We also give a proof of the fundamental theorem of infinite Galoistheory and show that every Galois group is a profinite group.. 1.1 Infinite Galois Theory In this section, we define the
Infinite Galois Theory
In this section, we introduce the Krull topology on the Galois group of an infinite Galois extension, providing a topological framework essential for understanding infinite Galois theory We then utilize this topology to prove the fundamental theorem of infinite Galois theory, establishing a powerful correspondence between subgroups of the Galois group and intermediate fields To set the stage, we briefly state the fundamental theorem of Galois theory for finite Galois extensions, which serves as the foundation for extending the result to the infinite case (see Theorem 5.1, page 51, [1]).
Theorem 1.1.1 LetΩbe a finite Galois extension ofK, with Galois groupG There is a bijection between the set of subfieldsE ofΩcontainingK and the set of subgroups
H of G, given byE 7−! Gal(Ω/E)and H 7−! Ω H Moreover, E is Galois overK if and only ifH is normal inG; if that is the case, thenG/H ∼=Gal(E/K).
In the case of an infinite Galois extension \(\Omega\) over \(K\), the classical theorem does not hold, and it's possible for distinct subgroups of the Galois group \(G\) to share the same fixed field This phenomenon highlights a key difference between finite and infinite Galois extensions, where the correspondence between subgroups and fixed fields becomes more complex Therefore, unlike finite cases, subgroups of \(G\) in infinite extensions do not always uniquely determine their fixed fields, as demonstrated by specific examples.
Example 1.1.2 We let Ω denote the field obtained from Q by adjoining all square roots of prime numbers, so thatΩis an infinite Galois extension overQ It is obvious that
For every prime number p, consider the automorphism σp in Gal(Ω/Q) defined by σp(√p) = −√p and σp(√q) = √q for all other primes q ≠ p The subgroup H generated by all such automorphisms σp is a proper subset of Gal(Ω/Q), since H does not include the automorphism σ that sends each √p to −√p simultaneously This highlights that H is a proper subgroup, illustrating the structure of the Galois group and its subgroups in the context of prime-based automorphisms.
Indeed, it is obvious thatQ⊆ Ω H Conversely, for anya∈Ω H , there exists a subfield
The field \(L\) is obtained from \(\mathbb{Q}\) by adjoining square roots of finitely many prime numbers, specifically \(p_1, \dots, p_n\), and contains the element \(a\) It is straightforward to verify that \(L\) is a finite Galois extension over \(\mathbb{Q}\), with automorphisms \(\sigma_{p_1}, \dots, \sigma_{p_n}\) generating the Galois group \(\operatorname{Gal}(L/\mathbb{Q})\) Since \(a\) belongs to the Galois closure \(\Omega_H\), it follows that \(a \in L\) and the Galois group acts transitively, leading to the conclusion that \(\Omega_H \subseteq \mathbb{Q}\) Therefore, we deduce that \(\Omega_H = \mathbb{Q}\), completing the proof.
In this section, we focus on a Galois extension Ω/K, which may be infinite, and analyze its Galois group G We assign a specific topology to G, transforming it into a topological group, facilitating deeper study of its structure and properties within algebraic and number theory contexts.
We denote byLthe set of finite Galois subextensions ofΩ, and set
B 1 ={Gal(Ω/L)|L∈L} The following lemma will be useful.
Lemma 1.1.3 Every finite set of elements in Ωis contained in some field L ∈ L In particular, we have
The proof demonstrates that the first statement logically implies the second To establish this, we consider the splitting field L of the minimal polynomials of elements a₁, a₂, , aₙ over the field K Since L/K is a finite Galois extension, it follows that L belongs to the set L, thereby confirming the lemma.
The following corollary is straightforward from the above lemma.
Proof Fori = 1,2, assume that Ni = Gal(Ω/Li) for someLi ∈ L It is easily seen that
N1∩N2 =Gal(Ω/L1)∩Gal(Ω/L2) = Gal(Ω/L1L2), whereL1L2 is the compositum ofL1 andL2 in Ω Since eachLi is finite Galois over
K, we get L1L2 is finite Galois over K, i.e., L1L2 ∈ L, hence N1 ∩N2 ∈ B 1 This proves our lemma.
Let X be a topological space By a neighborhood basis at x ∈ X, we mean a collection B x of neighborhoods of x such that for each neighborhood U of x, there existsV ∈ B xsuch thatV ⊆ U For eachσ ∈G, we set
From Lemma 1.1.5, it is straightforward to check that the setB σforms a neighborhood basis atσ for a topology onG(c.f Theorem 4.5, page 33, [3]).
Definition 1.1.6 The topology on G, in which B σ is a neighborhood basis at each σ∈G, is called theKrull topology.
When G is finite, the Krull topology simplifies to a discrete topology, making the Galois group a topological group under this structure Consequently, the Krull topology provides a natural framework for analyzing infinite Galois groups, which are endowed with this topology by default From this point onward, we assume that all Galois groups are equipped with the Krull topology, facilitating deeper insights into their topological properties and behaviors.
A totally disconnected space is a topological space where all connected components are single points In other words, the space is totally disconnected if the only nonempty connected subsets are singletons This property highlights the extreme level of disconnection within the space, making it a fundamental concept in topology For a detailed discussion, see Section [insert section reference].
29, [3]) We shall now study some properties of the topological groupG.
Theorem 1.1.7 The Galois groupGis a Hausdorff, compact and totally disconnected topological group.
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 = ∅, henceGis Hausdorff since σN and τ N are open neighborhoods ofσ andτ, respectively.
The most difficult part of this proof is to show that Gis compact From the Ty- chonoff theorem (c.f Theorem 17.8, page 120, [3]), the Cartesian product
Gal(L/K) is a compact group because every finite group with a discrete topology is compact The natural group homomorphism ϕ: G → P, defined by ϕ(σ) = (σ|L) for σ ∈ G, is used to establish a topological relationship between G and its image ϕ(G) We aim to prove that G is homeomorphic to ϕ(G) and that ϕ(G) is a closed subset of P Once this is confirmed, the compactness of P implies that ϕ(G) is compact, and consequently, G is also compact.
To demonstrate that G is homeomorphic to ϕ(G), it suffices to prove that ϕ is an open map onto ϕ(G) and that ϕ is a continuous injection (see Theorem 7.9, page 46, [3]) It is evident that ϕ: G → ϕ(G) is an open map, since for any open neighborhood σGal(Ω/L) ∈ Bσ, we have ϕ(σGal(Ω/L)) = ϕ(G) ∩ Y.
The subset Gal(L'/K) × {σ | L} forms an open set within ϕ(G) To establish injectivity, we consider σ in G such that σ acts as the identity on all L ∈ L, and Lemma 1.1.3 confirms that σ must be the identity element, ensuring that ϕ is injective For continuity, we analyze the natural projection pL from P to Gal(L/K) for each L ∈ L; demonstrating that the composition pL ◦ ϕ is continuous is sufficient, as supported by Theorem 8.8 on page 55 of [3] By extending an element τ ∈ Gal(L/K) to an element ˜τ in G, it becomes straightforward to show the necessary continuity conditions.
