Lesson plan

Ext functor

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext 1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer in 1934. It was named by Samuel Eilenberg and Saunders MacLane in 1942, and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in 1956.

Back to article

Article summary

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The cohomology of groups, Lie algebras, and associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext 1 classifies extensions of one module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer in 1934. It was named by Samuel Eilenberg and Saunders MacLane in 1942, and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring, Ext was defined by Henri Cartan and Eilenberg in 1956.

Key claims

No claims recorded.

Discussion questions

  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

Bibliography

No sources recorded.

Key claims and bibliography are generated from the same tracked claims and sources as the article page.