Tài liệu Luận văn thạc sĩ 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 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 và 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 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 1 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 2 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. 3 Table of Contents Declaration 1 Acknowledgement 2 Table of Contents 3 List of Symbols 7 Introduction 8 1 Profinite Groups 10 1.1 Infinite Galois Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Profinite Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 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 3 Galois Cohomology 86 3.1 Algebraic Affine Group Schemes . . . . . . . . . . . . . . . . . . . . 86 3.2 Galois Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.3 Galois Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4 The Embedding Problem 108 4.1 Group Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2 The Embedding Problem . . . . . . . . . . . . . . . . . . . . . . . . . 115 4 Conclusion 120 Reference 120 5 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 A⋊G 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. 6 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. 7 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. 8 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 considered as an important method in various branches of mathematics, including class field theory, theory of central simple algebras, theory of quadratic forms, ... In this thesis, we give an introduction to Galois cohomology, and then restrict our attention to the embedding problem for a profinite group. The central materials in the thesis have been taken from [2], [6], [7], [8], [14] and [16]. We presented them in a systematic way, logically consistent with the author’s point of view. The thesis consists of four chapters: In the first chapter, we introduce the concept of profinite groups and study their basic properties. We also give a proof of the fundamental theorem of infinite Galois theory and show that every Galois group is a profinite group. Most of the results developed here are needed for the subsequent chapters. In Chapter 2, we define cohomology groups and investigate their functorial properties. We then introduce the concept of universal δ -functors, and use this to prove some important results, such as Shapiro’s lemma. Induced modules allow us to apply the dimension shifting technique to prove the skew commutativity of cup products. The notion of cup products, introduced in this chapter, will be used in the last chapter to give a characterization of solvability of the embedding problem in a special case. The last section of the chapter is devoted to nonabelian cohomology. Corollary 2.6.13 in this section will be used to prove a main result of the next chapter. In Chapter 3, we first use the functorial approach to Galois cohomology. Then we introduce Galois descent and give a correspondence between equivalence classes of twisted forms and cohomology classes under some conditions which will be described explicitly. We also give a proof of Hilbert’s Theorem 90, and use this to introduce the Kummer exact sequence which will be used in the last chapter. The last chapter are dedicated to the embedding problem of a profinite group. We 9 give a connection between 2-cocycles and group extensions, and use this connection to prove Hoechsmann’s theorem which provides a characterization of solvability of the embedding problem. We end this chapter with a discussion of Heisenberg extensions. 