(pL ◦ϕ) −1 ({τ}) = ˜τGal(Ω/L) which is an open subset ofG, hencepL is continuous, as desired We have shown that
In order to show thatϕ(G)is closed in P, we set
This article discusses the set C, defined as C = {(σL)L∈L ∈ P | σL|L∩L′ = σL′|L∩L′ for all L, L′ ∈ L}, and asserts that ϕ(G) equals C It is clearly shown that ϕ(G) is a subset of C, and the reverse inclusion is established by constructing a mapping σ from Ω to Ω for any element in C According to Lemma 1.1.3, every element a ∈ Ω belongs to some L ∈ L, enabling the definition of σ(a) as σL(a), ensuring the coherence of the function σ across the entire domain This characterization highlights the fundamental relationship between the set C and the image ϕ(G) in the context of the algebraic structure discussed.
The article demonstrates that the map σ is well-defined and belongs to the group G, satisfying the relation ϕ(σ) = (σ_L)_L∈L, which confirms that the image of G under ϕ is precisely the set C It further shows that any element outside C, represented by (σ_L)_L∈L, can be separated from C using open sets, implying that the complement of C in P is open Consequently, this establishes that ϕ(G) = C is a closed subset of P, leading to the conclusion that G is compact.
Gis is shown to be totally disconnected by considering any two distinct elements σ and τ in the subset X of G Because σ ≠ τ, it follows that σ^{-1}τ ≠ 1, and Corollary 1.1.4 guarantees the existence of an N in B1 such that σ^{-1}τ is not in N, implying τ is not in σN This property demonstrates that the space G is totally disconnected, confirming the key topological characteristic of the group.
X = (σN ∩X)∪((G−σN)∩X) is a union of two disjoint, nonempty open subsets of X, hence X is not connected. Therefore,Gis totally disconnected This concludes the proof.
Proposition 1.1.8 LetH be a subgroup ofG Then
H=Gal(Ω/Ω H ), whereH is the closure ofH inG.
Proof It is obvious that H ⊆ Gal(Ω/Ω H ), so it suffices to show that Gal(Ω/Ω H ) is closed inGand Gal(Ω/Ω H )⊆H.
In order to show Gal(Ω/Ω H ) is closed in G, consider an arbitrary element σ ∈
G−Gal(Ω/Ω H ), i.e.,σ(a)6or somea∈Ω H Lemma 1.1.3 shows that there exists
In the given context, for an element a in L, the set σGal(Ω/L) forms an open neighborhood of a, which is contained within G−Gal(Ω/Ω H) This is because, for all τ in Gal(Ω/L), the action στ(a) differs from a, indicating that στ is not in Gal(Ω/Ω H) Consequently, G−Gal(Ω/Ω H) is an open set, and thus Gal(Ω/Ω H) is a closed subgroup within G, highlighting important topological properties of Galois groups in the extension.
In order to prove the inclusion Gal(Ω/Ω H )⊆H, for anyσ ∈Gal(Ω/Ω H )and open neighborhoodσGal(Ω/L)∈ B σ , we need to show thatσGal(Ω/L)∩H 6=∅ Setting
Profinite Groups
This section explores fundamental properties of inverse limits of topological groups, providing essential insights into their structure We introduce profinite groups and offer key characterizations that distinguish these compact, totally disconnected topological groups Additionally, the section discusses the interpretation of Galois groups as inverse limits, highlighting their significance in modern algebra and topology.
Definition 1.2.1 Adirected setis a nonempty setI together with a preorder⩽such that every pair of elements ofI has an upper bound, i.e., for everyi, j ∈I, there exists k ∈I such thati, j ⩽k.
It is easily seen that every finite subset of a directed setI has an upper bound. Example 1.2.2.
(1) Every nonempty totally ordered set is a directed set.
Let Ω be a Galois extension over a field K The set L consists of all finite Galois subextensions of Ω A preorder ⩽ on L is defined such that for any L, L′ ∈ L, L ⩽ L′ if and only if L is contained within L′ This structure is directed because, for any pair of subextensions L and L′, their compositum LL′ is also a finite Galois extension over K that contains both L and L′, thus satisfying L, L′ ⩽ LL′.
(3) LetGbe a topological group We letNdenote the set of open normal subgroups ofG We define a preorder ⩽ on Nas follows: for eachN, N ′ ∈ N, we define
N ⩽ N ′ if and only if N ′ ⊆ N It is clear that (N,⩽) is directed, since for anyN, N ′ ∈ N, the intersectionN ∩N ′ is an open normal subgroup of G, i.e.,
For the rest of this section, we assume that(I,⩽)is a directed set.
An inverse system of topological groups over an index set I consists of a family of topological groups (Gi) indexed by I, along with continuous homomorphisms ϕij: Gj → Gi for i ≤ j These homomorphisms satisfy specific compatibility conditions, ensuring the coherence of the system This structure plays a crucial role in understanding how topological groups can be approximated or studied via inverse limits in topology and algebra.
(ii) ϕik =ϕij ◦ϕjkfor alli⩽j ⩽ k.
An inverse system of topological groups (Gi, ϕij) over an index set I consists of a family of topological groups connected by continuous homomorphisms ϕij The inverse limit of this system is a topological group G paired with continuous homomorphisms ϕi : G → Gi for each i ∈ I, which collectively satisfy the universal property This universal property ensures that G serves as the most general group factoring through all the groups in the inverse system, maintaining the structure and compatibility dictated by the homomorphisms ϕij The inverse limit provides a way to reconstruct complex topological groups by coherently linking simpler components within the inverse system.
(ii) For every topological groupG ′ and every family of continuous homomorphisms ψi:G ′ !Gisatisfyingψi=ϕij ◦ψj for alli⩽j, there exists a unique continu- ous homomorphismψ:G ′ !Gmaking the following diagram commutative:
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
(1) There exists an inverse limit of the inverse system(Gi, ϕij).
This limit is unique because, for any two inverse limits (G, ϕi) and (G′, ψi) of the inverse system (Gi, ϕij), there exists a unique isomorphism of topological groups, ψ: G′ → G, satisfying the condition ϕi ∘ ψ = ψi for all i in the index set.
Let P represent the Cartesian product of the family (G_i)_{i∈I}, equipped with the product topology This construction ensures that P forms a topological group with group operations defined coordinatewise, facilitating a comprehensive understanding of product topologies in topological group theory.
G={(σi) i∈I ∈P |ϕij(σj) =σi fori⩽j } , we haveGis a topological subgroup ofP.
Fori∈I, we define ϕi: G−!Gi to be the restriction of the natural projection P −! Gi; it is easily seen that ϕi is a continuous homomorphism satisfyingϕi=ϕij ◦ϕj for alli⩽ j.
In order to show that(G, ϕi)is an inverse limit of the inverse system(Gi, ϕij), let
G ′ be a topological group , and let(ϕ i ) i∈I be a family of continuous homomorphisms ψi: G ′ −!Gisatisfyingψi=ϕij ◦ϕj for alli⩽ j Consider a map ψ: G ′ −!G given by τ 7−!(ψ i (τ)) i∈I
It is easily seen thatψ is the unique continuous homomorphism satisfyingϕi◦ψ =ψi. This proves part (1).
