Cohomological Invariants in Galois Cohomology
The book contains a discussion of invariants, which are analogues of characteristic classes in topology (Chern classes, Stiefel-Whitney classes, etc.) for Galois cohomology. Topological spaces are replaced here by schemes Spec(k) for a field k. Similarly, an analogue of the universal bundle in topology is the notion of versal torsor. Historically, the first example of cohomological invariants of the type considered in the book was the Hasse-Witt invariant of quadratic forms. The book consists of two parts. The first part grew out of lectures by J.-P. Serre at UCLA from 2001. It contains an introduction to Galois cohomology, together with various operations on it (including restriction, corestriction, inflation, etc). As an application, it classifies invariants of quadratic forms and ètale algebras with values in Galois cohomology modulo 2, or in the Witt ring. For G a simple and simply connected algebraic group, Rost proved the existence of a canonical and nontrivial invariant of G-torsors with values in Galois cohomology of dimension 3. The second part, written by Merkurjev, gives detailed proofs of the existence and basic properties of the Rost invariant.