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Random information on the term “EXT”:
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 Ext1 classifies extensions of one module by another.
In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (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 their 1956 book Homological Algebra.
Let R be a ring and let R-Mod be the category of modules over R. (One can take this to mean either left R-modules or right R-modules.) For a fixed R-module A, let T(B) = HomR(A, B) for B in R-Mod. (Here HomR(A, B) is the abelian group of R-linear maps from A to B; this is an R-module if R is commutative.) This is a left exact functor from R-Mod to the category of abelian groups Ab, and so it has right derived functors RiT. The Ext groups are the abelian groups defined by