(2) The proof is straightforward from the universal property of the inverse limit.
We denote the inverse limit of the inverse system (Gi, ϕij) by lim
Gi, or simply lim−Giwhen there is no ambiguity.
We next investigate some elementary properties of inverse limits.
Lemma 1.2.6 If (Gi, ϕij)is an inverse system of Hausdorff topological groups over
−Gi is a closed subgroup of the Cartesian product of the family (Gi) i∈I endowed with product topology.
Proof For simplicity of notation, we set
In order to show that lim
−Gi is closed in P, consider an arbitrary element (σi) i∈I ∈
−Gi, so that there exist i0, j0 ∈ I such that i0 ⩽ j o and ϕi 0 j 0(σj 0) 6= σi 0 Since
Gi 0 is Hausdorff, there exist two neighborhoods U ofϕi 0 j 0(σj 0)andV of σi 0 such that
Gi, it is clear that W is a open subset of P One verifies at once thatW is open neigh- borhood of (σi) I contained in P −lim
−Gi is open inP, i.e., lim−Giis closed in P This proves our lemma.
Proposition 1.2.7 Let(Gi, ϕij)be an inverse system of Hausdorff, compact topolog- ical groups overI Thenlim
−G i is also a Hausdorff, compact topological group. Proof We first set
Since Gi is Hausdorff for all i ∈ I, it follows that P is Hausdorff, hence so islim
(c.f Theorem 13.8, page 87, [3]) From the Tychonoff theorem, we haveP is compact sinceGi is compact fori∈ I Moreover, Lemma 1.2.6 shows thatlim
−Giis compact This concludes the proof.
Lemma 1.2.8 discusses an inverse system of topological groups, denoted by (Gi, ϕij), over an index set I It states that if there is a topological group G with compatible continuous surjections ψi: G → Gi for each i ∈ I, then these maps induce a continuous homomorphism ψ: G → lim Gi This result highlights how compatible surjections in an inverse system lead to a natural continuous homomorphism from a topological group G to the inverse limit of the system.
−Gi sendsGonto a dense subset oflim
Proof LetU be an open subset oflim
−Gi We need to show thatψ(G)∩ U 6=∅ From the definition of the product topology, we may assume that
In this proof, we consider an open subset \( U_{i_j} \subseteq G_{i_j} \) for \( 1 \leq j \leq n \), and select an element \( \tau = (\tau_i)_{i \in I} \in U \) Since the index set \( I \) is directed, there exists an \( i_0 \in I \) such that \( i_1, \ldots, i_n \leq i_0 \) Utilizing the surjectivity of \( \psi_{i_0} \), we find a \( \sigma \in G \) satisfying \( \psi_{i_0}(\sigma) = \tau_{i_0} \) This guarantees that \( \psi(\sigma) \in U \), and consequently \( \psi(\sigma) \in \psi(G) \cap U \), confirming that this intersection is non-empty This completes the proof of the lemma.
The following proposition will be used frequently in the sequel.
Proposition 1.2.9 states that for an inverse system of Hausdorff topological groups (Gi, ϕij) over an index set I, a compact topological group G with compatible continuous surjections ψi: G → Gi induces a continuous homomorphism ψ: G → lim G_i This result highlights the crucial relationship between compact topological groups and inverse limits in the context of Hausdorff topological groups It emphasizes that the structure of G can be fully understood through its projections onto the inverse system, preserving continuity and homomorphism properties Such a connection is fundamental in understanding the behavior of topological groups within inverse system frameworks, ensuring the induced map remains continuous and compatible with the group structure, which is essential for advanced studies in topological group theory and related fields.
−Gi is a closed map and a surjection.
To demonstrate that ψ is a closed map, we begin by considering an arbitrary closed subset F of G Due to the compactness of G, the subset F is also compact, which implies that the image ψ(F) is compact as well Building on the proof of Proposition 1.2.7, we further analyze the limit properties to establish that ψ maps closed sets to closed sets, confirming its status as a closed map in the topological context.
−Gi is Hausdorff, hence ψ(F) is a closed subset oflim
−Gi, showing thatψis a closed map In particular,ψ(G)is a closed subset oflim
For the surjectivity of ψ, Lemma 1.2.8 shows that ψ sends G to a dense subset of lim
−Gi since ψ(G) is closed in lim
−Gi, showing thatψ is surjective This concludes our proposition.
Profinite groups are an important concept in modern algebra, originating from the study of inverse limits of finite groups A finite discrete group is simply a finite group equipped with the discrete topology, which makes it a topological group by default Understanding finite discrete groups is essential, as they serve as the building blocks for profinite groups, which are defined as projective limits of these finite groups These structures play a crucial role in various areas of mathematics, including number theory and algebraic geometry.
Definition 1.2.10 A topological group is called a profinite groupif it is the inverse limit of some inverse system of finite discrete groups.
(1) Every finite discrete group is a profinite group.
For every prime number p, the additive group of p-adic integers, Z_p, is a profinite group, highlighting its rich algebraic structure This is demonstrated through the natural projections π_nm: Z/p^m Z → Z/p^n Z for n ≤ m, which establish an inverse system of finite discrete groups over the natural numbers These projections showcase how Z_p can be viewed as an inverse limit of finite cyclic groups, emphasizing its importance in number theory and topological algebra.
(3) The Pr ¨ufer group Zb is a profinite group More precisely, we define an order
In the given mathematical framework, the partial order on the set Z is defined by the divisibility relation: for all m, n ∈ Z, n ⩽ m if and only if n divides m This makes the set (Z, ⩽) a directed set, enabling the construction of inverse systems For each pair m, n ∈ Z with n ⩽ m, a natural projection map π_nm: Z/mZ → Z/nZ is considered These projection maps form an inverse system of finite discrete groups over Z, illustrating the hierarchical structure of quotient groups linked through divisibility relations.
The following theorem provides some useful characterizations of profinite groups.
Theorem 1.2.12 Let G be a topological group Then the following conditions are equivalent:
(2) Gis Hausdorff, compact and totally disconnected.
(3) Gis Hausdorff, compact and the setNof all open normal subgroups ofGforms a neighborhood basis of the identity element ofG.
To prove this theorem, we start by reviewing a key characterization of compactness A family of sets, denoted as \(A\), has the finite intersection property if and only if the intersection of any finite subfamily of \(A\) is nonempty This concept is essential in understanding compactness, and we reference a relevant theorem (see Theorem 17.4, page 118, [3]) without providing its proof.
Theorem 1.2.13 For a topological spaceX, the following are equivalent:
(2) Each familyAof closed subsets ofXwith finite intersection property has nonempty intersection.
By aclopensubset of a topological space, we mean a subset which is both closed and open.
Lemma 1.2.14 Let X be a Hausdorff, compact space Let x ∈ X and let (U t ) t∈T is the family of all clopen neighborhoods ofx Then the intersection
Proof For simplicity of notation, we set
We shall prove by contradiction Suppose that F and K are two nonempty closed subsets of A such that A = F ∪K and F ∩K = ∅, so that F and K are closed in
X sinceA is closed in X It follows thatF andK are compact since X is compact. Moreover, sinceXis Hausdorff, there exists two open subsetsU andV ofX such that
Note thatX−(U ∪ V)is closed in X Theorem 1.2.13 shows that there exists a finite subfamilyT ′ ofT such that
Ut, it is clear thatW is a clopen subset ofX sinceT ′ is finite.
