Branches of Mathematics
Homological Algebra
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Homological algebra is the branch of mathematics that studies homology, a technique for measuring how far a sequence of algebraic objects fails to be exact, in a general algebraic setting.
Facts
Central QuestionHow the technique of measuring exactness along a sequence of algebraic objects, first developed to compute topological invariants, can be carried into a general algebraic setting and applied wherever such sequences arise. 1 Key DebateHow general the setting for homological methods could be made. Alexander Grothendieck's 1957 Tohoku paper reformulated the subject using the abelian category concept, a level of generality that let the same machinery reach sheaves of abelian groups and structures far removed from the topological spaces homology was first built to measure. 1 Cross-Tradition Connections
Sources
1. Wikipedia: Homological Algebra
Wikimedia FoundationLead sectionQuote, Lead section
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
1. Wikipedia: Homological Algebra
Wikimedia FoundationExact sequences sectionQuote, Exact sequences section
A sequence of groups and homomorphisms may be either finite or infinite.
1. Wikipedia: Homological Algebra
Wikimedia FoundationFoundational aspects sectionQuote, Foundational aspects section
Tohoku: The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept.
Stone-Weierstrass Theorem (Wikipedia)
Wikipedialead paragraphQuote, lead paragraph
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
Stone-Weierstrass Theorem (Wikipedia)
WikipediaFoundational aspects sectionQuote, Foundational aspects section
The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept (to include sheaves of abelian groups).
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