Ext Functor - Definition and Computation

Definition and Computation

Let R be a ring and let Mod_R be the category of modules over R. Let B be in Mod_R and set T(B) = Hom_R(A,B), for fixed A in Mod_R. This is a left exact functor and thus has right derived functors RnT. The Ext functor is defined by

This can be calculated by taking any injective resolution

and computing

Then (RnT)(B) is the homology of this complex. Note that Hom_R(A,B) is excluded from the complex.

An alternative definition is given using the functor G(A)=Hom_R(A,B). For a fixed module B, this is a contravariant left exact functor, and thus we also have right derived functors RnG, and can define

This can be calculated by choosing any projective resolution

and proceeding dually by computing

Then (RnT)(A) is the homology of this complex. Again note that Hom_R(A,B) is excluded.

These two constructions turn out to yield isomorphic results, and so both may be used to calculate the Ext functor.