Without loss of generality, assume thatx ∈ U It is obvious thatW ∩ U is open in
X Moreover, it is easily seen that
W ∩ U = [X−(W ∩ V)]∩ W, hence W ∩ U is closed since W ∩ V is open in X, showing that W ∩ U is a clopen neighborhood ofx We then getA ⊆ W ∩ U ⊆ U, henceK ⊆A∩ V =∅, contrary to the hypothesis thatK is nonempty This concludes the proof.
We are now ready to prove Theorem 1.2.12.
Suppose G is the inverse limit of an inverse system of finite discrete groups; since finite discrete groups are Hausdorff, compact, and totally disconnected, G inherits these properties Specifically, Proposition 1.2.7 confirms that G is Hausdorff and compact, while the total disconnectedness of each Gᵢ ensures that their Cartesian product remains totally disconnected, thereby making G itself totally disconnected (see Theorem 29.3, p 210, [3]).
(2)⇒ (3) LetV be an open neighborhood of 1 We need to show thatV contains an open normal subgroup ofG Let(Ut)t∈T be the family of clopen neighborhoods of
1 SinceGis totally disconnected, Lemma 1.2.14 shows that
Note that G− V is closed in G; it follows from Theorem 1.2.13 that there exists a finite subfamilyT ′ ofT such that
Ut, it is clear thatU is a clopen subset ofGcontained inV sinceT ′ is finite.
We shall prove thatV contains an open subgroup ofG Set
Cohomology Groups
The aim of this section is to introduce the notion of G-modules, and define coho- mology groups by using homogeneous cochain complex and inhomegeneous cochain complex.
(1) AnabstractG-moduleis an abelian groupAtogether with an action
G×A−!A denoted by (σ, a)7−!σa such that σ(a+b) = σa+σb forσ ∈Ganda, b∈A.
(2) A discrete G-module(or simplyG-modulewhen that does not lead to confu- sion) is an abstractG-moduleAsuch that the actionG×A −!Ais continuous, whereAis endowed with the discrete topology.
Let A be an abstract G-module For each a ∈ A, we denote by StabG(a) the stabilizer ofa, meaning that
For each subgroupH ofG, we denote byA H the subgroup of fixed points of Aunder the action ofH, meaning that
A H ={a∈A|τ a=a,∀τ ∈H}. The following proposition will be used frequently in the sequel.
Proposition 2.1.2 LetAbe an abstractG-module Then the following conditions are equivalent:
(1) The actionG×A−!Ais continuous, i.e.,Ais aG-module.
(2) For eacha∈A, the stabilizerStab G (a)is an open subgroup ofG.
A U , whereU run through the set of all open subgroups ofG.
Consider an arbitrary element σ in the stabilizer subgroup StabG(a), with the property that σa = a Since the action map from G × A to A is continuous, there exists an open neighborhood U of σ within G such that τa = σa for all τ in U This implies that U is contained in StabG(a), making the stabilizer subgroup an open subgroup of G.
(2)⇒(3) It is straightforward since a∈A Stab G (a) for alla∈A.
For any pair (σ, a) in G × A, there exists an open subgroup U of G such that a ∈ U; consequently, the set σU × {a} forms an open neighborhood of (σ, a) Any point (τ, a) in this neighborhood satisfies τ = σu for some u ∈ U, which ensures τa = σ(ua) = σa, demonstrating that the G × A action on A is continuous This completes the proof.
Remark 2.1.3 highlights that the proof of the proposition does not rely on the abelian group structure of A, indicating that the result applies more broadly Specifically, the proposition remains valid for any discrete topological space on which a group G acts, extending its applicability beyond abelian groups This insight emphasizes the generality of the proposition within topological and group-action contexts.
(1) Every abelian groupAon whichGacts trivially is aG-module.
(2) Let Ω/K be a Galois extension with Galois group G Recall from Section 1.2 that G is a profinite group Suppose that G acts on Ω, regarded as an additive group, by the following action: σa=σ(a) forσ∈Ganda∈A.
Ω is a G-module because, for any element a ∈ Ω, the subextension K(a) is finite, and Theorem 1.1.9 confirms that the Galois group Gal(Ω/K(a)) is an open subgroup of G, which ensures that a is contained within Ω Gal(Ω/K(a)) Consequently, applying Proposition 2.1.2 guarantees that Ω is indeed a G-module, highlighting its structured algebraic properties within Galois theory.
Under the above action, one verifies at once that the multiplicative group Ω × and the group of allnth roots of unity inΩare alsoG-modules.
Let \(A\) be a \(G\)-module and \(H\) a closed subgroup of \(G\) Since \(H\) is a profinite group, the induced action of \(G\) on \(A\) naturally restricts to \(H\), making \(A\) an \(H\)-module This relationship highlights the structural interplay between \(G\)-modules and their subgroups, providing a foundational concept in the study of profinite groups and module theory.
Proposition 2.1.2.(2) shows that Stab G (a) is an open subgroup of G, henceStabH(a) is an open subgroup of H, i.e., A is a H-module by using Proposi- tion 2.1.2.(2) again.
(4) LetAbe aG-module, and letH be a closed normal subgroup ofG Recall from Proposition 1.2.18 thatG/H is a profinite group Suppose thatG/H acts onA H by the following action:
ThenA H is aG/H-module Indeed, letπ: G−!G/Hbe the natural projection; for eacha∈A H , one verifies easily that π −1 (Stab G/H (a)) = StabG(a).
Proposition 2.1.2.(2) shows that StabG(a) is an open subgroup of G, hence Stab G/H (a) is an open subgroup of G/H, i.e., A H is a G/H-module by using Proposition 2.1.2.(2) again.
(5) Let A be a G-module, and let ϕ: G ′ −! G is a continuous homomorphism of profinite groups Suppose thatG ′ acts onAby the following action: σ ′ a=ϕ(σ ′ )a forσ ′ ∈G ′ anda ∈A.
A is a G′-module because, for any a in A, there exists an open subgroup U of G such that a is in A U, as established in Proposition 2.1.2(3) This implies that ϕ⁻¹(U) is an open subgroup of G′ containing a, ensuring that A is equipped with a G′-module structure through the application of Proposition 2.1.2(3) repeatedly.
Definition 2.1.5 LetAandB beG-modules A G-homomorphismf: A−!B is a group homomorphism such that f(σa) =σf(a) forσ ∈Ganda∈A.
AG-isomorphismis aG-homomorphism which is also a bijection.
The class of G-modules and G-homomorphisms constitutes an abelian category that we denote by Mod(G) For the rest of this section, we assume that A is a G- module.
In our study, we define \( X_n \) as the set of all continuous maps \( \alpha: G^{n+1} \to A \) for \( n \geq 0 \), which naturally forms an abelian group under the standard addition operation A key concept introduced is that of a locally constant map: a function \( f: X \to S \) from a topological space \( X \) to a set \( S \), where \( f \) remains constant within some neighborhood around each point \( x \in X \) This property of locally constant maps plays an essential role in the analysis, as highlighted by the subsequent lemma that leverages this concept.
