homological algebra

homological algebra (uncountable)

1. (algebra) The branch of mathematics dealing with homology and in particular with homological functors and the algebraic structures they entail.

   The notion of chain complex is central to homological algebra.

   The development of homological algebra was closely intertwined with the emergence of category theory.

   • 2002, C. R. Leedham-Green, S. McKay, The Structure of Groups of Prime Power Order, Oxford University Press, page 171:

     Homological algebra for our purposes consists of studying certain abelian groups ${\displaystyle H^{n}(G,A)}$ and ${\displaystyle H_{n}(G,A)}$, known as the ${\displaystyle n}$th cohomology and homology groups of the group ${\displaystyle G}$ with coefficients in the ${\displaystyle G}$-module ${\displaystyle A}$.

   • 2007, Ralph Krömer, Tool and Object: A History and Philosophy of Category Theory‎, Springer (Birkhäuser), page 93:

     A similar shift of interpretation of CT took place at about the same time in another context, namely homological algebra.

   • 2014, Anant R. Shastri, Basic Algebraic Topology, Taylor & Francis (CRC Press / Chapman & Hall), page 169:

     In Section 4.1, we shall introduce some minimal amount of homological algebra needed to launch singular homology.

• homological invariant

• Chain complex on Wikipedia.Wikipedia
• Exact sequence on Wikipedia.Wikipedia
• Snake lemma on Wikipedia.Wikipedia
• Ext functor on Wikipedia.Wikipedia
• Tor functor on Wikipedia.Wikipedia
• Homological algebra on Encyclopedia of Mathematics