Problem 8 - Algebra

Question

For each \(G\) -module \(A \in \operatorname{Mod}(G)\), define \(\varepsilon_{A}: A \rightarrow M(G, A)\) by the condition \(\varepsilon_{A}(a)=\) the function \(f_{a}\) such that \(f_{x}(\sigma)=\sigma a\) for \(\sigma \in G\). Show that \(a \mapsto f_{a}\) is a \(G\) -module embedding, and that the exact sequence $$ 0 \rightarrow A \stackrel{E}{\rightarrow} M(G, A) \rightarrow X_{A}=\text { coker } \varepsilon_{A} \rightarrow 0 $$ splits over \(\mathbf{Z}\). (In fact, the map \(f \mapsto f(e)\) splits the left side arrow.)

Step-by-Step Solution

Answer

In summary, to show that the map \(a \mapsto f_a\) is a \(G\)-module embedding and that the given exact sequence splits over \(\mathbf{Z}\), we followed these steps: 1. Proved that \(\varepsilon_A\) is a \(G\)-module homomorphism by showing that it preserves both the \(G\)-action and addition. 2. Proved that \(\varepsilon_A\) is an embedding by showing that it is injective. 3. Showed that the exact sequence splits over the integers by demonstrating that the map \(S: M(G, A) \rightarrow A\) defined by \(S(f) = f(\operatorname{Id}_G)\) is a splitting map. By accomplishing these steps, we have shown that \(a \mapsto f_a\) is a \(G\)-module embedding and the exact sequence splits over \(\mathbf{Z}\).

Step 1: Prove ε_A is a G-module homomorphism

In order to show that ε_A is a G-module homomorphism, we need to check two properties: first, that it preserves the G-action, and second, that it preserves the addition operation of A. Let a, b ∈ A and σ ∈ G. Then: 1. ε_A(σa) = f_{σa}, and f_{σa}(τ) = τ(σa) for any τ ∈ G. 2. σ(ε_A(a)) = σf_a, and (σf_a)(τ) = σ(f_a(τσ^{-1})) = σ(τσ^{-1}a) = τ(σa) for any τ ∈ G. Since both expressions are equal to τ(σa), we have ε_A(σa) = σ(ε_A(a)), which means that ε_A preserves the G-action. Now let's check that it preserves addition: 1. ε_A(a + b) = f_{a + b}, and f_{a + b}(σ) = σ(a + b) = σa + σb for any σ ∈ G. 2. ε_A(a) + ε_A(b) = f_a + f_b, and (f_a + f_b)(σ) = f_a(σ) + f_b(σ) = σa + σb for any σ ∈ G. Both expressions are equal to σa + σb, so ε_A(a + b) = ε_A(a) + ε_A(b), which means that ε_A preserves addition. Therefore, ε_A is a G-module homomorphism.

Step 2: Prove ε_A is an embedding

To show that ε_A is an embedding, we need to prove that it is injective, meaning that if a ≠ b, then ε_A(a) ≠ ε_A(b). Suppose, to the contrary, that ε_A(a) = ε_A(b) for some a, b ∈ A with a ≠ b. Then: f_a(σ) = ε_A(a)(σ) = ε_A(b)(σ) = f_b(σ) for all σ ∈ G. But this implies that σa = σb for all σ ∈ G, which in turn implies that a = b, since G is a group and σ is invertible. This contradicts our assumption that a ≠ b, so ε_A must be injective, meaning it is an embedding.

Step 3: Show that the exact sequence splits over the integers

We are given that the map f ↦ f(e), where e is the identity element of G, splits the left side arrow. Consider the map S: M(G, A) → A defined by S(f) = f(e). We will show that S is a splitting map, i.e., that S(ε_A(a)) = a for all a ∈ A and ε_A(S(f)) = f for all f ∈ M(G, A). For any a ∈ A, we have S(ε_A(a)) = S(f_a) = f_a(e) = ea = a, so S is a left-inverse for ε_A. Now for any f ∈ M(G, A), we have ε_A(S(f)) = ε_A(f(e)) = f_{f(e)}, and by the definition of ε_A, we have that f_{f(e)}(σ) = σf(e) for all σ ∈ G. Since f is a G-module homomorphism, we have f(σ) = σf(e) for all σ ∈ G, so f_{f(e)} = f, and thus ε_A(S(f)) = f. Since S is both a left-inverse and a right-inverse for ε_A, the exact sequence splits over the integers, and we have completed the proof.

Key Concepts

Exact Sequence

An exact sequence is a special arrangement of modules and homomorphisms. It shows a sort of balance between them. In a typical sequence \[ 0 \rightarrow A \xrightarrow{E} B \xrightarrow{F} C \rightarrow 0 \] each map sends elements in such a way that the image of one map is the same as the kernel (preimage of zero) of the next map. The '0' at the beginning and end of the sequence highlights that the sequence starts and ends with the zero module. This shows that the map from \(A\) into \(B\) is injective, and the map from \(B\) onto \(C\) is surjective. When all these conditions hold, the sequence is called exact.

Module Embedding

Module embedding refers to a situation where one module is represented within another module without information loss. Essentially, it acts like inserting a smaller structure into a larger one while preserving all properties of the smaller structure. An embedding is a special type of homomorphism that is injective, which means it maps distinct elements of the first module to distinct elements of the second module.

• Injectivity: If an embedding from module \(A\) to module \(B\) is injective, it ensures no two different elements in \(A\) are mapped to the same element in \(B\).
• Preservation of structure: The embedding function must maintain the operations as they were in the original module.

These properties ensure that an embedded module fits perfectly inside a larger module, like a small piece fitting into a puzzle.

Group Action

In mathematics, a group action is a way of describing symmetries. It represents how elements of a group can "act" on elements of a set or module. Think of it as a way a group can manipulate or transform another mathematical object while retaining its inherent structure. For example, how transformations might change positions of shapes without altering their sizes or angles.

• Group Structure Maintenance: Each group operation maintains the algebraic structure of the set it's acting upon.
• Consistency: When groups act on modules, they ensure that the module behaves in a way compatible with the group’s behavior.

This powerful tool helps mathematicians explore objects and their symmetries across various fields. It essentially adds layers to how we can understand these objects mathematically.

Splitting Sequence

A splitting sequence in an exact sequence indicates that we can express a module as a direct sum of others, capturing a clear division of structures within the module. Specifically, a split exact sequence means there is a certain map, called a splitting map, between modules that effectively reverses part of the exact sequence, allowing us to neatly separate elements into different submodules.

• Split Map or Splitting Map: This is the function that "splits" the exact sequence, often by identifying a subgroup that complements the image of one map in the sequence.
• Direct Sum Representation: A splitting sequence permits the breakdown of modules into direct sums, enabling easier handling and understanding of their algebraic properties.

In the original exercise, the map \(f \mapsto f(e)\) serves as the splitting function, establishing that the sequence can be split into a direct combination of simpler structures.