2.3 Abelian categories and exact sequences

Abelian categories and exact sequences form the backbone of homological algebra. They provide a framework for studying algebraic structures, generalizing concepts like kernels and cokernels from concrete categories to abstract settings.

Exact sequences capture relationships between objects and morphisms in abelian categories. They're powerful tools for proving isomorphisms, computing cohomology groups, and understanding extensions, playing a crucial role in various areas of mathematics.

Abelian Categories and Key Properties

Foundational Concepts of Abelian Categories

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• Abelian categories generalize properties of categories of modules over rings and serve as a foundation for homological algebra
• Abstract mathematical structures capturing essential algebraic and homological properties
• Examples include categories of abelian groups, modules over a ring, and sheaves of abelian groups
• Provide a unified framework for studying various algebraic structures and their homological properties

Zero Objects and Binary Operations

• Zero object acts as both an initial and terminal object in an abelian category
  • Serves as an identity element for binary products and coproducts
  • Analogous to the trivial group or zero module in concrete categories
• Binary products generalize Cartesian products
  • Categorical product of two objects
  • Example: direct product of abelian groups
• Binary coproducts dual to binary products
  • Categorical sum (direct sum) of two objects
  • Example: direct sum of vector spaces

Kernels, Cokernels, and Morphism Factorization

• Kernels generalize set-theoretic kernels
  • Represent the "difference" between two morphisms
  • Example: kernel of a group homomorphism captures the subgroup of elements mapped to the identity
• Cokernels dual to kernels
  • Represent the "quotient" of the codomain by the image of a morphism
  • Example: cokernel of a linear transformation is the quotient space of the codomain by the image
• Image factorization of morphisms
  • Every morphism uniquely factors into a normal epimorphism followed by a normal monomorphism
  • Decomposes a morphism into its "onto" and "one-to-one" parts
  • Example: First Isomorphism Theorem in group theory as an instance of image factorization

Exact Sequences in Abelian Categories

Concept and Significance of Exact Sequences

• Exact sequence captures behavior of morphisms in abelian categories
  • Sequence of objects and morphisms where image of each morphism equals kernel of the next
  • Generalizes notion of group homomorphisms
• Provides powerful tool for studying algebraic structures
  • Encodes information about relationships between objects and morphisms
  • Example: exact sequence of groups reveals subgroup and quotient group relationships
• Short exact sequence consists of five terms
  • First and last terms are zero objects
  • Example: $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ represents B as an extension of C by A

Fundamental Theorems and Applications

• Snake Lemma relates exact sequences and connects kernels and cokernels
  • Fundamental result in homological algebra
  • Yields a long exact sequence from a commutative diagram of short exact sequences
• Long exact sequences arise from applying functors to short exact sequences
  • Crucial role in cohomology theories
  • Example: long exact sequence in singular homology relates homology groups of spaces in a pair
• Five Lemma and variations prove isomorphisms using exact sequences
  • Powerful tools for establishing isomorphisms in abelian categories
  • Example: proving isomorphism of homology groups using a map of long exact sequences

Short Exact Sequences and Extensions

Extensions and Their Classification

• Short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ represents extension of C by A
  • B is the "extended" object
  • Example: extension of Z/2Z by Z/3Z yields cyclic group of order 6
• Ext groups classify extensions
  • Measure non-triviality of short exact sequences up to equivalence
  • Example: Ext^1(Z/nZ, Z) classifies extensions of Z/nZ by Z
• Yoneda Ext group provides categorical interpretation of extensions
  • Formalizes equivalence classes of extensions
  • Connects extension theory to homological algebra

Properties and Operations on Extensions

• Split exact sequences correspond to trivial extensions
  • Middle object isomorphic to direct sum of end objects
  • Example: short exact sequence of vector spaces always splits
• Baer sum defines abelian group structure on equivalence classes of extensions
  • Allows addition of extensions
  • Example: Baer sum of two extensions of Z/2Z by Z/2Z yields a third extension
• Long exact sequence of Ext groups relates extensions to higher-dimensional homological algebra
  • Connects extensions to derived functors
  • Example: long exact sequence of Ext groups for a short exact sequence of modules

Applications of Abelian Categories and Exact Sequences

Homological Algebra Techniques

• Derive long exact sequences of derived functors (Tor and Ext)
  • Utilize properties of abelian categories and exact sequences
  • Example: long exact Tor sequence from a short exact sequence of modules
• Horseshoe Lemma constructs projective resolutions
  • Builds resolution of middle term in a short exact sequence
  • Crucial for computing derived functors
• Snake Lemma analyzes kernels and cokernels in diagram chasing
  • Powerful tool for homological computations
  • Example: connecting homomorphism in long exact sequence of homology

Advanced Applications and Connections

• Freyd-Mitchell Embedding Theorem proven using abelian category properties
  • Embeds small abelian categories into categories of modules
  • Allows concrete representation of abstract abelian categories
• Exact sequences compute cohomology groups
  • Applications in group cohomology and sheaf cohomology
  • Example: calculating group cohomology using a projective resolution
• Spectral sequences analyzed using abelian category machinery
  • Arise from exact couples
  • Example: Leray spectral sequence in algebraic topology
• Derived categories constructed from abelian categories
  • Study derived functors and triangulated categories
  • Example: derived functor cohomology as cohomology in derived category