Exploring Split Short Exact Sequences A Deep Dive Into Abstract Algebra

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In the realm of abstract algebra, particularly within the study of modules and exact sequences, the concept of a split short exact sequence holds significant importance. A short exact sequence is a sequence of modules and homomorphisms that satisfy certain exactness conditions, providing valuable insights into the relationships between the modules involved. The added condition of being split introduces further structure, leading to specific isomorphisms and decompositions. However, the nuances of these concepts can sometimes be subtle, particularly when considering the converse of certain implications. This article delves into the intricacies of split short exact sequences, addressing a specific question arising from Rotman's "Advanced Modern Algebra" and providing a comprehensive exploration of the topic.

Understanding Short Exact Sequences

Before diving into the specifics of splitting, it's crucial to establish a firm understanding of short exact sequences. A short exact sequence is a sequence of modules and homomorphisms of the form:

0 -> A --f--> B --g--> C -> 0

where A, B, and C are modules (over a ring), and f and g are homomorphisms. The sequence is said to be exact if the image of each homomorphism is equal to the kernel of the next. In other words:

  1. f is injective (i.e., the kernel of f is {0}).
  2. g is surjective (i.e., the image of g is C).
  3. The image of f is equal to the kernel of g (i.e., Im(f) = Ker(g)).

These three conditions ensure a specific relationship between the modules. The injectivity of f means that A can be considered as a submodule of B (up to isomorphism). The surjectivity of g means that every element in C has a preimage in B. The condition Im(f) = Ker(g) connects these two aspects, stating that the elements of B that map to zero in C are precisely those that come from A. This careful balance makes short exact sequences a powerful tool for analyzing module structure.

Split Short Exact Sequences: The Definition

A short exact sequence

0 -> A --f--> B --g--> C -> 0

is said to split if there exists a homomorphism h: C -> B such that g o h = id_C, where id_C is the identity map on C. In simpler terms, a split short exact sequence means there's a way to "lift" elements from C back to B in a way that, when followed by g, gives you back the original element in C. This condition has significant consequences for the structure of B. Alternatively, the sequence is said to split if there exists a homomorphism k: B -> A such that k o f = id_A, where id_A is the identity map on A.

The Splitting Lemma and Its Implications

The key result concerning split short exact sequences is the Splitting Lemma. This lemma states that the following conditions are equivalent for a short exact sequence:

0 -> A --f--> B --g--> C -> 0
  1. The sequence splits (i.e., there exists a homomorphism h: C -> B such that g o h = id_C).
  2. There exists a homomorphism k: B -> A such that k o f = id_A.
  3. B is isomorphic to the direct sum of A and C (i.e., B ≅ A ⊕ C).

The Splitting Lemma provides a powerful connection between the existence of splitting maps and the decomposition of the middle module B. The equivalence of these conditions means that if you can find a splitting map (either h or k), then B can be expressed as the direct sum of A and C. Conversely, if B is isomorphic to A ⊕ C, then the sequence splits. This isomorphism significantly simplifies the analysis of B, as it allows us to understand its structure in terms of the simpler modules A and C.

The Question of the Converse

Rotman's "Advanced Modern Algebra" highlights a crucial point regarding split short exact sequences: if a short exact sequence splits, then B is isomorphic to A ⊕ C, but the converse is not necessarily true. This means that even if B is isomorphic to A ⊕ C, it does not automatically imply that the short exact sequence splits. This is the crux of the question we aim to address. The isomorphism B ≅ A ⊕ C is a necessary condition for the sequence to split, but it is not sufficient. To truly understand this subtle distinction, we need to delve into a concrete example.

To further clarify, let's rephrase the central question: If B is isomorphic to A ⊕ C, what additional conditions are required to ensure that the short exact sequence

0 -> A --f--> B --g--> C -> 0

splits? What specific examples demonstrate that the isomorphism B ≅ A ⊕ C alone is insufficient to guarantee a split short exact sequence?

Counterexamples: When the Converse Fails

To demonstrate that the converse of the Splitting Lemma does not hold, we need to construct a specific example of a short exact sequence where B is isomorphic to A ⊕ C, but the sequence does not split. This means we need to find modules A, B, and C, and homomorphisms f and g, such that:

  1. The sequence 0 -> A --f--> B --g--> C -> 0 is exact.
  2. B is isomorphic to A ⊕ C.
  3. The sequence does not split.

A classic example to illustrate this is the following:

Let's consider the following short exact sequence:

0 -> Z --f--> Z --g--> Z/2Z -> 0

where Z represents the integers, and Z/2Z represents the integers modulo 2. The homomorphism f is defined by multiplication by 2 (i.e., f(x) = 2x), and the homomorphism g is the canonical projection map (i.e., g(x) = x mod 2).

