3.2 The snake lemma and its applications

The snake lemma is a powerful tool in homological algebra. It connects kernels and cokernels in commutative diagrams with exact rows, creating a long exact sequence. This lemma is crucial for understanding how different parts of algebraic structures relate to each other.

The snake lemma has wide-ranging applications in algebra. It's used to prove other important results like the five lemma and can be extended to more complex situations. Understanding this lemma is key to grasping how short and long exact sequences work together.

The Snake Lemma and Connecting Homomorphisms

Snake Lemma and its Components

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• States that given a commutative diagram with exact rows, there exists a long exact sequence connecting the kernels and cokernels of the vertical maps
• Involves the connecting homomorphism, which maps the cokernel of one vertical map to the kernel of the next vertical map
• Utilizes the ker-coker exact sequence, which is a long exact sequence that alternates between kernels and cokernels of the vertical maps in the commutative diagram
• Applies the zig-zag lemma, which states that if a composition of two maps is zero, then the image of the first map is contained in the kernel of the second map

Applications and Extensions

• Used to prove the five lemma, which states that if the first two and last two vertical maps are isomorphisms in a commutative diagram with exact rows, then the middle vertical map is also an isomorphism
• Can be generalized to the 3x3 lemma, which involves a 3x3 commutative diagram with exact rows and columns
• Applies to various algebraic structures, such as modules over a ring, abelian groups, and chain complexes
• Helps in understanding the relationship between short exact sequences and long exact sequences in homological algebra

Homological Algebra Fundamentals

Commutative Diagrams and Chain Complexes

• A commutative diagram is a diagram of objects and morphisms where all directed paths between two objects lead to the same result
• Commutativity is a crucial property in homological algebra, as it allows for the study of functorial properties and the construction of exact sequences
• A chain complex is a sequence of abelian groups or modules connected by homomorphisms (differentials) such that the composition of any two consecutive homomorphisms is zero
• Chain complexes are used to define homology groups, which measure the "holes" or "cycles modulo boundaries" in the complex

Homology and its Properties

• Homology is a functor that assigns to each chain complex a sequence of abelian groups (homology groups), which are the quotients of the kernel of one differential by the image of the previous differential
• Homology groups are invariants of the chain complex and provide information about its algebraic structure
• The homology functor is a covariant functor from the category of chain complexes to the category of graded abelian groups
• Homology satisfies the Eilenberg-Steenrod axioms, which characterize homology theories in algebraic topology and allow for the comparison of different homology theories