Cohomological invariants in Galois cohomology.

*(English)*Zbl 1159.12311
University Lecture Series 28. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3287-5/pbk). vii, 168 p. (2003).

In the present volume of the AMS University Lecture Series, the authors provide a systematic description of certain algebraic invariants that occur in the border area of several branches of contemporary mathematics, especially in number theory, arithmetic algebraic geometry, and in the theory of linear algebraic groups. Chiefly, these invariants appear as analogues for Galois cohomology of the well-established characteristic classes in algebraic topology, and it is hoped that they will prove similarly useful in the further study of various algebraic structures. Historically, early versions of some of the invariants discussed here arose in the classification theory of quadratic forms over a given field, ranging from the classical Hasse-Witt invariant (1937) to the cohomological invariants introduced later by J. Kr. Arason [J. Algebra 36, 448–491 (1975; Zbl 0314.12104) and Math. Z. 145, 139–143 (1975; Zbl 0297.13021)] and by M. Rost [C. R. Acad. Sci., Paris, Sér. I 313, No. 12, 823–827 (1991; Zbl 0756.17014)]. In this regard, the important books by J.-P. Serre [Cohomologie galoisienne. Cours au College de France, 1962-196, Lect. Notes Math. 5, Springer-Verlag, Berlin (1964; Zbl 0128.26303) and 5th ed. (1994; Zbl 0812.12002)] and by M.-A. Knus, A. S. Merkurjev, M. Rost and J.-P. Tignol [The book of involutions. With a preface by J. Tits. Colloquium Publications. American Mathematical Society (AMS). 44. Providence, RI: American Mathematical Society (AMS) (1998; Zbl 0955.16001)] should be viewed as direct forerunners of the recent work under review, apart from the numerous research papers on the subject published ever since. In fact, the work presented in the current volume propels the theory of cohomological invariants, in Galois cohomology much further, and this is the first time that most of the material appears in print.

As for the contents of the present research monograph,the text consists of two basically separate parts.

The first part, which occupies nearly two thirds of the book, is titled “Cohomological invariants, Witt invariants, and trace forms”. This is an expanded version of a series of lectures that J.-P. Serre gave at UCLA, Los Angeles, California, in 2001 as part of the Gill Distinguished Lecture Series, where the notes were taken and elaborated by the first co-author (S. Garibaldi). Appearing now in an enriched and updated version, these notes comprise nine chapters, in which invariants of both quadratic forms and etale algebras with values in Galois cohomology modulo 2 or in the Witt ring of a field, respectively, are constructed and classified. As a principal methodological tool appears here the notion of versal torsor, which may be seen as an analogue of the universal bundle in the theory of characteristic classes in algebraic topology. Chapter 1 introduces various functors with respect to which certain invariants are defined. Most of the numerous concrete examples have an interpretation in terms of \(G\)-torsors, where \(G\) is a smooth linear algebraic group over the ground field. Chapters 2 and 3 provide the relevant basics of the framework of Galois’ cohomology, whereas Chapter 4 discusses several specialization properties of cohomological invariants, the main one among them being an unpublished result of M. Rost. Chapter 5 derives the fundamental properties of the restriction and corestriction of these invariants, in perfect analogy with the corresponding constructions in group cohomology. Chapter 6 applies the foregoing results to the determination of the new invariants for quadratic forms, octonions, Albert algebras, and other related structures. Chapters 6–9 turn then to the case of étale algebras, with complete proofs of the main results announced earlier by J.-P.Serre in his 1993/1994 lectures at the College de France, Paris (cf. E. Bayer-Fluckinger, Galois cohomology and the trace form, Jahresber. Deutsch. Math.-Verein. 96, 35–55 (1994)]. This includes the fact that all cohomological and Witt invariants of étale algebras can be derived from the trace form, an explicit description of all possible trace forms of \(\text{rank\,}\leq 7\) and a proof that the Noether problem (on the rationality of fields of invariants) has a negative answer for certain linear algebraic groups over \(\mathbb{Q}\).

In three appendices to these lectures, the relevant correspondence of J.-P. Serre with M. Rost (1991), S. Garibaldi (2002) and B. Totaro (2002) is published, thereby illuminating the genesis of the theory exhibited in the present work.

The second part of the book comes with the title “Rost invariants of connected algebraic groups” and is written by A. Merkurjev, with the last section contributed by S. Garibaldi. In the sixteen sections of this part, a full account of the description of the degree 3 cohomological invariants (with coefficients in \(\mathbb{Q}/\mathbb{Z}(2)\)) for a semi-simple, simply connected linear algebraic group \(G\) is provided, and that for the first time in a systematic, detailed and complete manner. Most of the results delivered here have their origin in M. Rost’s pioneering approach [Doc. Math., J. DMV 1, 319–393 (1996; Zbl 0864.14002)], and they were already stated (without proof) in the foregoing monograph “The Book of Involutions” [Zbl 0955.16001] cited above. In particular, detailed proofs of the existence and functorial properties of the Rost invariant \(R_G\in H^3(\mathbb{Q}/\mathbb{Z}(2))\) of a \(G\)-torsor are given, together with a fundamental theorem of B. Kahn [Doc. Math., J. DMV 1, 395–416 (1996; Zbl 0883.19002)] relating \(K\)-cohomology and dimension-three Galois cohomology in this context, and with the complete determination of Dynkin indices for certain types of linear algebraic groups. Overall, this book is a highly valuable source for graduate students and researchers in algebraic number theory, arithmetic algebraic geometry, and algebraic group theory.

As for the contents of the present research monograph,the text consists of two basically separate parts.

The first part, which occupies nearly two thirds of the book, is titled “Cohomological invariants, Witt invariants, and trace forms”. This is an expanded version of a series of lectures that J.-P. Serre gave at UCLA, Los Angeles, California, in 2001 as part of the Gill Distinguished Lecture Series, where the notes were taken and elaborated by the first co-author (S. Garibaldi). Appearing now in an enriched and updated version, these notes comprise nine chapters, in which invariants of both quadratic forms and etale algebras with values in Galois cohomology modulo 2 or in the Witt ring of a field, respectively, are constructed and classified. As a principal methodological tool appears here the notion of versal torsor, which may be seen as an analogue of the universal bundle in the theory of characteristic classes in algebraic topology. Chapter 1 introduces various functors with respect to which certain invariants are defined. Most of the numerous concrete examples have an interpretation in terms of \(G\)-torsors, where \(G\) is a smooth linear algebraic group over the ground field. Chapters 2 and 3 provide the relevant basics of the framework of Galois’ cohomology, whereas Chapter 4 discusses several specialization properties of cohomological invariants, the main one among them being an unpublished result of M. Rost. Chapter 5 derives the fundamental properties of the restriction and corestriction of these invariants, in perfect analogy with the corresponding constructions in group cohomology. Chapter 6 applies the foregoing results to the determination of the new invariants for quadratic forms, octonions, Albert algebras, and other related structures. Chapters 6–9 turn then to the case of étale algebras, with complete proofs of the main results announced earlier by J.-P.Serre in his 1993/1994 lectures at the College de France, Paris (cf. E. Bayer-Fluckinger, Galois cohomology and the trace form, Jahresber. Deutsch. Math.-Verein. 96, 35–55 (1994)]. This includes the fact that all cohomological and Witt invariants of étale algebras can be derived from the trace form, an explicit description of all possible trace forms of \(\text{rank\,}\leq 7\) and a proof that the Noether problem (on the rationality of fields of invariants) has a negative answer for certain linear algebraic groups over \(\mathbb{Q}\).

In three appendices to these lectures, the relevant correspondence of J.-P. Serre with M. Rost (1991), S. Garibaldi (2002) and B. Totaro (2002) is published, thereby illuminating the genesis of the theory exhibited in the present work.

The second part of the book comes with the title “Rost invariants of connected algebraic groups” and is written by A. Merkurjev, with the last section contributed by S. Garibaldi. In the sixteen sections of this part, a full account of the description of the degree 3 cohomological invariants (with coefficients in \(\mathbb{Q}/\mathbb{Z}(2)\)) for a semi-simple, simply connected linear algebraic group \(G\) is provided, and that for the first time in a systematic, detailed and complete manner. Most of the results delivered here have their origin in M. Rost’s pioneering approach [Doc. Math., J. DMV 1, 319–393 (1996; Zbl 0864.14002)], and they were already stated (without proof) in the foregoing monograph “The Book of Involutions” [Zbl 0955.16001] cited above. In particular, detailed proofs of the existence and functorial properties of the Rost invariant \(R_G\in H^3(\mathbb{Q}/\mathbb{Z}(2))\) of a \(G\)-torsor are given, together with a fundamental theorem of B. Kahn [Doc. Math., J. DMV 1, 395–416 (1996; Zbl 0883.19002)] relating \(K\)-cohomology and dimension-three Galois cohomology in this context, and with the complete determination of Dynkin indices for certain types of linear algebraic groups. Overall, this book is a highly valuable source for graduate students and researchers in algebraic number theory, arithmetic algebraic geometry, and algebraic group theory.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

12G05 | Galois cohomology |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11E72 | Galois cohomology of linear algebraic groups |

13K05 | Witt vectors and related rings (MSC2000) |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |