{"title":"Weil Restriction and Algebraic Tori","authors":"","doi":"10.1090/mmono/246/08","DOIUrl":"https://doi.org/10.1090/mmono/246/08","url":null,"abstract":"","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132008003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a group. We can form the group ring Z[G] over G; by definition it is the set of formal finite sums ∑ aigi, where ai ∈ Z, gi ∈ G, and multiplication is defined in the obvious manner. We shall call an abelian group A a G-module if it is a left Z[G]-module. This means, of course, that there exists a homomorphismG→ AutZ(A). We can also makeA into a right Z[G]-module simply by writing ag := g−1a for a ∈ A, g ∈ G. This is important for tensor products. An example of a G-module is any abelian group with trivial action by G. For instance, we shall in the future denote by Z the integers with trivial G-action. Finally, if A and B are G-modules, then a G-homomorphism between them is a map φ : A→ B which is a Z[G] homomorphism. The set of G-homomorphisms between A and B is denoted by HomG(A,B). It is a left exact functor of A and B, covariant in B and contravariant in A. As usual its derived functors are denoted by Ext. Let A be a G-module. Then we define the cohomology groups as
{"title":"Group Cohomology","authors":"A. Mathew","doi":"10.1090/mmono/246/01","DOIUrl":"https://doi.org/10.1090/mmono/246/01","url":null,"abstract":"Let G be a group. We can form the group ring Z[G] over G; by definition it is the set of formal finite sums ∑ aigi, where ai ∈ Z, gi ∈ G, and multiplication is defined in the obvious manner. We shall call an abelian group A a G-module if it is a left Z[G]-module. This means, of course, that there exists a homomorphismG→ AutZ(A). We can also makeA into a right Z[G]-module simply by writing ag := g−1a for a ∈ A, g ∈ G. This is important for tensor products. An example of a G-module is any abelian group with trivial action by G. For instance, we shall in the future denote by Z the integers with trivial G-action. Finally, if A and B are G-modules, then a G-homomorphism between them is a map φ : A→ B which is a Z[G] homomorphism. The set of G-homomorphisms between A and B is denoted by HomG(A,B). It is a left exact functor of A and B, covariant in B and contravariant in A. As usual its derived functors are denoted by Ext. Let A be a G-module. Then we define the cohomology groups as","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116220199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Arithmetic of Two-dimensional\u0000 Quadratics","authors":"","doi":"10.1090/mmono/246/06","DOIUrl":"https://doi.org/10.1090/mmono/246/06","url":null,"abstract":"","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121238874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
: The Brauer-Manin obstruction is a refinement of the Hasse principle, which gives a sufficient condition for non-existence of rational points on an algebraic variety. In this talk I will introduce the Brauer group of an algebraic variety and explain the Brauer-Manin obstruction geometrically. Then we will see through an example that the Brauer-Manin obstruction is a strict refinement of the Hasse principle.
{"title":"Brauer-Manin Obstruction","authors":"Yeqin Liu Uic","doi":"10.1090/mmono/246/11","DOIUrl":"https://doi.org/10.1090/mmono/246/11","url":null,"abstract":": The Brauer-Manin obstruction is a refinement of the Hasse principle, which gives a sufficient condition for non-existence of rational points on an algebraic variety. In this talk I will introduce the Brauer group of an algebraic variety and explain the Brauer-Manin obstruction geometrically. Then we will see through an example that the Brauer-Manin obstruction is a strict refinement of the Hasse principle.","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121621367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Example of a Unirational Non-rational\u0000 Variety","authors":"","doi":"10.1090/mmono/246/05","DOIUrl":"https://doi.org/10.1090/mmono/246/05","url":null,"abstract":"","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116718076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
These lecture notes accompanied a minicourse on étale cohomology offered by the author at the University of Michigan in the summer of 2017. They are only a preliminary draft and should not be used as a reference.
这些讲义是作者于2017年夏天在密歇根大学(University of Michigan)开设的一门关于上同源性的迷你课程附带的。它们只是初步草案，不应用作参考。
{"title":"Étale Cohomology","authors":"David Schwein","doi":"10.1090/mmono/246/12","DOIUrl":"https://doi.org/10.1090/mmono/246/12","url":null,"abstract":"These lecture notes accompanied a minicourse on étale cohomology offered by the author at the University of Michigan in the summer of 2017. They are only a preliminary draft and should not be used as a reference.","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129724081","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.
{"title":"Unramified Brauer Group and Its\u0000 Applications","authors":"S. Gorchinskiy, C. Shramov","doi":"10.1090/mmono/246","DOIUrl":"https://doi.org/10.1090/mmono/246","url":null,"abstract":"This is a textbook on arithmetic geometry with special regard to unramified Brauer groups of algebraic varieties. The topics include Galois cohomology, Brauer groups, obstructions to stable rationality, arithmetic and geometry of quadrics, Weil restriction of scalars, algebraic tori, an example of a stably rational non-rational variety, Brauer-Manin obstruction. All material is split into locally trivial problems with detailed hints.","PeriodicalId":371565,"journal":{"name":"Translations of Mathematical\n Monographs","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130756330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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