10 Chapter 1 Profinite Groups In this chapter, we introduce the concept of profinite groups and investigate their basic properties. These groups are the main object of study which will be used in the subsequent chapters. We start with infinite Galois theory and aim to prove the fundamental theorem of infinite Galois theory. Then we give some characterizations of profinite groups and show that every Galois group is a profinite group. The materials in this chapter have been taken from [1] and [2]. We assume the reader is familiar with some basic results for topological groups, and is willing to accept other results belonging topology, which will be stated as needed, with references to the literature. 1.1 Infinite Galois Theory In this section, we define the Krull topology on the Galois group of an infinite Galois extension, and then use this topology to prove the fundamental theorem of infinite Galois theory. We first state without proof the fundamental theorem of Galois theory for finite Galois extensions (c.f. Theorem 5.1, page 51, [1]). Theorem 1.1.1. Let Ω be a finite Galois extension of K , with Galois group G. There is a bijection between the set of subfields E of Ω containing K and the set of subgroups H of G, given by E 7−! Gal(Ω/E) and H 7−! ΩH . Moreover, E is Galois over K if and only if H is normal in G; if that is the case, then G/H ∼ = Gal(E/K). However, when Ω is an infinite Galois extension over K , this theorem is not true anymore. It could happen that different subgroups of G have the same fixed field, as the following example: 11 Example 1.1.2. We let Ω denote the field obtained from Q by adjoining all square roots of prime numbers, so that Ω is an infinite Galois extension over Q. It is obvious that ΩGal(Ω/Q) = Q. On the other hand, for every prime number p, consider σp ∈ Gal(Ω/Q) given by √ √ √ √ σp ( p) = − p and σp ( q) = q for all prime numbers q 6= p. We denote by H the subgroup of Gal(Ω/Q) generated by σp for all prime numbers p. It then follows that H 6= Gal(Ω/Q) since H does not contain the element σ ∈ Gal(Ω/Q) given by √ √ σ( p) = − p for all prime numbers p. Moreover, we claim that ΩH = Q . Indeed, it is obvious that Q ⊆ ΩH . Conversely, for any a ∈ ΩH , there exists a subfield L obtained from Q by adjoining square roots of finitely many prime numbers such that a ∈ L. We may assume these prime numbers are p1 , . . . , pn . One verifies easily that L is a finite Galois extension over Q and σp1 |L , . . . , σpn |L are generators of Gal(L/Q). It follows that a ∈ LGal(L/Q) = Q since a ∈ ΩH , hence ΩH ⊆ Q, showing that ΩH = Q, as desired. For the rest of this section, we assume that Ω/K is a Galois extension which is not necessarily finite, and let G denote its Galois group. We shall endow G with a specific topology to obtain a topological group structure on G. We denote by L the set of finite Galois subextensions of Ω, and set B1 = {Gal(Ω/L) | L ∈ L}. The following lemma will be useful. Lemma 1.1.3. Every finite set of elements in Ω is contained in some field L ∈ L. In particular, we have Ω= [ L. L∈L Proof. It is obvious that the first statement implies the second. In order to prove the first statement, for any a1 , . . . , an ∈ Ω, consider the splitting field L of the minimal polynomials of ai over K for 1 6 i 6 n; it is easily seen that L/K is a finite Galois extension, hence L ∈ L. This proves our lemma. The following corollary is straightforward from the above lemma. 