Lemma 2.1.6 states that every continuous map \(\alpha: G^{n+1} \to A\) is a locally constant function that takes only finitely many values in \(A\) when \(n \geq 0\) Furthermore, there exists an open normal subgroup \(N\) of \(G\) such that \(\alpha\) remains constant on the cosets of \(N^{n+1}\) within \(G^{n+1}\) This highlights the locally constant nature of continuous maps in this context and their stabilization on certain subgroup cosets, which is essential for understanding their algebraic and topological properties.
Since A is discrete, each σ in G^{n+1} has a neighborhood U_σ where α remains constant, making α a locally constant map Consequently, the collection {U_σ | σ ∈ G^{n+1}} forms an open cover of G^{n+1} By Tychonoff's theorem, G^{n+1} is compact because G is compact, ensuring the existence of a finite subcover {U_τ | τ ∈ T} with T finite This implies that α takes only finitely many values in A.
To establish the second statement, consider that the collection of all open normal subgroups of G constitutes a neighborhood basis at the identity element According to the definition of the product topology, for each τ = (τ₀, , τₙ) ∈ T, it can be assumed that
U τ =τ0N0,τ ì ã ã ã ìτnNn,τ, whereNi,τ is an open normal subgroup ofGfor0⩽i⩽n Setting
Since T is finite, it follows that N is an open normal subgroup of G The key claim is that the function α remains constant on the cosets of N^{n+1} in G^{n+1} Specifically, for any coset σN^{n+1} with σ = (σ_0, , σ_n) in G^{n+1}, there exists τ = (τ_0, , τ_n) in T such that σ belongs to U_τ, meaning each σ_i is in τ_i Ni,τ By the definition of N, the inclusions σ_i N ⊆ σ_i Ni,τ ⊆ τ_i Ni,τ hold for all i, leading to the conclusion that σ N^{n+1} is contained within U_τ Consequently, α is constant on each coset of N^{n+1} in G^{n+1}.
For the sequel, we assume thatGacts onX n by the following formula:
Proposition 2.1.7 X n is aG-module forn⩾0
Proof Consider an arbitrary element α ∈ X n Lemma 2.1.6 shows that there exists an open normal subgroup N of G such that α is constant on the cosets of N n+1 in
In the context of the group \(G_{n+1}\), \(N_{n+1}\) is an open normal subgroup, making \(G_{n+1}\) a compact group with finitely many cosets of \(N_{n+1}\) A finite set \(T\) of representatives for these cosets is selected within \(G_{n+1}\), with each element \(\tau = (\tau_0, \ldots, \tau_n)\) in \(T\) satisfying \(\tau N_{n+1} = \tau_0 N \times \ldots \times \tau_n N\) This structure highlights the key relationship between the subgroup \(N_{n+1}\) and the quotient group \(G_{n+1}/N_{n+1}\), emphasizing the finiteness and the role of representatives in understanding the group's topology and algebraic properties.
Proposition 2.1.2.(2) establishes that Stab G (α(τ)) is an open subgroup of G for each τ in T, indicating that U is also an open subgroup of G since T is finite This leads to the conclusion that if α belongs to (X_n) U, then, by Proposition 2.1.2.(3), X_n is a G-module.
Indeed, let σ ∈ U For any (σ0, , σn) ∈ G n+1 , there existsτ = (τ0, , τn) ∈ T such that(σ0, , σn)∈τ N n+1 , i.e.,σi∈τiN for0⩽i⩽n From the definition ofU, we haveσ −1 ∈τiN τ i −1 for0⩽i⩽n, henceσ −1 σi∈τiN for0⩽ i⩽ n, i.e.,
(σα)(σ0 , σn) =σα(σ −1 σ0 , σ −1 σn) = α(σ −1 σ0 , σ −1 σn) =α(σ0 , σn) since α is constant on τ N n+1 , hence σα = α, showing that α ∈ (X n ) U , as desired. This concludes the proof.
Forn = 0, we define a map∂ 0 : A−!X 0 which carriesa ∈Ato the constant map α ∈X 0 given by α(σ) = a, and for n ⩾ 1, we define a map ∂ n : X n−1 −!X n which carriesα ∈X n−1 to∂ n α∈X n given by
The expression (−1)ᶦ α(σ₀, , σ̂ᶦ, , σₙ) demonstrates how the omission of the ith coordinate (indicated by the hat symbol) affects the overall form It is straightforward to observe that the boundary operator ∂ₙ functions as a G-homomorphism for all n ≥ 0 These G-homomorphisms are known as coboundary operators and are commonly denoted by ∂, provided there is no risk of confusion.
Proposition 2.1.8 The following sequence is exact
Proof We first show that the above sequence is a complex, i.e.,∂ n ◦∂ n−1 = 0 It is obvious that∂ 1 ◦∂ 0 = 0 For anyα∈X n−2 , we have
We thus obtain summands of the formα(σ0, ,σˆi, ,σˆj, , σn)with certain signs. Each summand arises twice: once of the form
(−1) i+j α(σ0, ,σˆi, ,σˆj, , σn) when firstσj and thenσiis omitted, and again of the from
(−1) i+j−1 α(σ0, ,σˆi, ,σˆj, , σn) when firstσiand thenσj is omitted Thus all summands cancel to give zero, showing that∂ n ◦∂ n−1 = 0for alln ⩾1.
In order to prove the exactness, we construct a chain homotopy as follows: for n= 0, we define a mapD 0 : X 0 −!Agiven byD 0 α =α(1), and forn ⩾1, we define a mapD n+1 : X n+1 −!X n which carriesα∈X n+1 toD n+1 α ∈X n given by
Then we obtain the following diagram:
Note that D n are group homomorphisms, but notG-homomorphisms in general An easy calculation shows that
If α ∈ ker(∂ n+1 ), then α = ∂ n (D n α) ∈ im(∂ n ), hence ker(∂ n+1 ) ⊆ im(∂ n ), showing that ker(∂ n+1 ) = im(∂ n ) since ∂ n ◦ ∂ n−1 = 0 for all n ⩾ 1 This concludes the proof.
The exact sequence in the above proposition is called thestandard resolution of
For every G-homomorphism f: A −! B, there is an induced homomorphism
A G −!B G , denoted by the same letterf We thus obtain a functor
− G : Mod(G)−!Ab given by A7−!A G which is left exact, as one verifies at once.
We now apply the above functor to the standard resolution For eachn⩾0, we set
C n (G, A) = X n (G, A) G , meaning thatC n (G, A)consists of all continuous mapsα: G n+1 −!Asuch that σα(σ0, , σn) = α(σσ0, , σσn) for allσ ∈G.
−!C 2 (G, A)−!ã ã ã which is no longer exact in general This complex is called thehomogeneous cochain complexofGwith coefficient inA Set
The elements ofC n (G, A), Z n (G, A)andB n (G, A)are calledhomogeneousn-cochains, n-cocycles andn-coboundaries, respectively.
