Split Exact Sequence

Definition

A split exact sequence is a specific type of exact sequence in which a short exact sequence of modules or chain complexes splits, meaning that there is a way to express the middle object as a direct sum of the other two objects. This concept is crucial because it implies that the sequence can be broken down into simpler components, making it easier to analyze and understand. The existence of a section (a right inverse) for the inclusion map provides an intuitive way to visualize how the modules relate to each other.

5 Must Know Facts For Your Next Test

1. For a short exact sequence $$0 \to A \to B \to C \to 0$$ to split, there must exist a homomorphism from C to B that acts as a right inverse to the projection onto C.
2. In practical terms, if you have a split exact sequence, you can think of the middle object as being made up of two 'pieces' that correspond to the first and last objects in the sequence.
3. The splitting property is particularly useful in category theory and allows for easier computations, such as finding projective resolutions.
4. Split exact sequences can be extended into long exact sequences, preserving the splitting property under certain conditions.
5. The notion of splitting leads to important results such as the fact that projective modules allow every short exact sequence to split.

Review Questions

• How does the concept of splitting in an exact sequence help in simplifying the study of chain complexes?
  • When an exact sequence splits, it allows us to treat the middle module as a direct sum of the other two modules. This simplification means we can analyze each part independently, making it easier to compute homology and understand relationships between modules. The split property essentially breaks down complex interactions into manageable pieces, enhancing clarity in computations and theoretical discussions.
• Discuss how the existence of sections relates to the properties of split exact sequences and their implications in homological algebra.
  • Sections provide a critical link in split exact sequences by acting as right inverses for inclusion maps. When a section exists, it indicates that we can 'lift' elements from the quotient back to the larger module without losing information. This property has significant implications in homological algebra because it guarantees that certain structures are preserved, such as dimensions and ranks, which helps in identifying projective modules within a category.
• Evaluate how understanding split exact sequences can influence our approach to constructing projective resolutions in homological algebra.
  • Understanding split exact sequences fundamentally alters our approach to constructing projective resolutions by allowing us to ensure that each stage in the resolution can be treated independently. If a short exact sequence splits, we can easily build larger resolutions by stacking these independent components without worrying about complications from interdependencies. This capability not only streamlines our resolution construction but also enhances our ability to apply tools like the five lemma and spectral sequences effectively in further studies.

Related terms

Exact Sequence: An exact sequence is a sequence of modules and homomorphisms between them such that the image of one homomorphism equals the kernel of the next.

Chain Complex: A chain complex is a sequence of abelian groups or modules connected by homomorphisms where the composition of consecutive maps is zero.

Direct Sum: The direct sum is an operation that combines two or more modules into a new module, where each component retains its individuality.