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Concept
Ext functor
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 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 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 Ab, and so it has right derived functors RiT. The Ext groups are the abelian groups defined by
for an integer i. By definition, this means: take any injective resolution
remove the term B, and form the cochain complex:
For each integer i, Ext_i(A, B) is the cohomology of this complex at position i. It is zero for i negative. For example, Ext_0(A, B) is the kernel of the map HomR(A, I0) → HomR(A, I1), which is isomorphic to HomR(A, B).
An alternative definition uses the functor G(A)=HomR(A, B), for a fixed R-module B. This is a contravariant functor, which can be viewed as a left exact functor from the (R-Mod)op to Ab. The Ext groups are defined as the right derived functors RiG:
That is, choose any projective resolution
remove the term A, and form the cochain complex:
Then Ext_i(A, B) is the cohomology of this complex at position i.
Official source
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