Basic techniques

• Cokernel
  Cokernel
  In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....
  
  
• Exact sequence
  Exact sequence
  An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
  
  
• Chain complex
  Chain complex
  In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
  
  
• Differential module
• Five lemma
  Five lemma
  In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
  
  
• Short five lemma
  Short five lemma
  In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma....
  
  
• Snake lemma
  Snake lemma
  The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology...
  
  
• Nine lemma
  Nine lemma
  In mathematics, the nine lemma is a statement about commutative diagrams and exact sequences valid in any abelian category, as well as in the category of groups. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...
  
  
• Extension (algebra)
  • Central extension
  • Splitting lemma
    Splitting lemma
    In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements for short exact sequence are equivalent....
    
    
• Projective module
  Projective module
  In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
  
  
• Injective module
  Injective module
  In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
  
  
• Projective resolution
• Injective resolution
• Koszul complex
  Koszul complex
  In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
  
  
• Exact functor
  Exact functor
  In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...
  
  
• Derived functor
  Derived functor
  In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
  
  
• Ext functor
  Ext functor
  In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
  
  
• Tor functor
  Tor functor
  In homological algebra, the Tor functors are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology....
  
  
• Filtration (abstract algebra)
  Filtration (abstract algebra)
  In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with the index i running over some index set I that is a totally ordered set, subject to the condition that if i ≤ j in I then Si ⊆ Sj...
  
  
• Spectral sequence
  Spectral sequence
  In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
  
  
• Abelian category
  Abelian category
  In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
  
• Triangulated category
  Triangulated category
  A triangulated category is a mathematical category satisfying some axioms that are based on the properties of the homotopy category of spectra, and the derived category of an abelian category. A t-category is a triangulated category with a t-structure.- History :The notion of a derived category...
  
  
• Derived category
  Derived category
  In mathematics, the derived category D of an abelian category C is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on C...
  
  

Applications

• Group cohomology
  Group cohomology
  In abstract algebra, homological algebra, algebraic topology and algebraic number theory, as well as in applications to group theory proper, group cohomology is a way to study groups using a sequence of functors H n. The study of fixed points of groups acting on modules and quotient modules...
  
  
• Galois cohomology
  Galois cohomology
  In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups...
  
  
• Lie algebra cohomology
  Lie algebra cohomology
  In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups...
  
  
• Sheaf cohomology
  Sheaf cohomology
  In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
  
  
• Whitehead problem
  Whitehead problem
  In group theory, a branch of abstract algebra, the Whitehead problem is the following question:Shelah proved that Whitehead's problem is undecidable within standard ZFC set theory.-Refinement:...
  
  
• Homological conjectures in commutative algebra
  Commutative algebra
  Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...