Ext

For a ring $R$ , and $R$ -module $A$ , we have a covariant functor $\Hom A-R$ . The functions $\Ext^n_R(A,-)$ are defined to be the right derived functors of $\Hom A-R: \Ext^n_R(A,-)=R^n\Hom A-R$ .

$\Ext$ gets its name from the existence of a natural bijection between elements of $\Ext^1_R(A,B)$ and extensions of $B$ by $A$ up to isomorphism of short exact sequences, where an extension of $B$ by $A$ is an exact sequence $$0\to B\to C\to A\to 0.$$ For example, $$\Ext^1_\Z(\Z/n\Z,\Z)\cong\Z/n\Z,$$ with $0$ corresponding to the trivial extension $0\to\Z\to\Z\oplus\Z/n\Z\to 0$ , and $m\neq 0$ corresponding to

A more modern interpretation of the $\Ext$ functors was given by S. Mac Lane, namely that there is a correspondence between $\Ext^n(A,B)$ with equivalence classes of exact sequences