Understanding Split Short Exact Sequences In Abstract Algebra

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#Introduction

In the realm of abstract algebra, particularly within the study of modules and exact sequences, the concept of split short exact sequences plays a crucial role. These sequences provide a framework for understanding the structure of modules and their relationships. A short exact sequence is a sequence of modules and homomorphisms that satisfy certain exactness conditions, essentially capturing how modules can be built from smaller components. The added condition of being 'split' gives us even more insight into the module structure, hinting at direct sum decompositions. However, a common point of confusion arises from the subtleties of the converse: While a split short exact sequence implies a direct sum decomposition, the existence of a direct sum decomposition does not automatically guarantee that the corresponding short exact sequence is split. This article delves into the intricacies of split short exact sequences, exploring the relationship between such sequences and direct sums. We aim to clarify why the converse of the theorem presented in Rotman's Advanced Modern Algebra does not hold universally, providing concrete examples and detailed explanations to solidify understanding. We will dissect the conditions required for a short exact sequence to split and highlight scenarios where modules might be isomorphic to direct sums without the sequence itself possessing the splitting property. This exploration will not only enhance the comprehension of abstract algebraic concepts but also equip readers with the tools to analyze module structures more effectively.

Definition of Short Exact Sequences

Before diving into the specifics of split short exact sequences, it is essential to define the foundational concept of a short exact sequence itself. A short exact sequence is a sequence of modules and homomorphisms, typically denoted as 0 → A →f B →g C → 0, where A, B, and C are modules over a ring R, and f and g are module homomorphisms. The '0' represents the zero module, which is the trivial module containing only the zero element. The arrows represent the homomorphisms mapping between these modules. This sequence must satisfy three critical conditions to be considered exact:

  1. f is injective: This means that the kernel of f, denoted as ker(f), contains only the zero element. In other words, if f(a) = 0 for some a in A, then a must be the zero element in A. Injectivity ensures that f maps distinct elements of A to distinct elements of B.
  2. g is surjective: This condition requires that the image of g, denoted as im(g), is equal to the entire module C. For every element c in C, there must exist an element b in B such that g(b) = c. Surjectivity implies that g maps onto the entire module C.
  3. im(f) = ker(g): This is the core of the exactness condition. The image of f, which consists of all elements in B that are the result of applying f to elements in A, must be precisely the kernel of g, which consists of all elements in B that are mapped to zero by g. This condition establishes a crucial link between the homomorphisms f and g, ensuring that the sequence flows seamlessly from one module to the next.

These three conditions together define a short exact sequence, which can be thought of as a way to decompose a module B into two parts, A and C, connected by the homomorphisms f and g. The module A is effectively a submodule of B (since f is injective), and C is isomorphic to the quotient module B/im(f) (due to the first isomorphism theorem). Understanding the definition of a short exact sequence is the bedrock for understanding split short exact sequences.

Split Short Exact Sequences: The Definition and Implications

Building upon the concept of short exact sequences, we now introduce the notion of a split short exact sequence. A short exact sequence 0 → A →f B →g C → 0 is said to be split if there exists a homomorphism s: C → B such that g ◦ s = idC, where idC is the identity homomorphism on C. In simpler terms, a short exact sequence splits if there is a map s from C back into B that, when composed with g, gives the identity map on C. This map s is called a section or a splitting homomorphism.

The existence of a splitting homomorphism has profound implications for the structure of the module B. Specifically, if the short exact sequence 0 → A →f B →g C → 0 splits, then B is isomorphic to the direct sum of A and C, denoted as B ≅ A ⊕ C. This means that every element in B can be uniquely expressed as a sum of an element from A and an element from C. The isomorphism provides a clear picture of how B is constructed from its submodules A and C.

To understand this isomorphism, consider the map φ: A ⊕ C → B defined by φ(a, c) = f(a) + s(c). It can be shown that φ is an isomorphism if the sequence splits. The injectivity of φ follows from the fact that if f(a) + s(c) = 0, then applying g yields g(f(a) + s(c)) = g(f(a)) + g(s(c)) = 0 + idC(c) = c. Thus, c = 0, and since f(a) = 0, the injectivity of f implies that a = 0. For surjectivity, consider any b in B. Let c = g(b), and consider the element b - s(c) in B. Applying g to this element gives g(b - s(c)) = g(b) - g(s(c)) = c - idC(c) = 0. Therefore, b - s(c) is in the kernel of g, which is equal to the image of f. Thus, there exists an a in A such that f(a) = b - s(c). Rearranging, we have b = f(a) + s(c) = φ(a, c), demonstrating that φ is surjective. Since φ is both injective and surjective, it is an isomorphism.

The fact that a split short exact sequence implies a direct sum decomposition is a powerful tool in module theory. It allows us to break down complex modules into simpler, more manageable components. However, the converse of this statement—that B ≅ A ⊕ C implies the sequence splits—is not always true, and understanding why this is the case is crucial.

The Converse and Counterexamples

As discussed earlier, the fundamental theorem states that if a short exact sequence 0 → A →f B →g C → 0 splits, then B is isomorphic to the direct sum A ⊕ C. However, the converse of this theorem does not hold universally. The mere fact that B is isomorphic to A ⊕ C does not guarantee that the short exact sequence is split. This subtle yet significant distinction is a common source of confusion in abstract algebra, and it is crucial to understand the conditions under which the converse fails.

To illustrate this point, let's consider a classic counterexample. Let Z denote the integers, and let Z_n denote the integers modulo n. Consider the short exact sequence:

0 → Z →f Z →g Z_2 → 0

where f is the multiplication by 2 (i.e., f(x) = 2x), and g is the reduction modulo 2. This sequence is exact because f is injective, g is surjective, and the image of f (the even integers) is precisely the kernel of g.

Now, consider Z ⊕ Z_2, the direct sum of the integers and the integers modulo 2. Clearly, Z ⊕ Z_2 is a module. The question is whether the original sequence splits even though the 'B' term, which is Z, might seem related to Z ⊕ Z_2. If the sequence were to split, there would need to exist a homomorphism s: Z_2 → Z such that g ◦ s = idZ_2. The integers modulo 2, Z_2, has two elements: {0, 1}. If s exists, it would need to map 0 to 0 (as homomorphisms must preserve the zero element), and it would need to map 1 to some integer in Z. Let's say s(1) = k for some integer k. Then, g(s(1)) = g(k), and for g ◦ s to be the identity on Z_2, we would need g(k) to be equal to 1 (mod 2). This is possible; we could choose k to be any odd integer.

However, here's where the problem arises. If s is a homomorphism, it must satisfy s(a + b) = s(a) + s(b) for all a, b in Z_2. In Z_2, 1 + 1 = 0. Therefore, s(1 + 1) = s(0) = 0. But if s(1) = k, then s(1) + s(1) = k + k = 2k. For s to be a homomorphism, we would need 2k = 0, which implies k = 0. This contradicts our earlier finding that k must be an odd integer for g ◦ s to be the identity on Z_2. Therefore, there is no such homomorphism s, and the sequence does not split.

Despite the sequence not splitting, consider the fact that Z is not isomorphic to Z ⊕ Z_2. If it were, there would be a bijection between the two sets, and they would have the same algebraic structure. However, every element in Z ⊕ Z_2 has order 1 or 2 (since 2(a, b) = (2a, 2b) = (2a, 0) and 2a can be 0 in Z if a = 0, and 2b is always 0 in Z_2), while Z has elements of infinite order. This difference in structure proves that Z is not isomorphic to Z ⊕ Z_2.

This counterexample highlights the critical point: while a split short exact sequence guarantees a direct sum decomposition, the isomorphism B ≅ A ⊕ C alone is insufficient to conclude that the sequence 0 → A →f B →g C → 0 splits. The existence of the splitting homomorphism s is a necessary condition for the sequence to split, and this condition is not automatically satisfied just because B is isomorphic to A ⊕ C. Other counterexamples can be constructed using similar principles, often involving torsion elements and the lack of suitable homomorphisms.

Conditions for Splitting

Having established that the isomorphism B ≅ A ⊕ C does not guarantee that a short exact sequence splits, it is crucial to understand the conditions under which a short exact sequence does indeed split. There are several equivalent conditions that can determine whether a short exact sequence 0 → A →f B →g C → 0 splits. These conditions provide different perspectives on the splitting phenomenon and offer practical tools for determining if a given sequence splits.

  1. Existence of a Section (Right Split): As defined earlier, the sequence splits if there exists a homomorphism s: C → B such that g ◦ s = idC. This condition, by definition, is the most direct way to check for splitting. If such a map s can be found, the sequence splits. The map s is often called a section or a right splitting.
  2. Existence of a Retraction (Left Split): A short exact sequence also splits if there exists a homomorphism r: B → A such that r ◦ f = idA, where idA is the identity homomorphism on A. This map r is called a retraction or a left splitting. The existence of a retraction provides an alternative criterion for splitting. It essentially means that there is a way to 'project' B back onto A such that the composition with the inclusion map f gives the identity on A.
  3. B is Isomorphic to A ⊕ C with Specific Maps: The sequence splits if and only if there exists an isomorphism φ: B → A ⊕ C such that the following diagram commutes:
0 → A →f B →g C → 0
         ||   ||   ||
0 → A →i A ⊕ C →π C → 0

where i(a) = (a, 0) is the canonical injection and π(a, c) = c is the canonical projection. This condition states that B is not just isomorphic to A ⊕ C, but the isomorphism respects the maps f and g in the original sequence. In other words, the inclusion of A into B corresponds to the inclusion of A into A ⊕ C, and the map from B to C corresponds to the projection from A ⊕ C onto C. This condition provides a more stringent requirement for splitting, ensuring that the isomorphism preserves the structure of the sequence.

These conditions are equivalent, meaning that if one condition holds, all other conditions hold as well. They provide a comprehensive set of tools for determining whether a short exact sequence splits. Understanding these conditions is essential for tackling problems involving module extensions and decompositions in abstract algebra.

Examples and Applications

To further illustrate the concept of split short exact sequences and their applications, let's delve into some examples. These examples will help solidify the theoretical concepts discussed and demonstrate how split short exact sequences arise in various algebraic contexts.

Example 1: Vector Spaces

Consider vector spaces over a field F. Let V be a vector space, and let W be a subspace of V. Then we can form a short exact sequence:

0 → W →i V →π V/W → 0

where i is the inclusion map (i(w) = w for all w in W), and π is the canonical projection map (π(v) = v + W for all v in V). In the context of vector spaces, this sequence always splits. To see why, we can use the fact that every subspace of a vector space has a complement. That is, there exists a subspace U of V such that V = W ⊕ U. We can define a splitting homomorphism s: V/W → V by choosing a linear map that sends each coset v + W to its component in U. Specifically, for each coset v + W, there is a unique u in U such that v = w + u for some w in W. We can define s(v + W) = u. This map is well-defined and linear, and it satisfies π ◦ s = idV/W, making the sequence split.

This example highlights a general principle: short exact sequences of vector spaces always split. This is a consequence of the fact that vector spaces are free modules over a field, and free modules have the property that every submodule is a direct summand.

Example 2: Free Modules

More generally, consider free modules over a ring R. A module F is free if it has a basis, meaning a set of elements that can generate the module and are linearly independent. If F is a free module and M is a submodule of F, then the short exact sequence

0 → M →i F →π F/M → 0

splits if and only if F/M is a projective module. This is a crucial result in module theory, linking the splitting of sequences to the properties of quotient modules.

Example 3: Non-Splitting Sequence Revisited

Recall the short exact sequence

0 → Z →f Z →g Z_2 → 0

where f(x) = 2x and g is the reduction modulo 2. We previously showed that this sequence does not split. This example serves as a reminder that not all short exact sequences split, and the conditions for splitting are not always straightforward to verify. The failure of this sequence to split is related to the torsion properties of the modules involved. Z has no torsion elements (elements of finite order), while Z_2 has elements of order 2. The interaction between these torsion properties prevents the existence of a splitting homomorphism.

Applications

Split short exact sequences have numerous applications in various areas of mathematics, including:

  • Module Theory: They are used to classify modules and understand their structure. Splitting sequences provide a way to decompose modules into simpler components, making their analysis more tractable.
  • Homological Algebra: They play a central role in the development of homological algebra, where they are used to define and study Ext functors, which measure the extent to which a sequence fails to split.
  • Group Theory: They arise in the study of group extensions, where a group G is expressed as an extension of a normal subgroup N by a quotient group G/N. The splitting of the corresponding sequence provides information about the structure of G.

Conclusion

In this article, we have explored the concept of split short exact sequences in abstract algebra. We began by defining short exact sequences and then introduced the notion of a splitting homomorphism, which leads to the decomposition of a module into a direct sum. We emphasized the crucial point that while a split short exact sequence implies a direct sum decomposition, the converse is not universally true. We provided a detailed counterexample to illustrate this point and discussed the equivalent conditions for a sequence to split, including the existence of a section, the existence of a retraction, and the commutativity of diagrams involving canonical injections and projections.

Through various examples, such as vector spaces and free modules, we demonstrated how split short exact sequences arise in different algebraic contexts. We also revisited a non-splitting sequence to underscore the importance of verifying the splitting conditions. Finally, we highlighted the applications of split short exact sequences in module theory, homological algebra, and group theory.

Understanding split short exact sequences is essential for gaining a deeper insight into module structures and their relationships. By grasping the nuances of splitting conditions and the limitations of the converse, students and researchers can effectively analyze algebraic structures and tackle more advanced problems in abstract algebra. The concept serves as a cornerstone for further exploration in homological algebra and related fields, providing a powerful tool for understanding the building blocks of algebraic structures.