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.
By definition, this means: take any injective resolution remove the term B, and form the cochain complex: For each integer
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 opposite category (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 ExtiR(A, B) is the cohomology of this complex at position i.
In fact, Cartan and Eilenberg showed that these constructions are independent of the choice of projective or injective resolution, and that both constructions yield the same Ext groups.
[2] Moreover, for a fixed ring R, Ext is a functor in each variable (contravariant in A, covariant in B).
For a non-commutative ring R, ExtiR(A, B) is only an abelian group, in general.
If R is an algebra over a ring S (which means in particular that S is commutative), then ExtiR(A, B) is at least an S-module.
Here are some of the basic properties and computations of Ext groups.
[3] The Ext groups derive their name from their relation to extensions of modules.
Given R-modules A and B, an extension of A by B is a short exact sequence of R-modules Two extensions are said to be equivalent (as extensions of A by B) if there is a commutative diagram: Note that the Five lemma implies that the middle arrow is an isomorphism.
[9] The trivial extension corresponds to the zero element of Ext1R(A, B).
The Baer sum is an explicit description of the abelian group structure on Ext1R(A, B), viewed as the set of equivalence classes of extensions of A by B.
, Then form the quotient module The Baer sum of E and E′ is the extension where the first map is
Nobuo Yoneda defined the abelian groups ExtnC(A, B) for objects A and B in any abelian category C; this agrees with the definition in terms of resolutions if C has enough projectives or enough injectives.
Next, Ext1C(A, B) is the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum.
Finally, the higher Ext groups ExtnC(A, B) are defined as equivalence classes of n-extensions, which are exact sequences under the equivalence relation generated by the relation that identifies two extensions if there are maps
The Baer sum of two n-extensions as above is formed by letting
[11] Then the Baer sum of the extensions is An important point is that Ext groups in an abelian category C can be viewed as sets of morphisms in a category associated to C, the derived category D(C).
[12] The objects of the derived category are complexes of objects in C. Specifically, one has where an object of C is viewed as a complex concentrated in degree zero, and [i] means shifting a complex i steps to the left.
From this interpretation, there is a bilinear map, sometimes called the Yoneda product: which is simply the composition of morphisms in the derived category.
For i = j = 0, the product is the composition of maps in the category C. In general, the product can be defined by splicing together two Yoneda extensions.
Alternatively, the Yoneda product can be defined in terms of resolutions.
(This is close to the definition of the derived category.)
For example, let R be a ring, with R-modules A, B, C, and let P, Q, and T be projective resolutions of A, B, C. Then ExtiR(A,B) can be identified with the group of chain homotopy classes of chain maps P → Q[i].
For example, this gives the ring structure on group cohomology