Morita duality

Remarks

A left $R$-module $M$ is reflexive if the evaluation map $M \to \Hom_S(\Hom_R(M, U), U)$ is an isomorphism, and likewise for right $S$-modules. Under a Morita duality, the functors $\Hom_R(-, U)$ and $\Hom_S(-, U)$ are mutually inverse contravariant equivalences between the reflexive modules on the two sides. The reflexive modules include all finitely generated modules and all finitely cogenerated modules. Example: for a field $k$, take $R = S = U = k$. The space $k$ is injective and is a cogenerator of $k$-vector spaces, the multiplication maps $k \to \End(k)$ are isomorphisms, and the duality is $V \mapsto V^*$ on finite-dimensional spaces. Non-example: $R = S = U = \Z$ fails clause 1, since $\Z$ is not an injective $\Z$-module; the map $2\Z \to \Z$, $2 \mapsto 1$, does not extend to $\Z$. Over a quasi-Frobenius ring $R$, the bimodule $U = R$ gives a Morita duality between $R$ and itself. Over a complete Noetherian local commutative ring with residue field $k$, the injective hull of $k$ gives a Morita duality, known as Matlis duality.