finitely generated module over a commutative Noetherian ring R. Then Ext commutes with localization, in the sense that for every multiplicatively closed set S in R, every R-module B, and every integer i, Ext and extensions Equivalence of extensions 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. An extension of A by B is called split if it is equivalent to the trivial extension There is a one-to-one correspondence between equivalence classes of extensions of A by B and elements of Ext(A, B). The trivial extension corresponds to the zero element of Ext(A, B). The Baer sum of extensions The Baer sum is an explicit description of the abelian group structure on Ext(A, B), viewed as the set of equivalence classes of extensions of A by B. Namely, given two extensions and first form the pullback over , Then form the quotient module The Baer sum of E and E′ is the extension where the first map is and the second is . Up to equivalence of extensions, the Baer sum is commutative and has the trivial extension as identity element. The negative of an extension 0 → B → E → A → 0 is the extension involving the same module E, but with the homomorphism B → E replaced by its negative. Construction of Ext in abelian categories Nobuo Yoneda defined the abelian groups Ext(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. First, Ext(A,B) = HomC(A, B). Next, Ext(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 Ext(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 for all m in {1, 2, ..., n} so that every resulting square commutes, that is, if there is a chain map ξ → ξ' which is the