# (Co)homology of Crossed Products by Weak Hopf Algebras

## TL;DR

This paper introduces a simplified mixed complex for computing cyclic homologies of crossed products by weak Hopf algebras, generalizing existing spectral sequences and providing new computational tools.

## Contribution

It develops a new, simpler mixed complex with filtrations and spectral sequences for cyclic homology of crossed products involving weak Hopf algebras, extending prior spectral sequence results.

## Key findings

- A new mixed complex simplifies cyclic homology computations.
- The spectral sequence generalizes previous results by CGG.
- Another filtration leads to a spectral sequence extending Feigin-Tsygan.

## Abstract

We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle $A\times_{\rho}^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$. This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When $f$ takes its values in a separable subalgebra of $A$ that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence.