Let \(A\) and \(G\) be abelian groups, and let \[ \cdots \to F_2 \to F_1 \to F_0 \to A \to 0 \] be a free resolution of \(A\). Applying \(\Hom(\wait, G)\) to \(\cdots\to F_1\to F_0\) gives a cochain complex \(0\to\Hom(F_0,G)\to\Hom(F_1,G)\to\cdots\), and the Ext group \(\Ext^n(A, G)\) is its \(n\)th cohomology group; it is independent of the choice of free resolution [@Hat02].
One has \(\Ext^0(A,G) \cong \Hom(A,G)\) and \(\Ext^n(A,G) = 0\) for \(n\geq 2\). For \(m\geq 1\), \[ \Ext^1(\ZZ, G) = 0, \qquad \Ext^1(\ZZ/m, G) \cong G/mG \] [@Hat02].