Ext Functor

Let \(\mathcal{A}\) be an abelian category. Then the ext bifunctor is defined as right derived functor of \(\mathrm{Hom}^{\mathrm{Ab}}_{\mathcal{A}}(-,-)\)

\begin{align*} \mathrm{Ext}_R^n \coloneqq \mathrm{Hom}_{\mathcal{R}}(-,-): \mathcal{A}^{\mathrm{op}} \times \mathcal{A} \rightarrow \mathrm{Ab} \end{align*}

see:

• covariant hom-functor as left-exact functor
• contravariant ab-hom functor as left exact functor