Snake lemma

Definition

The snake lemma is a fundamental result in homological algebra that describes the relationship between two exact sequences and a commutative diagram of abelian groups or modules. It essentially provides a way to construct a long exact sequence from short exact sequences, revealing how elements are related in the context of homology and cohomology theories.

5 Must Know Facts For Your Next Test

1. The snake lemma allows one to derive a long exact sequence in homology when given two short exact sequences, making it essential for connecting different algebraic structures.
2. In the context of pairs of spaces, the snake lemma helps to understand how the relative homology groups relate to the absolute homology groups of the spaces involved.
3. One significant application of the snake lemma is in deriving the long exact sequence of a pair, which connects the homology groups of a space with those of its subspaces.
4. The lemma is visualized through commutative diagrams, where arrows represent morphisms, demonstrating how elements transition from one group to another.
5. The snake lemma illustrates how exactness can be preserved through relationships between different sequences, emphasizing its utility in both algebraic and topological contexts.

Review Questions

• How does the snake lemma relate short exact sequences to long exact sequences, and what is its significance in homological algebra?
  • The snake lemma connects short exact sequences by demonstrating how they can be combined to form a long exact sequence. This process is crucial because it allows for a deeper understanding of the relationships between different algebraic structures, particularly in homological algebra. The significance lies in its ability to reveal information about kernel and image relationships across multiple sequences, enabling one to track how elements evolve through these transformations.
• Describe how the snake lemma can be applied when working with pairs of topological spaces and what it tells us about their relative homology.
  • When applying the snake lemma to pairs of topological spaces, it provides a long exact sequence that links the homology groups of the space and its subspace. This reveals how the relative homology group captures information about features in the larger space that are not present in the subspace. Specifically, it shows how cycles and boundaries interact across both spaces, offering insights into their topological properties and how they relate to one another.
• Evaluate the implications of the snake lemma for understanding topological invariants and their behavior under continuous mappings between spaces.
  • The implications of the snake lemma extend into understanding how topological invariants behave under continuous mappings. By establishing a connection between short and long exact sequences through this lemma, it allows mathematicians to analyze changes in homology groups as spaces are mapped onto each other. This evaluation helps reveal whether certain features are preserved or altered during continuous transformations, providing critical insights into the nature of topological spaces and their invariants.

Related terms

Exact Sequence: An exact sequence is a sequence of algebraic objects and morphisms between them such that the image of one morphism is equal to the kernel of the next.

Long Exact Sequence: A long exact sequence is an exact sequence that arises in various contexts, typically associated with a pair of topological spaces or a filtration, allowing the tracking of homological properties over extended relationships.

Homology: Homology is a mathematical concept that studies topological spaces by associating sequences of abelian groups or modules to these spaces, helping to classify them based on their shapes and structures.