Homotopy categories of unbounded complexes of projective modules

We develop in this paper a stable theory for projective complexes, by which we mean to consider a chain complex of finitely generated projective modules as an object of the factor category of the homotopy category modulo split complexes. As a result of the stable theory we are able to prove that a complex of finitely generated projective modules over a generically Gorenstein ring is exact if and only if its dual complex is exact. This shows the dependence of total reflexivity conditions for modules over a generically Gorenstein ring.