Exact Sequence

Definition

An exact sequence is a sequence of objects and morphisms in an abelian category such that the image of each morphism equals the kernel of the next. This concept helps capture the idea of continuity and preservation of structure through mappings, allowing us to analyze relationships between objects and their corresponding homomorphisms. It serves as a fundamental tool in homological algebra, providing insight into how objects can be constructed and deconstructed through exactness.

5 Must Know Facts For Your Next Test

1. In an exact sequence, the condition that the image of one morphism equals the kernel of the next ensures that there are no 'gaps' in the relationships between objects.
2. Exact sequences can be used to define derived functors, which provide essential tools for measuring deviations from exactness in functorial relationships.
3. They play a crucial role in the study of sheaf cohomology, where they help understand how global sections relate to local sections.
4. Exact sequences can be classified into short exact sequences and long exact sequences, with short sequences being particularly useful for understanding basic properties of modules.
5. The concept extends beyond abelian categories to triangulated categories and derived categories, highlighting its importance across different areas of mathematics.

Review Questions

• How does an exact sequence illustrate the relationship between kernels and images in an abelian category?
  • An exact sequence illustrates this relationship by requiring that the image of each morphism coincides with the kernel of the following morphism. This means that every time you have a mapping from one object to another, it precisely captures all elements that become zero when passing through to the next object. This property highlights how closely related these structures are, revealing connections between different parts of a mathematical framework.
• Discuss the significance of short exact sequences in understanding module theory and their applications.
  • Short exact sequences are particularly significant in module theory as they provide clear insights into how one module can be embedded into another and how it relates to quotient modules. They help establish foundational results such as the splitting lemma, which states that if a short exact sequence splits, then the module can be decomposed into a direct sum. This understanding is essential for determining homological properties and constructing new modules from existing ones.
• Evaluate how the concept of exact sequences contributes to developments in derived functors and homological algebra as a whole.
  • Exact sequences contribute significantly to derived functors and homological algebra by offering a framework to analyze how functors behave with respect to exactness. Derived functors are built from extensions of exact sequences, allowing mathematicians to quantify deviations from being exact. This connection has far-reaching implications in both algebraic topology and algebraic geometry, where understanding these relationships enables deeper insights into complex structures within various mathematical disciplines.

Related terms

Kernel:

The kernel of a morphism is the set of elements that map to the zero object, capturing the notion of solutions to equations within the context of homomorphisms.

Cokernel:

The cokernel is a construction that represents the quotient of the codomain by the image of a morphism, serving as a way to measure how much 'information' is lost in the mapping.

Short Exact Sequence: A short exact sequence is a specific type of exact sequence involving three objects and two morphisms, usually written as 0 → A → B → C → 0, indicating that A injects into B and B surjects onto C.