Group Cohomology and Algebraic Cycles Cambridge Tracts in Mathematics Series

Group cohomology reveals a deep relationship between algebra and topology, and its recent applications have provided important insights into the Hodge conjecture and algebraic geometry more broadly. This book presents a coherent suite of computational tools for the study of group cohomology and algebraic cycles. Early chapters synthesize background material from topology, algebraic geometry, and commutative algebra so readers do not have to form connections between the literatures on their own. Later chapters demonstrate Peter Symonds's influential proof of David Benson's regularity conjecture, offering several new variants and improvements. Complete with concrete examples and computations throughout, and a list of open problems for further study, this book will be valuable to graduate students and researchers in algebraic geometry and related fields.

Preface; 1. Group cohomology; 2. The Chow ring of a classifying space; 3. Depth and regularity; 4. Regularity of group cohomology; 5. Generators for the Chow ring; 6. Regularity of the Chow ring; 7. Bounds for p-groups; 8. The structure of group cohomology and the Chow ring; 9. Cohomology mod transfers is Cohen–Macaulay; 10. Bounds for group cohomology and the Chow ring modulo transfers; 11. Transferred Euler classes; 12. Detection theorems for cohomology and Chow rings; 13. Calculations; 14. Groups of order p⁴; 15. Geometric and topological filtrations; 16. The Eilenberg–Moore spectral sequence in motivic cohomology; 17. The Chow–Künneth conjecture; 18. Open problems.

Burt Totaro is a Professor of Mathematics at the University of California, Los Angeles. He was elected a Fellow of the Royal Society in 2009.