identity on A and B. The Baer sum of two n-extensions as above is formed by letting be the pullback of and over A, and be the pushout of and under B. Then the Baer sum of the extensions is The derived category and the Yoneda product 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). 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. The Yoneda product can also be described in more elementary terms. 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 Ext(A,B) can be identified with the group of chain homotopy classes of chain maps P → Q[i]. The Yoneda product is given by composing chain maps: By any of these interpretations, the Yoneda product is associative. As a result, is a graded ring, for any R-module A. For example, this gives the ring structure on group cohomology since this can be viewed as . Also by associativity of the Yoneda product: for any R-modules A and B, is a module over . Important special cases Group cohomology is defined by , where G is a group, M is a representation of G over the integers, and is the group ring of G. For an algebra A over a field k and an A-bimodule M, Hochschild cohomology is defined by Lie algebra cohomology is defined by , where is a Lie algebra over a commutative ring k, M is a -module, and is the universal enveloping algebra. For a topological space X, sheaf cohomology can be defined as Here Ext is taken in