The Snake Lemma is a fundamental result in homological algebra that provides a long exact sequence connecting the kernels and cokernels of morphisms between two short exact sequences. It is a powerful tool for analyzing the relationships between algebraic structures and their homomorphisms in various mathematical contexts, such as in the study of modules and abelian groups.

Snake Lemma

fundamental result

Homological algebra is a branch of mathematics that studies homology in a general algebraic setting, providing a framework for analyzing and solving problems in algebraic topology, algebraic geometry, and beyond. It uses chain complexes and exact sequences to explore the relationships between different algebraic structures, revealing deep insights into their properties and interactions.

homological algebra

A long exact sequence is a sequence of algebraic objects and morphisms between them that is exact at every position, meaning the image of one morphism equals the kernel of the next. It is a fundamental tool in homological algebra and algebraic topology, often used to relate different algebraic structures and compute invariants.

long exact sequence

kernels and cokernels

morphisms between two short exact sequences

Analyzing relationships

Algebraic structures are mathematical entities defined by a set equipped with one or more operations that satisfy specific axioms, providing a framework to study abstract properties of numbers and operations. They form the foundational basis for various branches of mathematics and computer science, allowing for the exploration of symmetry, structure, and transformations in diverse contexts.

algebraic structures

A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on them. It is fundamental in abstract algebra as it allows the comparison and study of different algebraic systems by examining how they can be transformed into one another while maintaining their essential properties.

homomorphisms

mathematical contexts

study of modules

An Abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying the properties of closure, associativity, identity, invertibility, and commutativity. Named after mathematician Niels Henrik Abel, these groups are foundational in abstract algebra and are characterized by the fact that the order of operation does not affect the result.

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