List of homological algebra topics explained

This is a list of homological algebra topics, by Wikipedia page.

Basic techniques

• Cokernel
• Exact sequence
• Chain complex
• Differential module
• Five lemma
• Short five lemma
• Snake lemma
• Nine lemma
• Extension (algebra)
  • Central extension
  • Splitting lemma
• Projective module
• Injective module
• Projective resolution
• Injective resolution
• Koszul complex
• Exact functor
• Derived functor
• Ext functor
• Tor functor
• Filtration (abstract algebra)
• Spectral sequence
• Abelian category
• Triangulated category
• Derived category

Applications

• Group cohomology
• Galois cohomology
• Lie algebra cohomology
• Sheaf cohomology
• Whitehead problem
• Homological conjectures in commutative algebra