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. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
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 Classification
Pure or Applied Connections
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Source Charles Weibel (Wikipedia)
Source Princeton University Department of Mathematics: John C. Moore
Source MacTutor History of Mathematics: Norman Steenrod
Source Wikipedia: Roger Lyndon
Sources
1. Wikipedia: Homological Algebra
Wikimedia FoundationLead section
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
Exact sequences section
A sequence of groups and homomorphisms may be either finite or infinite.
Foundational aspects
'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 (to include sheaves of abelian groups).
In Group: Categorical and Homological Algebra, lead paragraph
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting.
View the Source Charles Weibel (Wikipedia)
Includes: Charles Weibel, Lead, homological-algebraQuote, Includes: Charles Weibel, Lead, homological-algebra
is an American mathematician working on algebraic K-theory, algebraic geometry and homological algebra
Princeton University Department of Mathematics: John C. Moore
Princeton University Department of MathematicsIncludes: John Coleman Moore, Memorial, homological algebraQuote, Includes: John Coleman Moore, Memorial, homological algebra
His work on Homological Algebra with S. Ellenberg
MacTutor History of Mathematics: Norman Steenrod
MacTutor History of MathematicsIncludes: Norman Steenrod, Summary, homological algebraQuote, Includes: Norman Steenrod, Summary, homological algebra
was one of the founders of homological algebra
Wikipedia: Roger Lyndon
Includes: Roger Lyndon, LeadQuote, Includes: Roger Lyndon, Lead
Lyndon's Ph.D. thesis concerned group cohomology
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