🧬Homological Algebra Unit 3 – Exact Sequences and Diagram Chasing

Key Concepts and Definitions

• Exact sequences fundamental tool in homological algebra used to study relationships between algebraic objects (groups, rings, modules)
• Homomorphism map between two algebraic structures preserving the operations
  • Injective homomorphism one-to-one mapping where distinct elements in the domain map to distinct elements in the codomain
  • Surjective homomorphism onto mapping where every element in the codomain is mapped to by at least one element in the domain
• Isomorphism bijective homomorphism with an inverse homomorphism, establishes an equivalence between two algebraic structures
• Kernel set of all elements in the domain that map to the identity element in the codomain under a given homomorphism
• Cokernel quotient of the codomain by the image of the homomorphism
• Chain complex sequence of objects and homomorphisms where the composition of any two consecutive homomorphisms is zero
• Homology measures the extent to which a sequence fails to be exact by taking quotients of kernels and images

Exact Sequences: Basics and Types

• Exact sequence chain complex where the image of each homomorphism equals the kernel of the next homomorphism
  • Short exact sequence consists of three non-zero objects and two non-zero homomorphisms with the sequence $0 \to A \to B \to C \to 0$
  • Long exact sequence infinite sequence with exactness at each object, often arising from short exact sequences of chain complexes
• Split exact sequence short exact sequence where there exists a homomorphism from the third object back to the second object, giving a direct sum decomposition
• Exact sequences capture intricate relationships between objects and provide a way to study their structures and properties
• Exact sequences can be constructed using various techniques (mapping cones, mapping cylinders, cofibrations)
• Exact sequences are preserved under certain operations (tensor products, Hom functors) allowing for further analysis
• Exact sequences can be used to compute homology and cohomology groups, which provide invariants for topological spaces and algebraic structures

Diagram Chasing: Techniques and Applications

• Diagram chasing technique used to prove statements about commutative diagrams by "chasing" elements through the diagram
• Commutative diagram consists of objects and morphisms where all directed paths between two objects lead to the same result
• Diagram chasing often involves exact sequences and can be used to prove exactness, injectivity, or surjectivity of maps
• Five Lemma powerful tool in diagram chasing, states that if certain maps in a commutative diagram of exact sequences are isomorphisms, then another map must also be an isomorphism
• Snake Lemma another important result in diagram chasing, relates the kernels, cokernels, and homology of maps between short exact sequences
• Diagram chasing can be used to construct long exact sequences from short exact sequences of chain complexes
• Diagram chasing is essential for proving theorems in homological algebra (Mayer-Vietoris sequence, Künneth formula)

Homomorphisms and Isomorphisms in Sequences

• Homomorphisms play a crucial role in the construction and analysis of exact sequences
• Isomorphisms in exact sequences allow for the identification of objects up to equivalence
• Homomorphisms in exact sequences preserve the exactness property, enabling the study of relationships between objects
• Isomorphisms in short exact sequences split the sequence, giving a direct sum decomposition of the middle object
• Homomorphisms can be used to construct long exact sequences by taking the homology of a short exact sequence of chain complexes
• Isomorphisms in long exact sequences provide a way to compute homology groups of complex objects using simpler ones
• Homomorphisms and isomorphisms in exact sequences are essential for understanding the structure and properties of algebraic objects

Short and Long Exact Sequences

• Short exact sequences fundamental building blocks in homological algebra, capturing essential relationships between three objects
  • Split short exact sequences equivalent to direct sum decompositions, providing a way to break down objects into simpler components
• Long exact sequences powerful tools for studying the homology and cohomology of topological spaces and algebraic structures
  • Connecting homomorphisms in long exact sequences relate the homology groups of different objects, allowing for computations
• Exact sequences can be constructed from short exact sequences using various techniques (mapping cones, mapping cylinders)
• Long exact sequences can be derived from short exact sequences of chain complexes by taking homology
• Mayer-Vietoris sequence long exact sequence relating the homology of a space to the homology of its subspaces, useful for computations
• Long exact sequences in cohomology provide dual results to those in homology, capturing different aspects of the objects studied

Snake Lemma and Its Uses

• Snake Lemma powerful result in homological algebra relating the kernels, cokernels, and homology of maps between short exact sequences
  • Provides a long exact sequence connecting these objects, allowing for the study of their relationships
• Snake Lemma can be used to prove the Five Lemma, another important result in diagram chasing
• Snake Lemma is essential for constructing long exact sequences from short exact sequences of chain complexes
• Snake Lemma can be used to compute homology and cohomology groups of complex objects by breaking them down into simpler components
• Snake Lemma has applications in various areas of mathematics (algebraic topology, algebraic geometry, representation theory)
• Snake Lemma is a key tool in proving theorems and solving problems involving exact sequences and commutative diagrams

Connecting Homomorphisms

• Connecting homomorphisms maps that appear in long exact sequences, relating the homology or cohomology groups of different objects
• Connecting homomorphisms are constructed using the Snake Lemma, which provides a long exact sequence from maps between short exact sequences
• Connecting homomorphisms are essential for computing homology and cohomology groups of complex objects using simpler ones
• Connecting homomorphisms preserve the exactness of the long exact sequence, ensuring that the sequence captures the relevant relationships between the objects
• Connecting homomorphisms can be used to prove theorems and solve problems involving long exact sequences (Mayer-Vietoris sequence, Künneth formula)
• Understanding the properties and behavior of connecting homomorphisms is crucial for working with long exact sequences in homological algebra

Practice Problems and Examples

• Example: Prove that the sequence $0 \to \mathbb{Z} \xrightarrow{2} \mathbb{Z} \xrightarrow{\pi} \mathbb{Z}/2\mathbb{Z} \to 0$ is exact, where $2$ is multiplication by 2 and $\pi$ is the quotient map
• Problem: Show that the homology of a chain complex is zero if and only if the chain complex is exact
• Example: Use the Snake Lemma to derive the long exact sequence in homology associated with a short exact sequence of chain complexes
• Problem: Compute the homology groups of the torus using the Mayer-Vietoris sequence
• Example: Prove that the connecting homomorphism in the long exact sequence associated with the short exact sequence $0 \to A \to B \to C \to 0$ is a natural transformation
• Problem: Use diagram chasing to prove the Five Lemma for exact sequences
• Example: Construct a split short exact sequence and show that it is equivalent to a direct sum decomposition