Snake Lemma

Definition

The Snake Lemma is a fundamental result in homological algebra that describes how to obtain long exact sequences of homology or cohomology groups from a commutative diagram of abelian groups or modules. It reveals the relationships between the kernels and cokernels of morphisms, allowing one to deduce new information about the homological properties of objects involved. This lemma plays a crucial role in understanding induced cohomomorphisms, the long exact sequences that arise from pairs, and the construction of connecting homomorphisms.

5 Must Know Facts For Your Next Test

1. The Snake Lemma provides a way to derive long exact sequences from short exact sequences in a commutative diagram.
2. One of the main applications of the Snake Lemma is in deriving long exact sequences for relative cohomology, which helps in studying pairs of spaces.
3. The connecting homomorphism, which appears in the context of the Snake Lemma, connects the kernel of one map to the cokernel of another, illustrating how these concepts are intertwined.
4. In practical terms, the Snake Lemma can help identify whether a specific map is injective or surjective by analyzing the corresponding long exact sequence.
5. The lemma is particularly useful in algebraic topology and sheaf theory, where it facilitates calculations involving sheaves and their cohomology.

Review Questions

• How does the Snake Lemma facilitate the understanding of induced cohomomorphisms?
  • The Snake Lemma plays a critical role in understanding induced cohomomorphisms by allowing us to relate different cohomology groups through exact sequences. When we have a short exact sequence of groups or modules, applying the Snake Lemma yields a long exact sequence in cohomology. This helps in tracking how cohomology changes under morphisms, ultimately leading to clearer insights about induced maps and their properties.
• Discuss how the Snake Lemma leads to a long exact sequence for a pair and what that implies for their homological properties.
  • When applying the Snake Lemma to a pair of spaces, it generates a long exact sequence that connects their respective cohomology groups. This sequence shows how the relative cohomology groups are intertwined with absolute ones, indicating that there is a systematic way to analyze how adding or removing spaces affects their cohomological properties. It implies that knowledge about one space's cohomology can inform us about its pair's structure, enhancing our understanding of both individual and relative behaviors.
• Evaluate the significance of connecting homomorphisms as they relate to the Snake Lemma and long exact sequences.
  • Connecting homomorphisms are significant because they act as bridges within long exact sequences derived from the Snake Lemma. They connect kernels and cokernels, facilitating transitions between different levels of cohomological information. Evaluating these maps allows mathematicians to gain insights into whether certain maps are injective or surjective while revealing how various structures relate to each other. This ability to link disparate parts of algebraic structures emphasizes not only their individual characteristics but also their relationships through connectivity.

Related terms

Cohomology: A mathematical tool used to study topological spaces and algebraic structures through the use of cochain complexes and cohomology groups.

Exact Sequence: A sequence of abelian groups and homomorphisms between them where the image of one homomorphism equals the kernel of the next, often used to analyze the relationships among algebraic objects.

Homomorphism: A structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, which preserves the operations defined on those structures.