  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.
