%Name
Hans-Bj{\o}rn Foxby
%Title:
Duality and Equivalence
%Abstract
\newcommand{\Hom}{\mathrm{Hom}}
For complexes $X$ and $\Omega$ of modules over a ring $R$ we consider
the duality morphism $X\to\Hom_R(\Hom_R(X,\Omega),\Omega)$ together
with the two equivalence morphisms
$X\to\Hom_R(\Omega,\Omega\otimes_RX)$ and
$X\leftarrow\Omega\otimes_R\Hom_R(\Omega,X)$ as well as their
counterparts in the derived category. We present cases where these are
isomorphisms including: Hartshorne's Affine Duality. Duality with
respect to $R$ with connections to Auslander's G-dimension. Duality
with respect to a stable Koszul complex. The corresponding
equivalences.