12 Corollary 1.1.4. We have \ N ∈B1 N = {1}. Lemma 1.1.5. If N1 , N2 ∈ B1 , then N1 ∩ N2 ∈ B1 . Proof. For i = 1, 2, assume that Ni = Gal(Ω/Li ) for some Li ∈ L. It is easily seen that N1 ∩ N2 = Gal(Ω/L1 ) ∩ Gal(Ω/L2 ) = Gal(Ω/L1 L2 ), where L1 L2 is the compositum of L1 and L2 in Ω. Since each Li is finite Galois over K , we get L1 L2 is finite Galois over K , i.e., L1 L2 ∈ L, hence N1 ∩ N2 ∈ B1 . This proves our lemma. Let X be a topological space. By a neighborhood basis at x ∈ X , we mean a collection Bx of neighborhoods of x such that for each neighborhood U of x, there exists V ∈ Bx such that V ⊆ U . For each σ ∈ G, we set Bσ = {σ Gal(Ω/L) | L ∈ L}. From Lemma 1.1.5, it is straightforward to check that the set Bσ forms a neighborhood basis at σ for a topology on G (c.f. Theorem 4.5, page 33, [3]). Definition 1.1.6. The topology on G, in which Bσ is a neighborhood basis at each σ ∈ G, is called the Krull topology. It is easily seen that if G is finite then the Krull topology becomes the discrete topology. Moreover, G endowed with the Krull topology is a topological group. From now on, we assume that every Galois group is endowed with this topology. By a totally disconnected space, we mean a topological space X , whose connected components are the points, or equivalently, X is totally disconnected if and only if the only nonempty connected subsets of X are the one-point sets (c.f. Section 29, [3]). We shall now study some properties of the topological group G. Theorem 1.1.7. The Galois group G is a Hausdorff, compact and totally disconnected topological group. 13 Proof. We first show that G is Hausdorff. For any σ, τ ∈ G such that σ 6= τ , i.e., τ −1 σ 6= 1, Corollary 1.1.4 shows that there exists N ∈ B1 such that τ −1 σ 6∈ N , i.e., σN ∩ τ N = ∅, hence G is Hausdorff since σN and τ N are open neighborhoods of σ and τ , respectively. The most difficult part of this proof is to show that G is compact. From the Tychonoff theorem (c.f. Theorem 17.8, page 120, [3]), the Cartesian product P = Y Gal(L/K) L∈L is compact since every finite group Gal(L/K) endowed with discrete topology is compact. Consider the natural group homomorphism ϕ : G −! P defined by f (σ) = (σ|L )L∈L for σ ∈ G. We shall prove that G is homeomorphic to ϕ(G) and that ϕ(G) is closed in P . If that is the case, then from the compactness of P , we get ϕ(G) is compact, hence so is G. In order to show that G is homeomorphic to ϕ(G), it suffices to show that ϕ is an open map onto ϕ(G) and ϕ is a continuous injection (c.f. Theorem 7.9, page 46, [3]). It is clear that ϕ : G −! ϕ(G) is an open map, since for any open neighborhood σ Gal(Ω/L) ∈ Bσ , we have ϕ(σ Gal(Ω/L)) = ϕ(G) ∩ ( Y Gal(L′ /K) × {σ|L }) L ′ ∈L L′ 6=L which is an open subset of ϕ(G). For the injectivity, let σ ∈ G such that σ|L is the identity for all L ∈ L. Lemma 1.1.3 shows that σ = 1, so ϕ is injective. For the continuity, we let pL denote the natural projection from P to Gal(L/K) for L ∈ L. It suffices to show that pL ◦ ϕ is continuous (c.f. Theorem 8.8, page 55, [3]). For any τ ∈ Gal(L/K), we suppose τ̃ ∈ G is an extension of τ ; it is easily seen that (pL ◦ ϕ)−1 ({τ }) = τ̃ Gal(Ω/L) which is an open subset of G, hence pL is continuous, as desired. We have shown that G is homeomorphic to ϕ(G). In order to show that ϕ(G) is closed in P , we set C = {(σL )L∈L ∈ P | σL |L∩L′ = σL′ |L∩L′ for L, L′ ∈ L} and claim that ϕ(G) = C . Indeed, it is obvious that ϕ(G) ⊆ C . Conversely, for any (σL )L∈L ∈ C , we define a map σ : Ω −! Ω as follows: for each a ∈ Ω, Lemma 1.1.3 shows that a ∈ L for some L ∈ L. We then define σ(a) to be the element σL (a) ∈ Ω. 14 It follows immediately from the definition of C that σ is well-defined. Moreover, we have σ ∈ G and ϕ(σ) = (σL )L∈L , as one verifies at once. This shows that C ⊆ ϕ(G) and hence ϕ(G) = C , as desired. Let (σL )L∈L be an arbitrary element in P − C , so −1 ′ that σL |L∩L′ 6= σL′ |L∩L′ for some L, L′ ∈ L. One verifies easily that p−1 L (σL ) ∩ pL′ (σL ) is an open subset of P containing (σL )L∈L and disjoint from C , hence P − C is open, i.e., ϕ(G) = C is closed in P . This shows that G is compact. Finally, it remains to show that G is totally disconnected. Let X be an arbitrary subset of G containing two elements σ, τ such that σ 6= τ , i.e., σ −1 τ 6= 1. Corollary 1.1.4 shows that there exists N ∈ B1 such that σ −1 τ 6∈ N , i.e., τ 6∈ σN . Then it is easily seen that X = (σN ∩ X) ∪ ((G − σN ) ∩ X) is a union of two disjoint, nonempty open subsets of X , hence X is not connected. Therefore, G is totally disconnected. This concludes the proof. Proposition 1.1.8. Let H be a subgroup of G. Then H = Gal(Ω/ΩH ), where H is the closure of H in G. Proof. It is obvious that H ⊆ Gal(Ω/ΩH ), so it suffices to show that Gal(Ω/ΩH ) is closed in G and Gal(Ω/ΩH ) ⊆ H . In order to show Gal(Ω/ΩH ) is closed in G, consider an arbitrary element σ ∈ G − Gal(Ω/ΩH ), i.e., σ(a) 6= a for some a ∈ ΩH . Lemma 1.1.3 shows that there exists L ∈ L such that a ∈ L. It follows that σ Gal(Ω/L) is an open neighborhood of σ satisfying σ Gal(Ω/L) ⊆ G − Gal(Ω/ΩH ), since for all τ ∈ Gal(Ω/L) we have στ (a) = σ(a) 6= a, i.e., στ 6∈ Gal(Ω/ΩH ). This shows that G − Gal(Ω/ΩH ) is open, i.e., Gal(Ω/ΩH ) is closed in G. In order to prove the inclusion Gal(Ω/ΩH ) ⊆ H , for any σ ∈ Gal(Ω/ΩH ) and open neighborhood σ Gal(Ω/L) ∈ Bσ , we need to show that σ Gal(Ω/L) ∩ H 6= ∅. Setting H0 = {τ |L | τ ∈ H}, it is obvious that H0 is a subgroup of Gal(L/K). Note that L/K is a finite Galois extension and LH0 = L ∩ ΩH , as one verifies at once; the fundamental theorem for finite Galois extensions yields that H0 = Gal(L/L ∩ ΩH ). Moreover, it is easily seen 15 that σ|L ∈ Gal(L/L ∩ ΩH ), hence σ|L = τ |L for some τ ∈ H , i.e., σ −1 τ ∈ Gal(Ω/L). Therefore, τ ∈ σ Gal(Ω/L) ∩ H , showing that σ Gal(Ω/L) ∩ H 6= ∅. This concludes the proof. We are now ready to prove the fundamental theorem of infinite Galois theory. Theorem 1.1.9. Let Ω be a Galois extension over K , with Galois group G. There is a bijection between the set of subfields E of Ω containing K and the set of closed subgroups H of G, given by E 7−! Gal(Ω/E) and H 7−! ΩH . Moreover, (1) The field E is finite over K if and only if the closed subgroup H is open in G; (2) The field E is Galois over K if and only if the closed subgroup H is normal in G. If that is the case, then there is an isomorphism of topological groups G/H ∼ = Gal(E/K), where G/H is endowed with the quotient topology. Proof. Let E be a subfield of Ω containing K , so that Ω is Galois over E and E = ΩGal(Ω/E) . Moreover, Proposition 1.1.8 shows that Gal(Ω/E) is a closed subgroup of G. Conversely, let H be a closed subgroup of G; it follows from Proposition 1.1.8 that H = Gal(Ω/ΩH ). Therefore, two maps E 7−! Gal(Ω/E) and H 7−! ΩH define a bijection between the set of subfields E of Ω containing K and the set of closed subgroups H of G. (1) Let E be a finite subextension of Ω. Lemma 1.1.3 shows that E ⊆ L for some L ∈ L; for any σ ∈ H = Gal(Ω/E), it is easily seen that the open neighborhood σ Gal(Ω/L) of σ satisfying σ Gal(Ω/L) ⊆ H , hence H is an open subgroup of G. Conversely, let H be an open subgroup of G; it follows from the definition of Krull topology that there exists L ∈ L such that Gal(Ω/L) ⊆ H . Therefore, we get E = ΩH ⊆ L, showing that E is finite over K since L is finite over K . (2) The same argument as in the proof of the fundamental theorem for finite Galois extensions shows that field E