“…For an operator algebra 𝐴 (on a Hilbert space) it is pertinent to use operator modules to build a cohomology theory (for the definitions, see Section 2.2); of the numerous contributions we only mention [7], [16], [24], [32] and [34] here. In this paper, we focus on an appropriate definition of cohomological dimension and, in particular, answer a question raised by Helemskii [16] whether quantised global dimension zero is equivalent to 2 the algebra being classically semisimple; see Theorem 5.3 below. In contrast to the situation in ring theory, it appears necessary to limit ourselves to a relative cohomology theory since, otherwise, there exist too many monomorphisms (equivalently, epimorphisms) and the concepts of injectivity (respectively, projectivity) become too restrictive.…”