We introduce a notion of total acyclicity associated to a subcategory of an
abelian category and consider the Gorenstein objects they define. These
Gorenstein objects form a Frobenius category, whose induced stable category is
equivalent to the homotopy category of totally acyclic complexes. Applied to
the flat-cotorsion theory over a coherent ring, this provides a new description
of the category of cotorsion Gorenstein flat modules; one that puts it on equal
footing with the category of Gorenstein projective modules.Comment: Added Proposition 4.2, updated after review. Final version, to appear
in Contemp. Math.; 20 p