is Galois over K if and only if the closed subgroup H is normal in G. Moreover, the natural restriction ψ : G −! Gal(E, K) given by σ 7−! σ|E is a surjective group homomorphism with kernel H , hence G/H ∼ = Gal(E, K). It remains to show that the induced map ψ : G/H −! Gal(E/K) given by σH 7−! σ|E 16 is a homeomorphism. Theorem 1.1.7 shows that G is compact and Gal(E/K) is Hausdorff; it follows from the definition of the quotient topology that G/H is compact. Thus, it suffices to show that ψ is a continuous map (c.f. Theorem 17.14, page 123, [3]). For the continuity of ψ , we only need to verify the continuity of ψ (c.f. theorem 9.4, page 60, [3]). Let τ ∈ Gal(E/K) and let τ̃ ∈ G be an extension of τ . For any open neighborhood τ Gal(E/L) of τ where L is a finite Galois subextension of E , it is easily seen that ψ −1 (τ Gal(E/L)) = τ̃ Gal(Ω/L) which is an open subset of G, hence ψ is continuous, as desired. This concludes the proof. 1.2 Profinite Groups In this section, we define and study some elementary properties of inverse limits of topological groups. Then we introduce profinite groups and give some characterizations of these groups. The last part of this section is devoted to interpretations of Galois groups in terms of inverse limits. Definition 1.2.1. A directed set is a nonempty set I together with a preorder 6 such that every pair of elements of I has an upper bound, i.e., for every i, j ∈ I , there exists k ∈ I such that i, j 6 k . It is easily seen that every finite subset of a directed set I has an upper bound. Example 1.2.2. (1) Every nonempty totally ordered set is a directed set. (2) Let Ω be a Galois extension over a field K . Recall from Section 1.1 that L is the set of finite Galois subextensions of Ω. We define a preorder 6 on L as follows: for each L, L′ ∈ L, we define L 6 L′ if and only if L ⊆ L′ . It is clear that (L, 6) is directed, since for any L, L′ ∈ L, the compositum LL′ is a finite Galois extension over K , i.e., LL′ ∈ L satisfying L, L′ ⊆ LL′ , i.e., L, L′ 6 LL′ . (3) Let G be a topological group. We let N denote the set of open normal subgroups of G. We define a preorder 6 on N as follows: for each N, N ′ ∈ N, we define 17 N 6 N ′ if and only if N ′ ⊆ N . It is clear that (N, 6) is directed, since for any N, N ′ ∈ N, the intersection N ∩ N ′ is an open normal subgroup of G, i.e., N ∩ N ′ ∈ N satisfying N ∩ N ′ ⊆ N, N ′ , i.e., N, N ′ 6 N ∩ N ′ . For the rest of this section, we assume that (I, 6) is a directed set. Definition 1.2.3. An inverse system of topological groups over I is a pair (Gi , ϕij ) consisting of an indexed family of topological groups (Gi )i∈I together with a family of continuous homomorphisms ϕij : Gj ! Gi for i 6 j such that the following conditions are satisfied: (i) ϕii = 1Gi for each i ∈ I , (ii) ϕik = ϕij ◦ ϕjk for all i 6 j 6 k . Definition 1.2.4. Let (Gi , ϕij ) be an inverse system of topological groups over I . An inverse limit of the inverse system (Gi , ϕij ) is a pair (G, ϕi ) consisting of a topological group G and a family of continuous homomorphisms ϕi : G ! Gi for i ∈ I that satisfies the universal property in the following sense: (i) ϕi = ϕij ◦ ϕj for all i 6 j , (ii) For every topological group G′ and every family of continuous homomorphisms ψi : G′ ! Gi satisfying ψi = ϕij ◦ ψj for all i 6 j , there exists a unique continu- ous homomorphism ψ : G′ ! G making the following diagram commutative: G′ ψ ψj ψi G ϕj Gj ϕi ϕij Gi . The existence and the uniqueness of the inverse limit are given in the following proposition: Proposition 1.2.5. Let (Gi , ϕij ) be an inverse system of topological groups over I . Then