Cohomological Invariants in Galois Cohomology (University Lecture Series, Vol. 28)

This volume addresses algebraic invariants that occur in the confluence of several important areas of mathematics, including number theory, algebra, and arithmetic algebraic geometry. The invariants are analogues for Galois cohomology of the characteristic classes of topology, which have been extremely useful tools in both topology and geometry. It is hoped that these new invariants will prove similarly useful. Early versions of the invariants arose in the attempt to classify the quadratic forms over a given field. The authors are well-known experts in the field. Serre, in particular, is recognized as both a superb mathematician and a master author. His book on Galois cohomology from the 1960s was fundamental to the development of the theory. Merkurjev, also an expert mathematician and author, co-wrote The Book of Involutions (Volume 44 in the AMS Colloquium Publications series), an important work that contains preliminary descriptions of some of the main results on invariants described here. The book also includes letters between Serre and some of the principal developers of the theory. It will be of interest to graduate students and research mathematicians interested in number theory and Galois cohomology. This Volume Is Concerned With Algebraic Invariants, Such As The Stiefel-whitney Classes Of Quadratic Forms (with Values In Galois Cohomology Mod 2) And The Trace Form Of Etale Algebras (with Values In The Witt Ring). The Invariants Are Analogues For Galois Cohomology Of The Characteristic Classes Of Topology. Historically, One Of The First Examples Of Cohomological Invariants Of The Type Considered Here Was The Hasse-witt Invariant Of Quadratic Forms. The First Part Classifies Such Invariants In Several Cases. A Principal Tool Is The Notion Of Versal Torsor, Which Is An Analogue Of The Universal Bundle In Topology. The Second Part Gives Rost's Determination Of The Invariants Of G-torsors With Values In H[superscript 3](q/z(2)), When G Is A Semisimple, Simply Connected, Linear Group. This Part Gives Detailed Proofs Of The Existence And Basic Properties Of The Rost Invariant.--book Jacket. Cohomological Invariants, Witt Invariants, And Trace Forms / Jean-pierre Serre And Skip Garibaldi -- Rost Invariants Of Simply Connected Algebraic Groups / Alexander Merkurjev And Skip Garibaldi. Skip Garibaldi, Alexander Merkurjev, Jean-pierre Serre. Includes Bibliographical References (p. 159-163) And Indexes. This volume is concerned with algebraic invariants, such as the Stiefel-Whitney classes of quadratic forms (with values in Galois cohomology mod 2) and the trace form of étale algebras (with values in the Witt ring). The invariants are analogues for Galois cohomology of the characteristic classes of topology. Historically, one of the first examples of cohomological invariants of the type considered here was the Hasse-Witt invariant of quadratic forms. The first part classifies such invariants in several cases. A principal tool is the notion of versal torsor, which is an analogue of the universal bundle in topology. The second part gives Rost's determination of the invariants of $G$-torsors with values in $H^3(\mathbb{Q}/\mathbb{Z}(2))$, when $G$ is a semisimple, simply connected, linear group. This part gives detailed proofs of the existence and basic properties of the Rost invariant. This is the first time that most of this material appears in print. Addresses algebraic invariants that occur in the confluence of several important areas of mathematics, including number theory, algebra, and arithmetic algebraic geometry. This book states that the invariants are analogues for Galois cohomology of the characteristic classes of topology, which have been useful tools in both topology and geometry.