Construction of Ext in Abelian Categories
This identification enables us to define Ext1
Ab(A, B) even for abelian categories Ab without reference to projectives and injectives. We simply take Ext1
Ab(A, B) to be the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum. Similarly, we can define higher Ext groups Extn
Ab(A, B) as equivalence classes of n-extensions
under the equivalence relation generated by the relation that identifies two extensions
if there are maps Xm → X′m for all m in {1, 2, ..., n} so that every resulting square commutes.
The Baer sum of the two n-extensions above is formed by letting X′′
1 be the pullback of X′′
1 and X′
1 over A, and X′′
n be the pushout of Xn and X′
n under B quotiented by the skew diagonal copy of B. Then we define the Baer sum of the extensions to be
Read more about this topic: Ext Functor
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