Hopfological algebra and categorification at a root of unity: the first steps

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds. 1 Hopfological algebra Introduction. Two of the better known examples of triangulated categories are the category of complexes of modules over a ring A, up to chain homotopy, and the derived category D(A−mod). The first category is the quotient of the abelian category of complexes of A-modules by the ideal of