Definition 2.1.9 Thenth cohomology group of the homogeneous cochain complex
H n (G, A) =Z n (G, A)/B n (G, A) is called thenth cohomology group ofGwith coefficients in A.
For computational purposes, using inhomogeneous cochains is more convenient, with the definition that for n = 0, C₀(G, A) = A, and for n ≥ 1, Cₙ(G, A) represents the set of all continuous maps β: Gⁿ → A These cochain groups, Cₙ(G, A), form abelian groups under pointwise addition This structure allows us to construct a cochain complex that facilitates the study of group cohomology.
∂ n+1 : C n (G, A)−!C n+1 (G, A), which is called theinhomogeneous coboundary operator, is given by
Functoriality
In this section, we investigate the functorial property of cohomology groups and construct the cohomology exact sequence The main result of this section is Theorem 2.2.13.
A continuous homomorphism \( \varphi: G' \to G \) between profinite groups is key to understanding compatible module structures When considering modules \(A\) over \(G\) and \(A'\) over \(G'\), a group homomorphism \(f: A \to A'\) is said to be compatible with \(\varphi\) if, upon viewing \(A\) as a \(G'\)-module via \(\varphi\), \(f\) functions as a \(G'\)-homomorphism This compatibility condition is formalized by the relation \(f(\varphi(\sigma')a) = \sigma' f(a)\) for all \(\sigma' \in G'\) and \(a \in A\), ensuring that the actions of the groups and the homomorphism \(f\) are coherent.
(1) EveryG-homomorphism is compatible with the identity homomorphism1 G
(2) Let Ωbe a Galois extension over a field K, and letLbe a Galois subsextension of Ω Recall from Example 2.1.4.(2) that Ω and L are Gal(Ω/K)-module and Gal(L/K)-module, respectively Then the natural restriction
Gal(Ω/K)−!Gal(L/K) and the natural inclusionL−!Ωare compatible.
(3) LetAbe aG-module, and letHbe a closed subgroup ofG Recall from Example 2.1.4.(3) that A is a H-module Then the natural inclusion H −! G and the identity1A are compatible.
(4) Let A be a G-module, and let H be a closed normal subgroup of G Recall from Example 2.1.4.(4) thatA H is aG/H-module Then the natural projection
G−!G/H and the natural inclusionA H −!A are compatible.
For each pair of compatible homomorphismsϕ:G ′ −!Gandf: A−!A ′ , there is a homomorphism
This article discusses a homomorphism between cochain complexes defined by the formula C n (G, A) → C n (G′, A′), expressed as α7−!f ◦α◦ϕ n The map ϕ n acts on tuples by applying ϕ to each element, ensuring consistency across dimensions An important result is that these homomorphisms commute with the coboundary operator ∂, demonstrating their compatibility within the complex structure Consequently, these properties allow the formation of a homomorphism between the cohomology groups, preserving the algebraic invariants under the transformation.
A category can be formally defined where objects are pairs (G, A), with A being a G-module, and morphisms are pairs (ϕ, f) of compatible homomorphisms The induced homomorphisms (ϕ, f)*: Hⁿ(G, A) → Hⁿ(G′, A′) behave functorially; specifically, for morphisms (ϕ, f) from (G, A) to (G′, A′) and (ψ, g) from (G′, A′) to (G′′, A′′), the composition (ϕ ◦ ψ, g ◦ f) serves as a morphism from (G, A) to (G′′, A′′), ensuring the coherence of the functorial behavior in group cohomology.
(ϕ◦ψ, g◦f)∗ = (ψ, g)∗◦(ϕ, f)∗. Moreover, if(ϕ, f)is the identity, then so is(ϕ, f)∗
We shall now introduce some special homomorphisms of cohomology groups.
In the context of G-modules, when considering a closed subgroup H of G, the natural inclusion H → G and the identity on A are compatible, allowing us to define a restriction homomorphism res G H : H n (G, A) → H n (H, A) for all n ≥ 0 This restriction map, often simply called res, plays a crucial role in relating the cohomology of the group G to that of its subgroup H.
In terms of cochains these homomorphisms can be described as follows: letcbe a cohomology class inH n (G, A), and let α be a cocycle representingc; thenβ = α|H n is a cocycle representing res G H (c)∈H n (H, A).
The following proposition is straightforward from Remark 2.2.3.
Proposition 2.2.4 LetAbe aG-module, and letH, F be closed subgroups ofGsuch thatF ⊆H Then res H F ◦res G H =res G F
Inflation is a key concept in cohomology: for a G-module A and a closed normal subgroup H of G, the natural projection from G to G/H and the inclusion from A^H to A induce a homomorphism between their cohomology groups, known as the inflation map This inflation map, denoted as inf_G/H^G, connects H^n(G/H, A^H) to H^n(G, A) for all n ≥ 0, highlighting its importance in transferring cohomological information across group extensions.
In the context of cochains, these homomorphisms are described by considering a cohomology class c in Hⁿ(G/H, Aᴴ), with α as its representing cocycle A continuous map β: Gⁿ → A is defined by β(σ₁, , σₙ) = α(σ₁H, , σₙH), which serves as a cocycle representing the image of c under the inflation map from G/H to G.
The following proposition is straightforward from Remark 2.2.3.
Proposition 2.2.5 LetAbe aG-module, and letH, F be closed normal subgroups of
Gsuch thatF ⊆H Then inf G/F G ◦inf G/H G/F =inf G/H G
Before giving the main result of this section, let us introduce the notion of direct limit in the categoryAb We assume that(I,⩽)is a directed set in the sequel.
A direct system of abelian groups over an index set I consists of an indexed family of abelian groups (Ai) and a corresponding family of group homomorphisms (fij: Ai → Aj) for i ≤ j This system is defined such that the homomorphisms satisfy specific conditions, ensuring the coherence and structure of the system Understanding these foundational elements is essential for studying advanced algebraic concepts and their applications in mathematical research.
(ii) f ik =f jk ◦fij for alli⩽j ⩽k.
A direct limit of a direct system of abelian groups (Ai, fij) over an index set I is defined as a pair (A, fi), where A is an abelian group and each fi: Ai → A is a group homomorphism This construction satisfies a universal property: for any other abelian group B with homomorphisms from each Ai, there exists a unique homomorphism from A to B that makes all the necessary diagrams commute Understanding direct limits in abelian groups is fundamental in algebra and homological algebra, providing a way to assemble and analyze larger structures from directed systems.
(i) fi =fj◦fij for alli⩽j,
(ii) For any abelian group B and any family of group homomorphismsgi: Ai ! B satisfyinggi=gj ◦fij for alli⩽j, there exists a unique group homomorphism g: A!B making the following diagram commute:
The existence and the uniqueness of the direct limit in the categoryAbis given in the following proposition:
Proposition 2.2.8 Let(Ai, fij)be a direct system of abelian groups overI Then
(1) There exists a direct limit of the direct system(Ai, fij).
This limit is unique in its properties: if (A, fi) and (B, gi) are two limits of the direct system (Ai, fij), then there exists a unique isomorphism g: A → B such that g ◦ fi = gi for all i in the index set I.
