15.2 Abelian categories and homological algebra

Abelian categories elevate standard categories with additional structure, including zero objects, binary products, and coproducts. They're crucial in homological algebra, providing a framework for studying chain complexes and homology. This structure allows for deeper mathematical insights across various fields.

Derived functors extend functors between abelian categories to chain complexes, measuring how well they preserve exact sequences. They're vital in algebraic geometry, where coherent sheaves on schemes form abelian categories. This allows for powerful homological methods in studying geometric properties.

Abelian Categories

Definition of abelian categories

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• Abelian categories possess additional structure beyond a standard category
  • Additive category has a zero object, binary products and coproducts, and abelian group structure on hom-sets (morphisms between objects)
  • Every morphism has a kernel and cokernel (allows for studying exact sequences)
  • Every monomorphism is the kernel of its cokernel (injective morphisms are characterized by a universal property)
  • Every epimorphism is the cokernel of its kernel (surjective morphisms are characterized by a universal property)
• Kernels and cokernels play a crucial role
  • Kernel of a morphism $f: A \rightarrow B$ is a morphism $k: K \rightarrow A$ satisfying $f \circ k = 0$ and is universal with this property (initial object in the category of morphisms that compose with $f$ to give zero)
  • Cokernel of a morphism $f: A \rightarrow B$ is a morphism $c: B \rightarrow C$ satisfying $c \circ f = 0$ and is universal with this property (terminal object in the category of morphisms that compose with $f$ to give zero)
• Abelian categories have additional useful properties
  • Every morphism has a unique epi-mono factorization (can be decomposed into a surjective morphism followed by an injective morphism)
  • Pushouts and pullbacks exist and are closely related to kernels and cokernels (allow for constructing new objects and morphisms from existing ones)

Role in homological algebra

• Homological algebra studies sequences of morphisms and their compositions
  • Chain complexes are sequences of objects and morphisms where the composition of any two consecutive morphisms is zero (allows for studying homology)
  • Homology measures the extent to which the composition of morphisms fails to be exact (captures "holes" or "cycles" in the sequence)
• Abelian categories provide a suitable framework for homological algebra
  • Existence of kernels and cokernels allows for constructing chain complexes and studying homology (can define boundary maps and homology groups)
  • Properties of abelian categories ensure that homological constructions behave well (functoriality, long exact sequences, spectral sequences)
• Homological methods have applications in various areas of mathematics
  • Algebraic topology uses homology and cohomology of topological spaces (singular homology, de Rham cohomology)
  • Algebraic geometry employs homological methods to study sheaves and schemes (sheaf cohomology, derived categories)
  • Representation theory uses homological techniques to classify and understand representations (group cohomology, Lie algebra cohomology)

Derived Functors and Applications

Construction of derived functors

• Derived functors extend functors between abelian categories to chain complexes
  • Measure the failure of a functor to be exact (preserve short exact sequences)
  • Constructed using resolutions, which are chain complexes that approximate objects in a certain sense (projective or injective resolutions)
• Ext functor is a derived functor of the Hom functor
  • Measures the failure of the Hom functor to be exact in the second argument (contravariant argument)
  • Constructed using injective resolutions (replace objects by injective objects in a functorial way)
  • $Ext^n(A,B)$ is the $n$-th homology of the complex $Hom(A,I^\bullet)$, where $I^\bullet$ is an injective resolution of $B$
• Tor functor is a derived functor of the tensor product functor
  • Measures the failure of the tensor product functor to be exact (in either argument)
  • Constructed using projective resolutions (replace objects by projective objects in a functorial way)
  • $Tor_n(A,B)$ is the $n$-th homology of the complex $P_\bullet \otimes B$, where $P_\bullet$ is a projective resolution of $A$

Applications in algebraic geometry

• Coherent sheaves on a scheme form an abelian category
  • Allows for applying homological methods to study schemes and sheaves (derived categories, derived functors)
  • Homological invariants, such as Ext and Tor, provide information about the geometry of the scheme (dimensions, singularities, intersections)
• Serre duality is a fundamental result in algebraic geometry
  • Relates the Ext functors of a coherent sheaf and its dual (on a smooth projective variety over a field)
  • Provides a duality between cohomology groups of complementary degrees (generalizes Poincaré duality in topology)
• Derived categories are an important tool in modern algebraic geometry
  • Obtained by localizing the category of chain complexes at quasi-isomorphisms (morphisms that induce isomorphisms on homology)
  • Allow for studying schemes and sheaves up to homological equivalence (captures more information than the abelian category of coherent sheaves)
  • Derived functors, such as the derived pullback and pushforward, are well-behaved on derived categories (satisfy adjunction and base change properties)