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We find a finite free resolution of the counit of the free unitary quantumgroups of van Daele and Wang and, more generally, Bichon's universalcosovereign Hopf algebras with a generic parameter matrix. This allows us tocompute Hochschild cohomology with 1-dimensional coefficients for all theseHopf algebras. In fact, the resolutions can be endowed with a Yetter-Drinfeldstructure. General results of Bichon then allow us to compute also thecorresponding bialgebra cohomologies.  Finding the resolution rests on two pillars. We take as a starting point theresolution for the free orthogonal quantum group presented by Collins,H\"artel, and Thom or its algebraic generalization to quantum symmetry groupsof bilinear forms due to Bichon. Then we make use of the fact that the freeunitary quantum groups and some of its non-Kac versions can be realized as aglued free product of a (non-Kac) free orthogonal quantum group with $\mathbbZ_2$, the finite group of order 2. To obtain the resolution also for moregeneral universal cosovereign Hopf algebras, we extend Gromada's proof fromcompact quantum groups to the framework of matrix Hopf algebras. As a byproductof this approach, we also obtain a projective resolution for the freelymodified bistochastic quantum groups.  Only a special subclass of free unitary quantum groups and universalcosovereign Hopf algebras decompose as a glued free product in the describedway. In order to verify that the sequence we found is a free resolution ingeneral (as long as the parameter matrix is generic, two conditions which areautomatically fulfilled in the free unitary quantum group case), we use thetheory of Hopf bi-Galois objects and Bichon's results on monoidal equivalencesbetween the categories of Yetter-Drinfeld modules over universal cosovereignHopf algebras for different parameter matrices.

Free resolutions for free unitary quantum groups and universal cosovereign Hopf algebras