  We consider "Hopfological" techniques as in \cite{Ko} but for infinite
dimensional Hopf algebras, under the assumption of being co-Frobenius. In
particular, $H=k[{\mathbb Z}]\#k[x]/x^2$ is the first example, whose
corepresentations category is d.g. vector spaces. Motivated by this example we
define the "Homology functor" (we prove it is homological) for any co-Frobenius
algebra, with coefficients in $H$-comodules, that recover usual homology of a
complex when $H=k[{\mathbb Z}]\#k[x]/x^2$. Another easy example of co-Frobenius
Hopf algebra gives the category of "mixed complexes" and we see (by computing
an example) that this homology theory differs from cyclic homology, although
there exists a long exact sequence analogous to the SBI-sequence. Finally,
because we have a tensor triangulated category, its $K_0$ is a ring, and we
prove a "last part of a localization exact sequence" for $K_0$ that allows us
to compute -or describe- $K_0$ of some family of examples, giving light of what
kind of rings can be categorified using this techniques.