We define the set A as the disjoint union of the family (A_i)_{i∈I} and establish an equivalence relation on A: for elements a_i ∈ A_i and a_j ∈ A_j, we say that a_i is equivalent to a_j (a_i ∼ a_j) if there exists an index k in I such that i, j ≤ k and the images under the connecting functions satisfy f_{ik}(a_i) = f_{jk}(a_j) This equivalence relation leads to the formation of a quotient set that consolidates the structure across the family of sets, laying the foundation for constructing direct limits in category theory.
An abelian group \(A = \varinjlim A_i\) is formed under a specific operation where, for each \(a, b \in A\), representatives \(a_i \in A_i\) and \(b_j \in A_j\) are chosen Because the index set \(I\) is directed, there exists a \(k \in I\) such that \(i, j \leq k\), allowing us to define the sum \(a + b\) as the equivalence class of \(\varphi_{ik}(a_i) + \varphi_{jk}(b_j)\) in \(A_k\) The identity element of \(A\) is represented by the equivalence class of the identity elements in the groups \(A_i\), ensuring the group structure is well-defined and consistent across the directed system.
For each element i in the index set I, we define a map fi: Ai → A, which assigns every element ai in Ai to its corresponding equivalence class in A This map fi is a group homomorphism and satisfies the compatibility condition fi = fj ◦ fij for all i ≤ j, ensuring a coherent structure within the group theory framework.
To demonstrate that (A, fi) is a direct limit of the direct system (Ai, fij), we consider an abelian group B with a family of group homomorphisms gi: Ai → B satisfying the compatibility condition gi = gj ◦ fij for all i ≤ j We then define a map g: A → B by assigning to each a ∈ A the value g(a), which is given by evaluating the homomorphism gi at a representative ai ∈ Ai This construction ensures that g(a) is well-defined and independent of the chosen representative, as any alternative representative leads to the same value due to the compatibility conditions, confirming that (A, fi) indeed serves as the direct limit of the system.
Moreover, one verifies easily that g is the unique group homomorphism satisfying g◦fi =gifor alli∈I This proves (1).
(2) The proof is straightforward from the universal property of direct limit.
We denote the direct limit of the direct system(Ai, fij)by lim
−!Ai when there is no ambiguity.
In Example 2.2.9, let A be a G-module, and recall that the set N of all open normal subgroups of G, together with the preorder ⩽, forms a directed set For each pair of subgroups N ⩽ N′, with N′ ⊆ N, there is a natural inclusion map f_N_N′: A_N → A_N′ These inclusions create a direct system of abelian groups indexed by N, and the inverse limit lim−→_N A_N captures the structure of the G-module in this context.
A N , where the corresponding canonical homomorphisms are natural inclusions.
A morphism of direct systems of abelian groups, as defined in Definition 2.2.10, consists of a family of homomorphisms \( s_i: A_i \to B_i \) for each index \( i \in I \) This family must satisfy the compatibility condition \( s_j \circ f_{ij} = g_{ij} \circ s_i \) for all \( i \leq j \), ensuring the structure-preserving nature of the morphism across the directed systems.
On verifies immediately that the class of direct systems of abelian groups over I and morphisms of direct systems constitutes an abelian category.
−!Bi, gi)are direct limits of direct systems(Ai, fij) and(Bi, gij), respectively For each morphism(si) i∈I from(Ai, fij)to(Bi, gij), we get an abelian grouplim
−!B i and a family of group homomorphismsg i ◦s i : A i −! lim
Universal Delta-functors
This section introduces the concept of universal δ-functors, a key tool in homological algebra, building on the previously discussed category Mod(G) of G-modules and the additive functors Hⁿ(G,−) from Mod(G) to Ab We demonstrate that the family of functors Hⁿ(G,−) for n ≥ 0 forms a universal δ-functor, which plays a crucial role in deriving important results Throughout, the discussion assumes that A and B are abelian categories, ensuring the proper framework for these constructions.
Definition 2.3.1 A cohomologicalδ-functor (or simply δ-functor when that does not lead to confusion) from A to B is a family F of covariant additive functors
F n : A−!Bforn⩾0, and to each short exact sequence
0−!A−!B −!C −!0 inAan associated family of connecting morphisms δ n : F n (C)−!F n+1 (A) forn⩾0, satisfying the following conditions:
(i) For each short exact sequence as above, there is a long exact sequence
−!F n+1 (A)−!ã ã ã , (ii) For every commutative diagram
0 A ′ B ′ C ′ 0 inAwith exact rows, the following diagram is commutative forn⩾0
We denote the connecting morphism simply byδ when that does not lead to con- fusion.
Example 2.3.2 Theorem 2.2.16 and Proposition 2.2.20 show that the family
H(G,−) = (H n (G,−))n ⩾ 0 is aδ-functor from the categoryMod(G)to the categoryAb.
Definition 2.3.3 LetF andGbeδ-functors fromAtoB AmorphismfromF toG is a family η of natural transformations η n : F n −! G n forn ⩾ 0 such that for each short exact sequence
0−!A−!B −!C −!0 inA, the following diagram is commutative forn⩾0
One verifies easily that the class of δ-functors from A to B and morphisms of δ-functors constitutes a category.
A δ-functor F from category A to category B is considered universal if it satisfies a specific universal property Specifically, for any other δ-functor G from A to B and any natural transformation η₀: F₀ → G₀, there exists a unique extension of η₀ to a morphism of δ-functors This means that there is a unique family of natural transformations ηₙ: Fₙ → Gₙ for all n ≥ 0, which collectively form a morphism of δ-functors This universality property characterizes the δ-functor as the initial or most canonical among all δ-functors with similar properties.
The following proposition is straightforward from the above definition.
Proposition 2.3.5 LetF, Gbe universalδ-functors, and letH be aδ-functor Let η 0 : F 0 −!G 0 , à 0 : G 0 −!H 0 , ρ 0 :F 0 −!H 0 be natural transformations, and let η: F −!G, à: G−!H, ρ: F −!H be their extensions to morphisms ofδ-functors.
(1) Ifη 0 is the identity, then so isη.
(3) Ifη 0 is a natural isomorphism, thenηis an isomorphism.
Definition 2.3.6 Let F be a δ-functor from A to B We say that F is erasable by a subclass M of objects in A if for each A in A, there exists MA ∈ M and a momomorphism ε A :A −!M A such thatF n (M A ) = 0for alln >0.
In our discussion, we focus on δ-functors valued in the category Ab of abelian groups A key result is a theorem that provides a valuable characterization of universal δ-functors, highlighting their fundamental properties and significance in homological algebra.
Theorem 2.3.7 Let F be aδ-functor from A to Ab If F is erasable by the class of injectives of A , thenF is universal.
Proof Since F is erasable by the class of injectives of A, for each object A ∈ A, there exists an injective object MA and a monomorphism εA: A −! MA such that
F n (MA) = 0for alln >0, hence we obtain a short exact sequence
0−!A−! ε A M A −!X A −!0, (2.5) whereXA =coker(εA) LetGbe aδ-functor fromAtoB, and letη 0 :F 0 −!G 0 be a natural transformation We need to show that there exists a unique familyηof natural transformationsη n : F n −!G n forn⩾0, which is a morphism ofδ-functors.
