🧬Homological Algebra Unit 6 – Derived Functors and Resolutions

Key Concepts and Definitions

• Homological algebra studies algebraic structures using homology and cohomology theories
• Resolutions provide a way to replace an object with a complex of simpler objects
  • Injective resolutions consist of injective objects
  • Projective resolutions consist of projective objects
• Derived functors measure the failure of a functor to be exact
• Tor and Ext are derived functors of the tensor product and Hom functors, respectively
• Tor measures the deviation of the tensor product from being right exact
• Ext measures the deviation of the Hom functor from being left exact
• Spectral sequences are computational tools that relate different homology theories

Injective and Projective Resolutions

• Injective resolutions are exact sequences of injective objects that approximate a given object
  • Every object in an abelian category has an injective resolution
• Projective resolutions are exact sequences of projective objects that approximate a given object
  • Every object in an abelian category with enough projectives has a projective resolution
• Resolutions allow for the computation of derived functors
• Injective resolutions are used to compute right derived functors
• Projective resolutions are used to compute left derived functors
• The existence of resolutions is guaranteed by the axioms of abelian categories
• Resolutions are not unique, but they are unique up to homotopy equivalence

Derived Functors: Introduction and Motivation

• Derived functors are a way to extend functors that are not exact to the derived category
• They measure the failure of a functor to preserve exact sequences
• Derived functors are defined using resolutions and the original functor
• The motivation for derived functors comes from the need to study homological invariants
• Derived functors provide a way to extract homological information from non-exact functors
• The derived category is a localization of the category of chain complexes with respect to quasi-isomorphisms
• Derived functors are well-defined on the derived category

Left and Right Derived Functors

• Left derived functors are computed using projective resolutions
  • They measure the failure of a functor to be right exact
• Right derived functors are computed using injective resolutions
  • They measure the failure of a functor to be left exact
• The $i$-th left derived functor of $F$ is denoted by $L_iF$
• The $i$-th right derived functor of $F$ is denoted by $R^iF$
• Derived functors form a long exact sequence when applied to a short exact sequence
• The long exact sequence of derived functors is a powerful tool in homological algebra

Tor and Ext Functors

• Tor (Tensor product derived functor) is the left derived functor of the tensor product
  • $Tor_i(A,B)$ measures the failure of the tensor product to be right exact
• Ext (Ext functor) is the right derived functor of the Hom functor
  • $Ext^i(A,B)$ measures the failure of the Hom functor to be left exact
• Tor and Ext can be computed using projective and injective resolutions, respectively
• Tor and Ext have interpretations in terms of homology and cohomology of chain complexes
• Tor and Ext satisfy various properties, such as long exact sequences and base change
• Tor and Ext are important invariants in homological algebra and algebraic topology

Applications in Homological Algebra

• Derived functors are used to define and study homological invariants
  • Group cohomology is defined using the derived functors of the invariants functor
  • Lie algebra cohomology is defined using the derived functors of the Lie algebra cohomology functor
• Derived functors are used to prove homological properties of rings and modules
  • The global dimension of a ring is defined using the vanishing of Tor and Ext
  • The projective dimension and injective dimension of modules are defined using resolutions
• Spectral sequences are often constructed using derived functors
  • The Grothendieck spectral sequence relates the derived functors of composed functors
• Derived categories and derived functors are central to modern homological algebra

Computational Techniques and Examples

• Projective and injective resolutions can be constructed explicitly for some objects
  • Free resolutions are projective resolutions of modules over a ring
  • Injective resolutions of abelian groups can be constructed using divisible groups
• Tor and Ext can be computed using the bar resolution and the cobar resolution
• Spectral sequences provide a way to compute derived functors step-by-step
  • The Künneth spectral sequence computes the homology of a tensor product
  • The Grothendieck spectral sequence computes the derived functors of a composition
• Derived functors can be computed using the comparison theorem for resolutions
• Examples of derived functor computations include group cohomology and Lie algebra cohomology

Connections to Other Areas of Mathematics

• Derived functors and homological algebra have applications in various areas of mathematics
  • In algebraic topology, derived functors are used to define and study homology and cohomology theories
  • In algebraic geometry, derived categories and derived functors are used to study coherent sheaves and complexes
• Derived functors provide a unified framework for studying homological invariants
  • The Tor and Ext functors generalize homology and cohomology for modules over a ring
  • The derived category of a ring generalizes the category of chain complexes
• Homological algebra has connections to representation theory and mathematical physics
  • Group cohomology is used to classify group extensions and study representation theory
  • Lie algebra cohomology is used to study deformations and quantizations of Lie algebras
• The language and techniques of homological algebra are used in many areas of modern mathematics