Symmetric Auslander and Bass categories

<jats:title>Abstract</jats:title><jats:p>We define the symmetric Auslander category<jats:monospace>A</jats:monospace><jats:sup>s</jats:sup>(<jats:italic>R</jats:italic>) to consist of complexes of projective modules whose left- and right-tails are equal to the left- and right-tails of totally acyclic complexes of projective modules.</jats:p><jats:p>The symmetric Auslander category contains<jats:monospace>A</jats:monospace>(<jats:italic>R</jats:italic>), the ordinary Auslander category. It is well known that<jats:monospace>A</jats:monospace>(<jats:italic>R</jats:italic>) is intimately related to Gorenstein projective modules, and our main result is that<jats:monospace>A</jats:monospace><jats:sup>s</jats:sup>(<jats:italic>R</jats:italic>) is similarly related to what can reasonably be called Gorenstein projective homomorphisms. Namely, there is an equivalence of triangulated categories<jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0305004110000629_eqnU1" /></jats:disp-formula>where<jats:underline><jats:monospace>GMor</jats:monospace></jats:underline>(<jats:italic>R</jats:italic>) is the stable category of Gorenstein projective objects in the abelian category<jats:monospace>Mor</jats:monospace>(<jats:italic>R</jats:italic>) of homomorphisms of<jats:italic>R</jats:italic>-modules.</jats:p><jats:p>This result is set in the wider context of a theory for<jats:monospace>A</jats:monospace><jats:sup>s</jats:sup>(<jats:italic>R</jats:italic>) and<jats:monospace>B</jats:monospace><jats:sup>s</jats:sup>(<jats:italic>R</jats:italic>), the symmetric Bass category which is defined dually.</jats:p>