We study the cohomology of isometric group actions on (super) reflexive Banach spaces with a focus on the relation between finite conjugacy classes and split exactness of cochain complexes. In particular, we show that, if a uniformly convex Banach module has no almost invariant vectors under

the FC-centre of the acting group, then the associated cochain complex is split exact. Other similar rigidity results are established that are related to prior work of Bader-Furman-Gelander-Monod, Bader-Rosendal-Sauer, and Nowak.