000			03166 a2200265 4500
008			240808b2021 |||||||| |||| 00| 0 eng d
020			_a9783030742478 (HB)
041			_aeng
080			_a51 _bTHE
100			_aThelene, Jean-Louis Colliot
245			_aThe Brauer-Grothendieck Group
260			_bSpringer _c2021 _aSwitzerland
300			_axv, 453p.
490			_aErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, 0071-1136 _v71
504			_aIncludes Bibliography (427-448) and Index
505			_a1. Galois Cohomology 2. Étale Cohomology 3. Brauer Groups of Schemes 4. Comparison of the Two Brauer Groups, II 5. Varieties Over a Field 6. Birational Invariance 7. Severi-Brauer Varieties and hypersurfaces 8. Singular Schemes and Varieties 9. Varieties with a Group Action 10. Schemes Over Local Rings and Fields 11. Families of Varieties 12. Rationality in a Family 13. The Brauer-Manin Set and the formal lemma 14. Rational Points in the Brauer-Manin Set 15. The Brauer-Manin Obstruction for Zero-Cycles 16. The Tate Conjecture, Abelian Varieties and K3 Surfaces
520			_aThis monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer-Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong's proof of Gabber's theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer-Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer-Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry
650			_aArithmetical algebraic geometry
650			_aBrauer group
650			_aCohomology operations
650			_aGrothendieck groups
690			_aMathematics
700			_aSkorobogatov, Alexei N.
942			_cBK
999			_c60503 _d60503