Verifying Exactness

First, we need to verify that this sequence is indeed a short exact sequence:

  1. f is injective: If f(x) = 2x = 0, then x = 0. Thus, the kernel of f is {0}, and f is injective.
  2. g is surjective: For any element in Z/2Z (either 0 or 1), there exists a preimage in Z. For example, g(0) = 0 and g(1) = 1. Thus, g is surjective.
  3. Im(f) = Ker(g): The image of f consists of all even integers (2Z). The kernel of g consists of all integers that are congruent to 0 modulo 2, which are also the even integers. Thus, Im(f) = Ker(g).

Since all three conditions are met, the sequence is exact.

Isomorphism to Direct Sum

In this case, A = Z, B = Z, and C = Z/2Z. While it might seem counterintuitive at first, B = Z is not isomorphic to A ⊕ C = Z ⊕ Z/2Z. Z is an infinite cyclic group, and Z ⊕ Z/2Z is not cyclic because of the torsion element in Z/2Z. However, this initial perception highlights the subtlety of the situation.

To modify this example to fit the condition B ≅ A ⊕ C, consider modules over a different ring. Instead of working with integers, let's consider modules over the ring of integers modulo 4, denoted as Z/4Z. Let A = 2Z/4Z, B = Z/4Z, and C = Z/2Z. The sequence becomes:

0 -> 2Z/4Z --f--> Z/4Z --g--> Z/2Z -> 0

where f is the inclusion map and g maps an element in Z/4Z to its residue modulo 2. In this context, B = Z/4Z is isomorphic to A ⊕ C = 2Z/4Z ⊕ Z/2Z. Specifically, 2Z/4Z = {0, 2}, which is isomorphic to Z/2Z. Thus, A ⊕ C is isomorphic to Z/2Z ⊕ Z/2Z.

Why It Doesn't Split

Now, let's demonstrate why this sequence does not split. To do this, we'll show that there is no homomorphism h: Z/2Z -> Z/4Z such that g o h = id_(Z/2Z).

Suppose such a homomorphism h exists. Let's say h(1) = x, where x is an element in Z/4Z. Since g o h = id_(Z/2Z), we must have g(h(1)) = g(x) = 1. This means x must be an odd integer in Z/4Z, so x can be either 1 or 3.

However, in Z/2Z, 1 + 1 = 0. Therefore, h(1 + 1) = h(0) = 0. If h is a homomorphism, then h(1 + 1) = h(1) + h(1) = x + x. So, we have x + x = 0 in Z/4Z. If x = 1, then 1 + 1 = 2 ≠ 0 in Z/4Z. If x = 3, then 3 + 3 = 6 ≡ 2 ≠ 0 in Z/4Z. In both cases, we reach a contradiction. Therefore, there is no such homomorphism h, and the sequence does not split.

The Importance of the Ring

This example highlights a crucial point: the splitting of a short exact sequence is highly dependent on the underlying ring. The module structures and the properties of the homomorphisms are intertwined with the ring's characteristics. In the case of integers, the lack of divisibility properties prevents the existence of a splitting map, even when the isomorphism B ≅ A ⊕ C holds. In other words, this sequence does not split because Z is a cyclic group of order 4, while Z/2Z ⊕ Z/2Z is a non-cyclic group of order 4. Since these groups are not isomorphic, the sequence does not split.

Generalizations and Further Exploration

The example we've discussed serves as a concrete illustration of why the converse of the Splitting Lemma fails. However, it also prompts us to consider what additional conditions might be necessary to ensure that a short exact sequence splits when B ≅ A ⊕ C.

One such condition arises in the context of free modules. If C is a free module, then any short exact sequence of the form

0 -> A --f--> B --g--> C -> 0

will split. This is because the freeness of C allows us to construct a splitting map h: C -> B by lifting a basis of C to elements in B. However, even this condition is not universally applicable, and the question of splitting remains a nuanced one.

Further exploration of this topic involves delving into concepts such as projective modules, injective modules, and the Baer sum, which provide a more comprehensive framework for understanding split short exact sequences and their properties. Understanding the subtleties of split short exact sequences is crucial for deeper explorations in abstract algebra, particularly in areas like homological algebra and module theory. The example discussed here serves as a foundation for understanding the more general principles governing the splitting of exact sequences and the structure of modules.

Conclusion

In summary, while a split short exact sequence implies that B is isomorphic to A ⊕ C, the converse is not necessarily true. The example we've explored demonstrates that the isomorphism B ≅ A ⊕ C alone is not sufficient to guarantee that a short exact sequence will split. The splitting property depends on the specific homomorphisms and the underlying ring structure. This understanding is crucial for navigating the intricacies of abstract algebra and related fields, highlighting the importance of careful analysis and the use of counterexamples to refine our understanding of mathematical concepts. The exploration of split short exact sequences not only deepens our understanding of module theory but also provides valuable insights into the subtle relationships between algebraic structures.