To establish the existence of η, we begin by constructing natural transformations η_i : F_i → G_i for 0 ≤ i ≤ n−1 that commute with δ in each short exact sequence Since η_{n−1} is a natural transformation, it induces a commutative diagram for any object A in A, derived from the short exact sequence (2.5) This ensures the coherence and naturality of η across the entire sequence, facilitating the extension of the transformation to η_n.
The surjectivity of δ n−1 F is established based on the assumption that F n (MA) ≠ 0 for all n > 0 Additionally, the commutativity of the left square implies that the kernel of δ F n−1 is contained within the kernel of the composition δ n−1 G ◦ η n−1 X, highlighting important structural properties of the mappings involved. -**Sponsor**Struggling to rewrite your article while keeping SEO in mind? It's tough! With [Article Generation](https://pollinations.ai/redirect-nexad/AYMlWtuS?user_id=983577), you can instantly create 2,000-word, SEO-optimized articles Imagine saving over $2,500 a month compared to hiring a writer, all while effortlessly generating meaningful content from complex source material It's like having a dedicated content team without the hassle!
A unique homomorphism η_A^n: F_n(A) → G_n(A) exists, ensuring the commutativity of the right square To complete the proof, it is necessary to demonstrate that the family of homomorphisms η_n: F_n(A) → G_n(A) forms a natural transformation that commutes with δ for each short exact sequence When this is established, induction guarantees the extension of η_0 to a morphism of δ-functors.
In order to show thatη n is a natural transformation, letf:A −!B be a morphism inA, so that we obtain a commutative diagram
0 B M B X B 0 ε A f m x ε B with exact rows The existence ofmfollows from the assumption thatM B is injective. The existence of x follows from the commutativity of the left square Consider the following diagram:
To demonstrate that the right square is commutative, we rely on the fact that the top and bottom squares are inherently commutative by the definition of δ-functors Additionally, the front and back squares follow directly from the properties of the natural transformation ηₙ The left square's commutativity is guaranteed by ηₙ₋₁ being a natural transformation A straightforward diagram chase confirms the overall commutativity of the entire diagram, ensuring the desired property holds.
In order to show thatη n commutes withδ, let
0−!A−!B −!C −!0 be a short exact sequence inA, so that we obtain a commutative diagram
1 A b c ε A with exact rows The existence ofbfollows from the assumption thatM A is injective.The existence of c follows from the commutativity of the left square Consider the following diagram:
To demonstrate that the right square is commutative, we leverage the fact that the top and bottom triangles are inherently commutative by the definition of δ-functors Additionally, the left square's commutativity follows from ηₙ₋₁ being a natural transformation, ensuring consistency across morphisms The front square's commutativity is established directly by the definition of ηₙ Finally, a straightforward diagram chase confirms that the right square is indeed commutative, completing the proof.
We have shown that there is an extensionηofη 0 to a morphism ofδ-functors.
For the uniqueness of η, let à be another extension of η 0 to a morphism of δ- functors Suppose thatη n−1 =à n−1 For any objectAinA, we obtain from the short exact sequence (2.5) a commutative diagram
The proof demonstrates that there exists a unique homomorphism \(F_n(A) \to G_n(A)\) that makes the right square commute, establishing the equality \(\eta_A^n = \alpha_A^n\) Consequently, this shows that \(\eta^n = \alpha^n\), and by induction, it concludes that \(\eta = \alpha\).
Using the above theorem, we shall prove that the δ-functor H(G,−) in Example 2.3.2 is universal To do so, we need some auxiliary results.
Before presenting the next proposition, it is essential to clarify some key notations Let G₀ be an abstract group, and denote its group ring over the integers by Z[G₀] Additionally, we consider Mod(G₀) as the category of Z[G₀]-modules, providing a foundational framework for the subsequent mathematical analysis.
Proposition 2.3.8 The categoryMod(G)has enough injectives.
Proof LetAbe an object inMod(G) We need to show that there exists an injective
G-module MA and a monomorphism εA: A −! MA of G-modules Let G0 be the abstract group underlyingG, so thatA∈Mod(G 0 ) It is well-known that the category
Mod(G0)has enough injectives (c.f Theorem 4.1, page 784, [12]), hence there exists an injectiveZ[G 0 ]-moduleM and a monomorphismεA: A−!M inMod(G 0 ) Set
In the study of group theory, the subset MA, where U runs through all open subgroups of G, is verified to be a subgroup of M, with an induced action of G0 on MA This structure demonstrates that MA is an abstract subgroup, highlighting its significance in understanding subgroup interactions and group actions within the broader context of algebraic structures.
G0-module Moreover, it is easily seen that
M A U , whereU run through all open subgroups ofG, hence Proposition 2.1.2.(3) shows that
M A ∈ Mod(G) It remains to show that M A is an injective G-module andε A (A) ⊆
M A If that is the case, thenε A induces a monomorphism A −! M A of G-modules, as desired.
To demonstrate that MA is an injective G-module, consider a monomorphism f: B → C of G-modules and a G-homomorphism g: B → MA Since f is a monomorphism in Mod(G0) and g is a Z[G0]-linear map, we analyze the corresponding commutative diagram to establish the injectivity of MA as a G-module.
Since M is an injective Z[G0]-module, there exists a Z[G0]-linear map h: C → M such that h ◦ f = g For any element c in C, an open subgroup U of G can be found with c ∈ C_U, which implies h(c) ∈ M_U, demonstrating that h(C) is contained within M_A Consequently, h induces a G-homomorphism from C to M_A, also denoted by h, which satisfies the property h ◦ f = g.
MA is an injective G-module Similarly, one verifies at once that εA(A) ⊆ MA This concludes the proof.
Proposition 2.3.9 LetN be a closed normal subgroup ofG, and letQbe an injective
G-module ThenQ N is injective inMod(G/N).
In the proof, consider a monomorphism \( f: B \to C \) in \(\operatorname{Mod}(G/N)\) and a \( Z[G/N] \)-linear map \( g: B \to Q \) Recall from Example 2.1.4.(5) that both \( B \) and \( C \) are \( G \)-modules via the natural projection \( G \to G/N \), which ensures that \( f \) is also a monomorphism of \( G \)-modules and that \( g \) is a \( G \)-homomorphism This setup allows us to construct the following commutative diagram, highlighting the relationships between these modules and homomorphisms within the context of \( G \)-module theory.
Similar arguments as in the proof of Proposition 2.3.8 show that there exists aZ[G/N]- linear maph: C−!Q N such thath◦f =g, henceQ N is injective inMod(G/N), as desired This concludes the proof.
We are now ready to prove the main result of this section.
Theorem 2.3.10 Theδ-functorH(G,−)is universal.
In the category of modules over a group G, any object A can be embedded into an injective G-module MA via a monomorphism εA, as established by Proposition 2.3.8 Furthermore, Proposition 2.3.9 states that for any open normal subgroup N of G, the module MA N is injective in the category Mod(G/N), implying that the higher cohomology groups Hⁿ(G/N, MA N) vanish for all n > 0 This result highlights the advantageous properties of injective modules in the context of group cohomology, as discussed on page 92 of reference [13].