postscript (.eps) files of each figure plus the postscript and PDF versions of each lecture note)