MINISTRY OF EDUCATION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM[.]
MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM MASTER THESIS IN MATHEMATICS Hanoi - 2021 Luan van BỘ GIÁO DỤC VIỆN HÀN LÂM VÀ ĐÀO TẠO KHOA HỌC VÀ CÔNG NGHỆ VN HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ Lê Khắc Nhuận ĐỐI ĐỒNG ĐIỀU GALOIS VÀ ỨNG DỤNG VÀO BÀI TOÁN NHÚNG Chuyên ngành: Đại số Lý thuyết số Mã số: 8460104 LUẬN VĂN THẠC SĨ NGÀNH TOÁN HỌC NGƯỜI HƯỚNG DẪN KHOA HỌC: PGS TS Nguyễn Duy Tân Hà Nội - 2021 Luan van MINISTRY OF EDUCATION VIETNAM ACADEMY AND TRAINING OF SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY Le Khac Nhuan GALOIS COHOMOLOGY AND ITS APPLICATIONS IN THE EMBEDDING PROBLEM Major: Algebra and Number theory Code: 8460104 MASTER THESIS IN MATHEMATICS SUPERVISOR: Assoc Prof Nguyen Duy Tan Hanoi - 2021 Luan van Declaration I declare that this thesis titled ”Galois cohomology and its applications in the embedding problem” and the work presented in it are my own Wherever the work of others are involved, every effort is made to indicate this clearly, with due reference to the literature I confirm that this thesis has not been previously included in a thesis or dissertation submitted for a degree or any other qualification at this graduate university or any other institution I will take responsibility for the above declaration Hanoi, 20th September 2021 Signature of Student Le Khac Nhuan Luan van Acknowledgement First of all, I would like to express my deepest gratitude to Assoc Prof Nguyen Duy Tan was willing to supervise me despite my impoverished undergraduate mathematical background Thanks to his topic suggestion and his supervision, I have learnt so many things about Galois cohomology in particular and mathematics in general His guidance helped me all the time of writing this thesis I could not have imagined having a better supervisor for my MSc study Secondly, I would like to thank my lecturers, especially Dr Nguyen Dang Hop and Dr Nguyen Chu Gia Vuong, for many valuable mathematical lessons and insights I would like also to thank the Graduate University of Science and Technology and the Institute of Mathematics for providing a fertile mathematical environment Special thanks goes to Nguyen Van Quyet and Quan Thi Hoai Thu who all helped me in numerous ways during my study in Hanoi To Quyet in particular I owe you one for being a supportive friend as well as keeping me feel accompanied and conversations on various issues besides mathematics during my leisure times I would like also to thank my girlfriend Pham Thi Minh Tam, who has been by my side throughout my MSc, for all her love and support Last but not least, my thanks goes to my parents and siblings for always believing in me and encouraging me to follow my dreams Luan van Table of Contents Declaration Acknowledgement Table of Contents List of Symbols Introduction Profinite Groups 10 1.1 Infinite Galois Theory 10 1.2 Profinite Groups 16 Cohomology of Profinite Groups 30 2.1 Cohomology Groups 30 2.2 Functoriality 39 2.3 Universal Delta-functors 54 2.4 Induced Modules 62 2.5 Cup Products 68 2.6 Nonabelian Cohomology 74 Galois Cohomology 86 3.1 Algebraic Affine Group Schemes 86 3.2 Galois Cohomology 91 3.3 Galois Descent 99 The Embedding Problem 108 4.1 Group Extensions 109 4.2 The Embedding Problem 115 Luan van Conclusion 120 Reference 120 Luan van List of Symbols (G : H) index of a subgroup H in a group G 1A , id identity homomorphism A−B difference of a set B from a set A A∈A a object A in a category A A⊗B tensor product of abelian groups A and B AoG semidirect product of A by G AG subgroup of fixed points of A under the action of G Aα twisted group of A using cocycle α EXT (G, A) set of equivalence classes of group extensions of A by G Gk absolute Galois group of a field k H3 (Z/pZ) Heisenberg group of degree p Ks separable closure of a field K MGH (A) induced module of a H -module A T Fx (Ω/K) set of K -equivalence of Ω/K -twisted forms of x T ORS(A) set of isomorphisms of A-torsors ασ image of σ under a cocycle α Aut(A) group of automorphisms of a group A ∪ cup product δ, δ n connecting homomorphism Luan van Gal(Ω/K) Galois group of a Galois extension Ω/K Z[G0 ] group ring of G0 over Z L set of finite Galois subextensions of a Galois extension N set of normal open subgroups of a profinite group E(F, ϕ, π) embedding problem for a profinite group F with respect to epimorphisms ϕ and π µn (Ω) group of nth roots of unity in a field Ω ΩH fixed field of a subgroup H Ω× multiplicative group of a field Ω K algebraic closure of a field K σa, σa image of a under the action of σ StabG (a) stabilizer of a in G GLn (Ω) general linear group of degree n over a field Ω lim Ai −! direct limit lim Gi − inverse limit c1/p pth root of c f∗ homomorphism of cohomology groups with respect to f k[G] representing k -algebra of an affine group scheme G over k B n (G, A) set of homogeneous n-coboundaries of a G-module A C n (G, A) set of homogeneous n-cochains of a G-module A H n (G, A) nth cohomology group of G with coefficient in A Z n (G, A) set of homogeneous n-cocycles of a G-module A n B (G, A) n C (G, A) set of inhomogeneous n-coboundaries of a G-module A set of inhomogeneous n-cochains of a G-module A Luan van n Z (G, A) ∂, ∂ n set of inhomogeneous n-cocycles of a G-module A inhomogeneous coboundary operator ∂, ∂ n coboundary operator inf inflation res restriction Ab category of abelian groups k-Alg category of unital commutative k -algebras Ek category of field extensions over k Grp category of groups Mod(G) category of G-modules Set∗ category of pointed sets Set category of sets Mod(G0 ) category of Z[G0 ]-modules coker(f ) cokernel of a homomorphism f im(f ) image of a homomorphism f ker(f ) kernel of a homomorphism f µn nth root of unity group scheme GLn general linear group scheme Ga additive group scheme Gm multiplicative group scheme SLn special linear group scheme Luan van ... LÂM VÀ ĐÀO TẠO KHOA HỌC VÀ CÔNG NGHỆ VN HỌC VIỆN KHOA HỌC VÀ CÔNG NGHỆ Lê Khắc Nhuận ĐỐI ĐỒNG ĐIỀU GALOIS VÀ ỨNG DỤNG VÀO BÀI TOÁN NHÚNG Chuyên ngành: Đại số Lý thuyết số Mã số: 8460104 LUẬN VĂN... connecting homomorphism Luan van Gal(Ω/K) Galois group of a Galois extension Ω/K Z[G0 ] group ring of G0 over Z L set of finite Galois subextensions of a Galois extension N set of normal open subgroups... group scheme Luan van Introduction Galois cohomology is the study of the group cohomology of Galois modules, i.e., G-modules, with G being the Galois group of some Galois extension This subject is