Mayer--Vietoris and Twisted \v{C}ech Spectral Sequences for C$^*$-Algebras with Free Quantum Group Coefficients

TL;DR

This paper develops a spectral sequence approach to compute K-theory for C*-algebras with quantum group coefficients, introducing a twist mechanism that reveals obstructions to Morita triviality.

Contribution

It introduces a twisted Mayer--Vietoris/cech spectral sequence framework for quantum group crossed products, incorporating a novel twist via a cech cocycle and automorphism.

Findings

The spectral sequence explicitly describes K-theory in terms of intersections.

The twist induces a nontrivial differential related to the automorphism on K-theory.

In certain cases, the differential is an isomorphism, revealing K-theoretic obstructions.

Abstract

We formulate a Mayer--Vietoris/\v{C}ech viewpoint on $K$-theory for crossed products by discrete quantum groups, emphasizing how local-to-global gluing data interacts with quantum-group coefficients. Starting from a $G$-equivariant ideal cover and the associated Mayer--Vietoris six-term exact sequence, we package the resulting $K$-theoretic computation into a \v{C}ech-type spectral sequence whose $E^1$-page is explicitly described by iterated intersections. We then introduce a minimal ``twisted'' gluing mechanism controlled by a $\mathbb Z/2$-valued \v{C}ech $2$-cocycle and an involutive automorphism of the coefficient algebra. Under a Kirchberg--UCT hypothesis on the quantum-group crossed-product coefficient, the twist produces a nontrivial differential $d_2$ identified as $\varphi_*-\mathrm{id}$ on coefficient $K$-theory. In a concrete regime where the coefficient $K$-groups are cyclic (e.g.\ order $3$), the differential becomes an isomorphism and forces a $K$-theoretic obstruction to Morita triviality. This yields a conceptual mechanism for producing non-Morita-trivial twisted C$^*$-algebras with quantum-group crossed-product fibers, detected purely by Mayer--Vietoris/\v{C}ech data.