# Hopfological algebra for infinite dimensional Hopf algebras

## TL;DR

This paper extends Hopfological techniques to infinite dimensional co-Frobenius Hopf algebras, introducing a new homology theory and analyzing its properties, including examples and K-theoretic implications.

## Contribution

It defines a homology functor for co-Frobenius Hopf algebras, generalizing classical homology, and explores its applications and K-theory in this broader context.

## Key findings

- The homology functor is homological for co-Frobenius Hopf algebras.
- The homology theory differs from cyclic homology but has an SBI-sequence.
- K_0 of the category can be computed using a localization sequence.

## Abstract

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.