Exact sequences and the Snake Lemma are key tools in homological algebra. They help us understand relationships between different mathematical objects by connecting their homology groups. These concepts are crucial for analyzing complex structures and solving problems in algebraic topology.

The Snake Lemma, in particular, allows us to create long exact sequences from short ones. This technique is super useful for computing homology groups and making connections between seemingly unrelated mathematical objects. It's like a secret weapon for tackling tricky algebraic problems!

Exact Sequences

Short and Long Exact Sequences

• A short exact sequence consists of three objects and two morphisms in an abelian category
  • The image of the first morphism equals the kernel of the second morphism
  • The first morphism is injective while the second morphism is surjective
• A long exact sequence is an infinite exact sequence of objects and morphisms in an abelian category
  • Often arises from short exact sequences of chain complexes
  • Connects homology groups of different degrees
• The connecting homomorphism in a long exact sequence measures the failure of exactness in the corresponding short exact sequence of chain complexes
  • Defined using a diagram chase
  • Plays a crucial role in relating homology groups of different chain complexes

Splitting Lemmas and Applications

• Splitting lemmas provide conditions under which a short exact sequence splits
  • A split exact sequence is isomorphic to a direct sum of the first and third objects
  • Splitting occurs when there exists a section (right inverse) or a retraction (left inverse) of one of the morphisms
• The long exact sequence in homology relates the homology groups of a chain complex to those of a subcomplex and the corresponding quotient complex
  • Constructed using the Snake Lemma or the Zigzag Lemma
  • Allows for computation of homology groups in terms of simpler chain complexes (Mayer-Vietoris sequence)

Snake Lemma for Homology

Statement and Diagram Chase

• The Snake Lemma studies the relationships between homology groups in a commutative diagram of abelian groups or modules with exact rows
  • Constructs a long exact sequence involving the kernels and cokernels of the vertical maps
  • The connecting homomorphism is defined using a diagram chase
• The Snake Lemma is named after the zigzag pattern formed by the kernels, images, and cokernels in the diagram
  • The "snake" connects the objects in the top row to those in the bottom row
  • Provides a way to relate the homology groups of different chain complexes

Applications to Long Exact Sequences

• The Snake Lemma can be used to derive the long exact sequence in homology associated with a short exact sequence of chain complexes
  • The short exact sequence of chain complexes induces short exact sequences of homology groups
  • The Snake Lemma connects these short exact sequences into a long exact sequence
• The long exact sequence in homology is a powerful tool for computing homology groups
  • Relates the homology of a chain complex to the homology of simpler chain complexes (subcomplex and quotient complex)
  • Allows for inductive arguments and the comparison of homology groups

Five Lemma for Exactness

Statement and Commutative Diagrams

• The Five Lemma is a result about commutative diagrams in an abelian category
  • If certain maps are isomorphisms and the rows are exact, then the remaining map is also an isomorphism
  • Provides a way to prove that a certain sequence is exact by comparing it to a known exact sequence
• The Five Lemma is often used in conjunction with other diagram lemmas (Snake Lemma, Short Five Lemma)
  • Helps to establish the exactness of sequences in homological algebra
  • Can be used to prove the functoriality of long exact sequences

Variants and Generalizations

• Variants of the Five Lemma handle similar situations with different numbers of objects and morphisms
  • The Four Lemma deals with four objects and three morphisms
  • The Six Lemma deals with six objects and five morphisms
• The Five Lemma can be generalized to longer exact sequences and more complex commutative diagrams
  • The Nine Lemma and the 3x3 Lemma are examples of such generalizations
  • These generalizations are useful in the study of spectral sequences and derived categories

Exact Sequences and Homological Algebra

Central Role of Exact Sequences

• Homological algebra studies sequences of abelian groups or modules connected by homomorphisms
  • Focuses on exact sequences and derived functors
  • Provides a framework for studying the structure of mathematical objects
• Exact sequences, particularly long exact sequences, encode important structural information about the objects involved
  • Relate the homology or cohomology groups of different objects
  • Allow for the computation of invariants using simpler objects

Tools and Techniques

• The Snake Lemma and the Five Lemma are key tools in homological algebra
  • Used to work with exact sequences and prove exactness
  • Enable the construction and study of long exact sequences
• Derived functors, such as Ext and Tor, measure the failure of a functor to be exact
  • Can be studied using long exact sequences
  • Provide additional invariants and structural information
• Spectral sequences are collections of exact sequences related by differentials
  • Provide a powerful computational tool in homological algebra
  • Enable the calculation of homology and cohomology groups in complex situations (Serre spectral sequence, Adams spectral sequence)